Advanced Graph Neural Networks for Smart Mining: A Systematic Literature Review of Equivariant, Topological, Symplectic, and Generative Models

Abstract

The transition of the mining industry towards Industry 5.0 demands predictive models capable of strictly adhering to physical laws and modeling complex, non-Euclidean geometries—capabilities often lacking in standard graph neural networks. This systematic review, conducted under the PRISMA 2020 protocol, analyzes the emergence of “Era 5” architectures by synthesizing 96 high-impact studies from 2019 to 2026, focusing on Clifford (geometric algebra) GNNs, simplicial and cell complex neural networks, symplectic/Hamiltonian GNNs, and generative flow networks (GFlowNets). The analysis demonstrates that Clifford architectures provide superior rotational equivariance for robotic control; Simplicial networks capture high-order topological interactions critical for geomechanics; Symplectic GNNs ensure energy conservation for stable long-term simulation of structural dynamics; and GFlowNets offer a novel paradigm for generative mine planning. We conclude that shifting from data-driven approximations to these mathematically rigorous, structure-preserving architectures is fundamental for developing reliable, physics-informed digital twins that optimize structural integrity and operational efficiency in complex industrial environments.

1. Introduction

1.1. Background and Motivation

Deep learning has rapidly evolved from architectures designed for regular Euclidean domains (e.g., convolutional neural networks) to models that operate on more general structures and symmetries, including sequences (RNNs/LSTMs) and attention-based models (Transformers) [1,2,3]. This evolution has been driven by a structural fact: an increasing fraction of scientific and engineering data are supported on irregular, non-Euclidean domains—relations, meshes, multi-body interactions, and physical networks—where neighborhood relations and permutation invariance are fundamental [4,5,6].

In this setting, graph neural networks (GNNs) have become a central paradigm for learning from relational data by representing the system as a graph $G=(V,E)$ with node features $x_i$ and, optionally, edge features $e_{ij}$ [7,8,9]. A standard formulation is message passing (MPNN). For each layer ℓ, a node i aggregates information from its neighborhood $\mathcal{N}\left(i\right)$ through messages $m_{ij}^{(𝓁)}$ and updates a latent state $h_i^{(𝓁)}$:

$$\begin{array}{cc}\hfill m_{ij}^{(𝓁)}& ={\varphi }_e^{(𝓁)} \left(h_i^{(𝓁)}, h_j^{(𝓁)}, e_{ij}\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill h_i^{(𝓁+1)}& ={\varphi }_h^{(𝓁)} \left(h_i^{(𝓁)}, {\mathrm{Agg}}_{j\in \mathcal{N}\left(i\right)} m_{ij}^{(𝓁)}\right),\hfill \end{array}$$

(2)

where $\mathrm{Agg}$ is permutation invariant (e.g., sum/mean/max) [10,11,12]. This construction explains the strong empirical performance of GNNs in domains such as molecular modeling, particle dynamics, and complex systems [6,13,14].

In this notation, the initial latent state is obtained from the raw node features via an (optional) input encoder, i.e., $h_i^{\left(0\right)}={\varphi }_{\mathrm{in}}\left(x_i\right)$, where ${\varphi }_{\mathrm{in}}$ can be the identity map when dimensions match or a learnable linear/MLP projection otherwise.

Despite their flexibility, MPNNs are constrained by a well-known theoretical limit: under mild assumptions, their discriminative power is upper bounded by the one-dimensional Weisfeiler–Lehman (1-WL) test [15,16,17]. Here, “mild assumptions” refers to the standard MPNN regime with shared local message-passing updates and permutation-invariant multiset aggregation (and readout), without injecting unique node identifiers or higher-order substructure features; under this widely used formulation, MPNNs cannot distinguish graphs beyond what 1-WL can separate. The 1-WL refinement can be written as the following color update rule:

$$c_i^{(𝓁+1)}=\mathrm{HASH}\left(c_i^{(𝓁)},\left\{\{c_j^{(𝓁)}:j\in \mathcal{N}\left(i\right)\}\right\}\right),$$

(3)

MPNNs implement a continuous relaxation in which $\mathrm{HASH}$ is replaced by learnable functions over multisets [13,16]. As a consequence, improved performance often requires more than incremental architectural tuning. It is typically achieved by changing the representation and the inductive bias, for example, by (i) enforcing equivariance to continuous symmetries, (ii) modeling higher-order interactions beyond edges, (iii) embedding conservation structure in dynamics, and/or (iv) shifting from point prediction to generative sampling in combinatorial spaces [18].

Modern mining systems combine continuous physics (stress–strain, airflow and water flow, vibrations), complex geometry (3D perception, moving frames), and discrete decision layers (network design, sequencing, logistics). This coupling is naturally expressed as constrained optimization:

$$\underset{\pi \in \Pi }{max} \mathcal{J}\left(\pi \right) \mathrm{subject} \mathrm{to} g_k\left(\pi \right)\le 0, k=1,\dots ,K,$$

(4)

where $\pi$ denotes a plan/design/sequence, $\mathcal{J}$ encodes operational value, and $\left\{g_k\right\}$ encode geotechnical, operational, and safety constraints. Systematic evidence shows growing adoption of machine learning in mining and geoscience, but it also highlights failure modes when models remain purely correlational and do not respect physical or geometric structure [19,20,21]. In parallel, digital twin programs emphasize predictive models that are not black boxes but are mathematically grounded, traceable, and trustworthy for lifecycle decision support [22,23,24,25]. This motivates the study of GNNs that are constrained by symmetry, topology, and conservation principles.

The originality of this study lies in providing a unified systematic synthesis of mathematically grounded graph learning for engineering-oriented digital twins, integrating four streams that are often reviewed in isolation: (i) E(n)-equivariant GNNs for geometric symmetry, (ii) higher-order (simplicial/cellular) operators for non-pairwise interactions and topological consistency, (iii) symplectic/Hamiltonian graph learning for long-horizon stability of dynamical rollouts, and (iv) generative decision frameworks (e.g., GFlowNets) for constrained design and planning. Beyond categorization, we quantify the current research landscape using a final corpus of $n=96$ studies, where simplicial/cell-complex networks represent 22.9%, Clifford GNNs 18.8%, symplectic/Hamiltonian GNNs 16.7%, GFlowNets 14.6%, theoretical foundations 14.6%, and E(n)-equivariant GNNs 12.5%. These results support the significance of the proposed synthesis: they motivate principled inductive-bias selection and deployment-oriented evaluation (stability, robustness, and constraint satisfaction), which are essential for safety-critical applications such as smart mining and learning-enabled industrial digital twins.

1.2. Mathematically Grounded Graph Learning as a Synthesis Label

Recent research has increasingly converged on structure-aware neural architectures that embed explicit mathematical structure into graph learning [4,5,26]. In this review, the term mathematically grounded GNNs is used as a synthesis label (introduced by the authors) for this convergence. The common objective is to preserve key invariances and invariants by construction, rather than relying only on data-driven approximation.

To formalize this perspective, equivariance is generally defined with respect to (possibly different) group representations on the input and output spaces. Let ${\rho }_{\mathrm{in}}:G\to \mathrm{GL}\left(\mathcal{X}\right)$ and ${\rho }_{\mathrm{out}}:G\to \mathrm{GL}\left(\mathcal{Y}\right)$ denote these representations, respectively. The equivariance constraint then reads

$$f_{\theta }\left({\rho }_{\mathrm{in}}\left(g\right)x\right)={\rho }_{\mathrm{out}}\left(g\right)f_{\theta }\left(x\right), \forall g\in G.$$

(5)

While the commonly written form $f_{\theta }(g·x)=g·f_{\theta }\left(x\right)$ corresponds to the special case where the same (or canonically induced) action is used on both domains, many engineering settings require ${\rho }_{\mathrm{out}}$ to match the specific semantic type of the prediction, such as invariant scalars, equivariant vectors, or higher-order tensors.

In Equation (5), G denotes a symmetry group (e.g., translations and rotations, such as the Euclidean group $E\left(n\right)$) acting on the input space. For any $g\in G$, the notation $g·x$ represents the group action applied to an input x, i.e., a transformed version of x; thus, $f_{\theta }(g·x)$ is the same model evaluated on the transformed input. This notation formalizes symmetry constraints: $f_{\theta }$ is G-equivariant if transforming the input induces a corresponding transformation of the output, typically $f_{\theta }(g·x)={\rho }_{\mathrm{out}}\left(g\right) f_{\theta }\left(x\right)$ (often written as $f_{\theta }(g·x)=g.f_{\theta }\left(x\right)$ when the output transforms under the same action), and it is G-invariant if $f_{\theta }(g·x)=f_{\theta }\left(x\right)$.

This criterion underlies group-equivariant convolutional networks and modern $E\left(n\right)$/$SE\left(3\right)$-equivariant architectures used in physics and geometry [27,28,29,30,31]. In mathematically grounded graph learning, this idea is extended to include higher-order/topological consistency and structure-preserving dynamics, and it is complemented by principled generative modeling on discrete spaces.

Accordingly, this systematic review focuses on four research directions that instantiate this perspective and are directly relevant to mining-scale physical systems:

1.

Geometric Algebra (Clifford) equivariant GNNs. Vector features are lifted to multivectors in a Clifford algebra $\mathrm{Cl}(p,q)$, enabling unified representations of scalars, vectors, and oriented subspaces. Rotations are represented by rotors R acting by conjugation,

$$x^{\prime }=R x \tilde{R},\hspace{1em}\hspace{1em}R\tilde{R}=1,$$

(6)

which improves compositional stability and avoids fragile coordinate parameterizations. This direction has produced practical equivariant architectures and efficient implementations [32,33,34], and it connects to broader developments in equivariant representation learning [27,28].

2.

Simplicial and cell complex neural networks (topological deep learning). To capture polyadic interactions beyond edges, the underlying domain is lifted from graphs to simplicial or cellular complexes. Boundary operators $B_k$ and the Hodge Laplacians,

$$L_k=B_k^{\top }{B}_k+B_{k+1}{B}_{k+1}^{\top },$$

(7)

enable information flow across dimensions, so that cycles and cavities can act as carriers of signal. This has been formalized in topological message passing frameworks and surveys [17,26,35,36,37,38].

Let $C_k$ denote the space of k-chains (formal linear combinations of oriented k-simplices) in a simplicial complex. The boundary operator $B_k:C_k\to C_{k-1}$ maps each oriented k-simplex to the signed sum of its $(k-1)$-dimensional faces. In matrix form, $B_k$ is the oriented incidence matrix between $(k-1)$-simplices (rows) and k-simplices (columns), with entries ${\left(B_k\right)}_{ij}\in \{-1,0,+1\}$ indicating whether the $(k-1)$-simplex i is a face of the k-simplex j and whether the induced orientation agrees ($+1$) or disagrees ($-1$) with the chosen reference orientation (and 0 otherwise). A fundamental identity is $B_k{B}_{k+1}=0$ (the boundary of a boundary is empty), which ensures a valid chain complex and underlies topological consistency. These operators also induce the (Hodge) k-Laplacian, e.g., $L_k=B_k^{\top }{B}_k+B_{k+1}{B}_{k+1}^{\top }$, which is commonly used for higher-order message passing on simplicial and cellular complexes.

3.

Symplectic/Hamiltonian graph learning for dynamics. Standard neural predictors can accumulate error and violate invariants in long-horizon simulation. Hamiltonian and symplectic networks instead learn a Hamiltonian $H(q,p)$ and constrain dynamics in phase space $z=(q,p)$ by Hamilton’s equations,

$$\dot{q}={\displaystyle \frac{\partial H}{\partial p}},\hspace{1em}\hspace{1em}\dot{p}=-{\displaystyle \frac{\partial H}{\partial q}},$$

(8)

While Hamiltonian/symplectic models target conservative dynamics, many industrial processes are forced and/or dissipative (e.g., actuation, friction, damping). A common extension is to decompose the dynamics into a conservative Hamiltonian component plus a non-conservative forcing/dissipation term, $\dot{z}=J\nabla H_{\theta }\left(z\right)+f_{\theta }(z,t)$, so that the symplectic structure is preserved by construction for the conservative part, while $f_{\theta }$ captures losses and external inputs. In practice, one can integrate the conservative component with a symplectic step and handle $f_{\theta }$ additively (e.g., via operator splitting), which yields controlled long-horizon rollouts even when energy is not strictly conserved. For strongly dissipative regimes, closely related structure-preserving formulations (e.g., port-Hamiltonian and conformal-symplectic variants) can be more appropriate and are increasingly explored in the recent literature.

These models are typically coupled with symplectic integration to preserve the symplectic form. This structure-preserving approach improves qualitative fidelity and stability [39,40,41,42]. It aligns with physics-informed modeling objectives in scientific machine learning [43,44]. Related dynamical-systems analyses of delayed and fractional-order neural networks further highlight the need for explicit stability guarantees in engineering-oriented learning. For instance, Xing et al. [45] derive stability and Hopf bifurcation conditions for a high-dimensional double-ring neural network with multiple time delays, whereas Jia et al. [46] study finite-time synchronization in uncertain fractional-order delayed memristive neural networks using adaptive sliding-mode control.

4.

Generative Flow Networks (GFlowNets) on graphs. Many engineering problems require diverse high-quality solutions rather than a single optimum. GFlowNets learn a stochastic construction policy that induces a distribution over terminal objects x proportional to a nonnegative reward,

$$P_{\theta }\left(x\right)\propto R\left(x\right).$$

(9)

This enables principled exploration in combinatorial spaces such as graphs and sequences [47,48,49,50].

1.3. Scope and Contribution of This Review

The reviewed literature is organized around the above four directions and their mathematical foundations. Emphasis is placed on (i) which structural constraints are enforced (symmetry, topology, conservation, or probabilistic flow), (ii) how these constraints address known limitations of MPNNs (including the 1-WL ceiling), and (iii) why these constraints matter for high-stakes, geometry-rich industrial domains such as mining and digital twins [19,23,24]. The resulting taxonomy is intended to support principled model selection and to clarify open mathematical questions that remain central to the next generation of reliable graph-based learning systems.

1.4. Objectives, Research Questions, and Scope (PRISMA 2020)

The PRISMA 2020 framework is adopted to ensure transparency, reproducibility, and traceability in (i) the formulation of research questions, (ii) the search strategy, (iii) eligibility criteria, (iv) study selection (identification–screening–inclusion), and (v) evidence extraction and synthesis. In this manuscript, “Q1” and “Q2” are used only as bibliometric source-quality filters (first and second quartile) during identification, so that the final corpus is grounded on peer-reviewed, high-impact venues, and bias from non-refereed sources is reduced.

To compare methodological families consistently during the narrative synthesis, we established a common set of indicators applicable across different architectures. For computational complexity, we extracted asymptotic or proxy measures such as the per-layer operator cost regarding graph/complex size (number of nodes, edges, or cells), representation dimensionality (e.g., multivector grades in Clifford GNNs), and the overhead of higher-order operators like Hodge Laplacians. Regarding scalability, we recorded the largest demonstrated scales (nodes, cells, or sequence lengths) alongside memory constraints and implementation details that enable large-scale training, such as sparsity or kernel optimizations. For numerical stability, we summarized indicators relevant to long-horizon rollouts, including sensitivity to the integration step size $\Delta t$, error accumulation over time, and the enforcement of structure-preserving constraints like symplecticity or energy consistency. As reporting practices vary across the literature, these indicators serve as standardized qualitative comparators, complemented by quantitative data whenever explicitly provided by the primary studies.

• Research questions (RQs).

The review is organized around four questions, which are addressed across the four methodological families mapped in Figure 1:

• RQ1 (Mathematical structure): Which mathematical structures (group actions, geometric algebras, simplicial/cellular complexes, symplectic geometry, and generative flow principles) formally define each family, and which guarantees are implied (equivariance, expressivity, conservation, diversity)?
• RQ2 (Architectures and training): Which architectural designs and training objectives dominate recent structure-aware graph learning, and which trade-offs arise in complexity, scalability, and numerical stability?
• RQ3 (Mining relevance): Which mining value-chain problems (e.g., geomechanics, ventilation, planning, predictive maintenance, robotics) align naturally with each family and under which modeling assumptions?
• RQ4 (Gaps and deployment): Which open challenges remain for industrial adoption (end-to-end composition, validation, interpretability, and real-time deployment)?

The search window (2019–2026) is selected to capture the maturation of modern GNNs (equivariance, simulation, scalability) and the recent consolidation of mathematically grounded graph learning as a research frontier [26]. Under PRISMA 2020 and Q1/Q2 source filtering, the review analyzes at least 96 high-impact papers, providing a representative evidence base for current methods and their projected impact on mining-scale physical systems and digital twins.

The specific contributions are as follows:

1.

A taxonomy of advanced GNN architectures (Figure 1) organized by dominant mathematical principle and by the type of constraint enforced (symmetry, topology, conservation, generation).

2.

A formal synthesis of the mechanisms that explain performance gains over standard MPNNs: group equivariance, topological operators, symplectic preservation, and generative flow balance.

3.

A mining applicability map that links industrial requirements (geometry, safety constraints, long-horizon stability, combinatorial decision-making) to suitable inductive biases and model classes [19,21,23].

4.

Identification of technology gaps and future directions oriented to implementation (scalability, hybrid composition, operational validation, and traceability).

The remainder of the paper is structured as follows. The Materials and Methods section reports the PRISMA 2020 protocol (search, eligibility, selection, and extraction). The Results section synthesizes the four families with their mathematical foundations and representative architectures. The Discussion section compares the approaches, highlights synergies and limitations for mining deployment, and summarizes open problems. The Conclusions section consolidates implications and outlines an applied research agenda.

2. Materials and Methods

2.1. PRISMA 2020 Protocol

This systematic literature review (SLR) followed the PRISMA 2020 statement (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) as the reporting and process standard [51,52]. PRISMA 2020 provides an evidence-based framework for conducting and reporting systematic reviews, supporting transparency, methodological integrity, and reproducibility. The workflow was structured into four main phases: identification, screening, eligibility, and inclusion, which are described below.

2.2. Search Strategy

The identification phase was designed as an exhaustive and structured search for relevant studies. To maximize coverage and reduce publication bias, multiple high-impact scholarly databases and preprint repositories were queried.

Searches were conducted between 15 January and 28 January 2026. For reproducibility, the database queries were executed on 16 January 2026 in Web of Science (WoS) and Scopus. In addition, arXiv was searched within the same period using the corresponding platform-specific syntax and the field restrictions (title/abstract/keywords) described in Section 2.4. The interval 15–28 January 2026 corresponds to the full identification workflow, including query execution, deduplication, screening, eligibility checks, and consolidation of records.

To minimize omissions, backward and forward citation checking was applied to (i) the final included studies and (ii) seminal or highly cited papers representative of each architecture family. Records retrieved through citation checking were screened using the same inclusion and exclusion criteria as database records and were incorporated into the identification counts when eligible.

2.3. Information Sources

The selected sources were chosen for broad coverage across computer science, mathematics, physics, and engineering:

• Journal databases: Web of Science (WoS) and Scopus.
• Conference proceedings: NeurIPS, ICML, ICLR, AAAI, IJCAI.
• Preprint servers: arXiv (used primarily for highly cited papers or works later published in peer-reviewed venues).

2.4. Query Strings

Dedicated query strings were designed for each of the four advanced GNN families considered in this review, utilizing Boolean operators (AND, OR) in English across title, abstract, and keyword fields. To ensure full reproducibility, we implemented a strategy based on OR-connected synonym blocks—such as “graph deep learning” or “geometric machine learning”—and concept-level expansions for terms like “symplectic” or “Hamiltonian”. Field restrictions were applied to maximize precision (TITLE/ABSTRACT/KEYWORDS in Scopus; TS = topic in Web of Science), with blocks being iteratively refined to balance recall and specificity. The full Web of Science and Scopus query strings, including all synonym families for higher-order topology (e.g., “simplicial” or “CW complex”), equivariance, and structure-preserving dynamics, are provided in Appendix A. Representative examples of these queries include the following:

• (‘Clifford algebra’ OR ‘geometric algebra’) AND (‘graph neural network’ OR ‘deep learning’)
• (‘simplicial complex’ OR ‘cell complex’) AND ‘neural network’
• ‘symplectic’ AND (‘graph neural network’ OR ‘neural network’) AND ‘Hamiltonian’
• (‘GFlowNet’ OR ‘generative flow network’) AND ‘graph’

3. Inclusion and Exclusion Criteria

To ensure both quality and relevance, strict eligibility criteria were defined and applied during the screening and eligibility phases.

3.1. Inclusion Criteria

1.

Publication quality: Papers published in Q1 or Q2 journals (according to Journal Citation Reports or Scimago Journal Rank) or accepted at Tier-1 conferences. In this review, Tier-1 refers to flagship, highly selective peer-reviewed venues in ML/AI—specifically NeurIPS, ICML, ICLR, AAAI, and IJCAI—operationalized using external conference-ranking standards (such as CORE A* and A) as a transparency aid for venue quality.

2.

Language: Publications written in scientific English only. This criterion was introduced to ensure consistent full-text interpretability during the extraction of technical assumptions, equations, and reproducibility details, thereby reducing translation-induced uncertainty in fine-grained algorithmic formulations. Operationally, a study was included if a complete English full text was available; works written primarily in other languages were excluded if only metadata or abstracts were accessible in English, representing a scope-based decision rather than a judgment of research quality.

3.

Time window: Papers published between 1 January 2019 and 19 January 2026. While the quantitative analysis covers the full period, the narrative synthesis places a qualitative emphasis on recent studies (2024–2026) to capture the immediate state of the art. This ensures that the discussion focuses on the latest architectural frontiers without excluding foundational eligible works from 2019–2023 from the corpus.

4.

Topic relevance: The study explicitly developed, analyzed, or applied at least one of the four advanced GNN families defined in this review.

5.

Availability: Full text access was available.

3.2. Exclusion Criteria

1.

Studies without a verifiable peer-review process (with the exception of arXiv preprints that had accumulated substantial citations and were regarded as seminal by the community).

2.

Duplicate publications reporting the same study.

3.

Studies with unclear or non-reproducible methodology. To reduce subjectivity, we operationalized this exclusion criterion using a minimal reproducibility checklist during full-text screening. A study was considered reproducible if it met the following minimum reporting requirements: (i) a clearly specified method (model architecture or algorithmic procedure) and training objective, (ii) a sufficiently described dataset or data-generating process (source, preprocessing, and evaluation split/setup), and (iii) an evaluable experimental protocol (metrics and baselines or comparative references). Additional reproducibility signals were recorded when available (code or repository link, hyperparameters/training details, ablation or sensitivity analyses), but lack of these signals alone was not sufficient for exclusion. Studies were excluded under this criterion only when the minimum requirements (i)–(iii) were not met to a degree that prevented reconstructing the method or verifying the main claims.

4.

Papers that only mentioned the target architectures superficially, without substantive development.

5.

JCR/SJR quartile snapshot and disagreement rule: Whenever journal quartiles are referenced (“according to JCR or SJR”), we use a fixed quartile snapshot aligned with the review cut-off date (the 2025 releases of JCR and SJR, reflecting 2024 citation metrics). If JCR quartiles are available for a venue, JCR is used; if JCR is unavailable, SJR is used. When both quartiles are available but differ (e.g., due to different subject-category assignments), both values are recorded for transparency and the more conservative quartile (i.e., the numerically larger quartile value, e.g., Q2 over Q1) is adopted for screening and any single-quartile labeling to avoid optimistic bias.

3.3. Study Selection Process

Study selection was conducted in two stages by a team of reviewers. First, titles and abstracts of all identified records were screened to remove clearly irrelevant studies. Second, full texts of the remaining articles were assessed against the eligibility criteria. Disagreements were resolved through discussion and consensus. The complete workflow was documented using a PRISMA flow diagram (see Figure 2). As summarized in Figure 2 (PRISMA 2020), we identified 970 records in total (847 from indexed databases and 123 from complementary sources). After removing duplicates, 612 records were screened by title and abstract, leading to the exclusion of 389 items. We assessed 223 full-text articles for eligibility and excluded 127 mainly due to access limitations (34), out-of-scope content (52), insufficient methodological rigor (28), or missing DOI/traceability (13). The final corpus comprised 96 studies, which were included in both the qualitative synthesis and the quantitative summaries.

3.4. Data Extraction and Evidence Synthesis

For each included study, key information was extracted using a standardized template. Extracted fields included authors, publication year, venue, GNN family, mathematical foundations, methodology, datasets, main application domain, and key findings.

Extraction validation and cross-check procedure: To improve reliability, we complemented the extraction template with a structured validation protocol. First, each study was extracted using the predefined fields (authors, venue, datasets, methodology, and key results). Second, we performed a dedicated second-pass cross-check focusing on high-impact items that drive the narrative synthesis: method-family assignment, main claims, datasets, and evaluation setup. Ambiguities or inconsistencies were resolved by re-reading the primary source (Appendix A) and updating the record with a short clarification note. Whenever quantitative values were reported (e.g., maximum graph/complex size, runtime or memory constraints), they were transcribed verbatim and verified against the original tables/figures to minimize interpretation drift.

To operationalize the architectural comparison (RQ2), we defined specific performance indicators for data extraction: (i) asymptotic complexity, assessed via the scaling of message-passing operations with respect to input dimension; (ii) demonstrated scalability, measured by the maximum graph size or simplex order reported in experiments; and (iii) numerical stability, evaluated through energy conservation metrics and sensitivity to integration step sizes in long-horizon simulations.

Evidence synthesis was performed narratively, grouping studies by the four architecture families to support thematic and comparative analysis. Particular emphasis was placed on mathematical foundations and on implications for mining-scale physical systems and digital twin applications, consistent with the scope of this review.

Figure 3 contextualizes the temporal dynamics of the selected literature. The distribution shows a sustained growth from 2019 (4 studies) to a peak in 2024 (22 studies), followed by a still-high volume in 2025 (20 studies), indicating rapidly increasing interest in advanced geometry/topology/physics-aware graph learning. The 2026 count (4 studies) corresponds to a partial-year window consistent with the review cut-off date and should therefore be interpreted as an early signal rather than a full-year trend.

Figure 4 reports how the final set of 96 studies distributes across the architecture families analyzed in this review. Simplicial/cell-complex neural networks represent the largest share (22/96, 22.9%), followed by Clifford (geometric algebra) GNNs (18/96, 18.8%) and symplectic/Hamiltonian GNNs (16/96, 16.7%). GFlowNets and theoretical foundations each account for 14/96 (14.6%), while E(n)-equivariant GNNs contribute 12/96 (12.5%). This breakdown motivates the ordering of Section 4 and supports the emphasis on inductive biases that improve long-horizon generalization and out-of-distribution robustness.

4. Results

This section synthesizes and formalizes the methodological outcomes of the review. Four transitions are documented: (i) the move from vector-valued GNNs to geometric-algebraic representations based on Clifford (geometric) algebra to encode symmetries and higher-grade geometric objects; (ii) the shift from prediction to reward-guided generation through generative flow networks; (iii) the topological generalization from graphs to simplicial and cellular complexes to represent higher-order interactions; and (iv) the explicit incorporation of Hamiltonian and symplectic structure to support long-horizon stability in learned dynamics. Across these directions, a shared objective is the explicit design of geometric, topological, and physical inductive biases to reduce out-of-distribution error and improve temporal extrapolation [4,5,44].

4.1. Clifford GNNs (Geometric Algebra GNNs)

4.1.1. Foundations: From Vector Spaces to Geometric Algebra

Conventional GNNs operate on a graph $G=(V,E)$ with node features $h_i\in {\mathbb{R}}^d$ and, optionally, edge features $e_{ij}\in {\mathbb{R}}^{d_e}$. In the general message-passing form, a layer is written as

$$\begin{array}{cc}\hfill m_{ij}^{(𝓁)}& ={\varphi }_e^{(𝓁)} \left(h_i^{(𝓁)},h_j^{(𝓁)},e_{ij}\right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill h_i^{(𝓁+1)}& ={\varphi }_h^{(𝓁)} \left(h_i^{(𝓁)},{\mathrm{Aɢɢ}}_{j\in \mathcal{N}\left(i\right)} m_{ij}^{(𝓁)}\right),\hfill \end{array}$$

(11)

where ${\varphi }_e,{\varphi }_h$ are parametrized functions (typically MLPs), and $\mathrm{Aɢɢ}$ is permutation invariant (sum/mean/max) [7,10,16]. This formulation is effective for relational learning [11,13,53]. A limitation is induced by representing geometry through components in ${\mathbb{R}}^d$. Higher-grade geometric entities (oriented planes, oriented volumes) and continuous transformations (rotations, frame changes) are then handled through external parameterizations or are learned implicitly. Sample efficiency can be reduced and numerical instabilities can be introduced in settings where rigid motions dominate the data-generating process [4,5,28].

Geometric deep learning addresses this issue by incorporating problem symmetries through invariance and equivariance to group actions [4,5,27]. For data in ${\mathbb{R}}^3$, $E\left(3\right)$/$SE\left(3\right)$-equivariant architectures have been proposed, including EGNN, tensor field networks, and SE(3)-Transformers [29,30,31]. In many cases, features are still organized as vectors or tensor irreducible representations. This choice can be restrictive when the domain is naturally described using oriented objects and their compositions, such as discontinuity planes in geomechanics, instantaneous rotation axes, or oriented areas [5,54,55].

Geometric algebra, formalized as a Clifford algebra $\mathrm{Cl}(p,q)$ associated with a vector space $V\cong {\mathbb{R}}^{p,q}$ equipped with a bilinear form $g(·,·)$, provides a unified representation for scalars, vectors, and k-vectors (blades) within a graded algebra:

$$\mathrm{Cl}(p,q) \cong \underset{k=0}{\overset{n}{⨁}}{\mathsf{\Lambda }}^{k}V, n=p+q.$$

(12)

Let ${\left\{e_{\mu }\right\}}_{\mu =1}^n$ be an orthonormal basis with $e_{\mu }^2=g_{\mu \mu }\in \{+1,-1\}$. The generating relation is

$$e_{\mu }{e}_{\nu }+e_{\nu }{e}_{\mu }=2g_{\mu \nu }.$$

(13)

The central operation is the geometric product. For vectors $a,b\in V$, it decomposes as

$$ab=a·b+a\wedge b,$$

(14)

where $a·b={\langle ab\rangle }_0$ is the inner product (scalar part) and $a\wedge b={\langle ab\rangle }_2$ is the exterior product (a bivector) representing the oriented plane spanned by a and b. A general multivector is written as

$$A=\sum _{k=0}^n{\langle A\rangle }_k, {\langle A\rangle }_k\in {\mathsf{\Lambda }}^{k}V,$$

(15)

and the grade projections ${\langle ·\rangle }_k$ isolate scalar, vector, bivector, and higher-grade components.

This formalism connects directly to Lie theory. In Euclidean ${\mathbb{R}}^n$, bivectors generate the Lie algebra $\mathfrak{so}\left(n\right)$, and rotations are represented by rotors

$$R=\exp \left(-\frac{1}{2}B\right), B\in {\mathsf{\Lambda }}^{2}V,$$

(16)

acting by conjugation on vectors $x\in V$:

$$x^{\prime }=R x \tilde{R}, R\tilde{R}=1,$$

(17)

where $\widetilde{(·)}$ denotes reversion. Coordinate singularities associated with Euler angles are avoided, and rotation composition is expressed directly through group multiplication in $\mathrm{Spin}\left(n\right)$ (or $\mathrm{Pin}(p,q)$ for non-Euclidean signatures) [56,57,58]. In learning settings, direct modeling in $\mathrm{Cl}(p,q)$ can reduce geometric overhead and can improve training stability when rigid transformations and frame changes are central to the task [4,5,58].

Figure 5 summarizes the operational idea. Instead of propagating only vectors in ${\mathbb{R}}^d$, multivectors are propagated with channels per grade, and geometric products and contractions are used as fundamental operators. The geometric structure is then preserved at the representation level [5,32,33].

4.1.2. Architectures for Clifford GNNs

In Clifford GNNs, node features $h_i\in {\mathbb{R}}^d$ are replaced by multivectors $H_i\in \mathrm{Cl}{(p,q)}^c$ with c channels. Each channel contains $2^n$ components, one per basis blade. For equivariant design, let g be a symmetry-group element (for example, in $\mathrm{Pin}(p,q)$). The induced action on multivectors is

$$\rho \left(g\right):A\mapsto gA\tilde{g}.$$

(18)

A layer $\mathcal{F}$ is equivariant if

$$\mathcal{F}\left(\rho \left(g\right)H, g·X\right)=\rho \left(g\right) \mathcal{F}(H,X) \forall g,$$

(19)

which ensures that symmetry is respected and the generalization burden is reduced [27,28,59]. In practice, equivariance is enforced by restricting internal operations to combinations of (i) grade-wise sums, (ii) geometric products, and (iii) grade projections ${\langle ·\rangle }_k$, so that outputs transform as multivectors under (19) [32,33,58].

A message-passing scheme in $\mathrm{Cl}(p,q)$ can be constructed using relative vectors $\Delta x_{ij}=x_j-x_i\in V$ and scalar invariants $r_{ij}^2={\langle \Delta x_{ij}\Delta x_{ij}\rangle }_0$. Multivector messages can be defined as

$$M_{ij}^{(𝓁)} = {\mathsf{\Phi }}_e^{(𝓁)} \left(H_i^{(𝓁)}, H_j^{(𝓁)}, \Delta x_{ij}, r_{ij}^2\right),$$

(20)

and node updates can be written as

$$H_i^{(𝓁+1)} = {\mathsf{\Phi }}_h^{(𝓁)} \left(H_i^{(𝓁)}, \sum _{j\in \mathcal{N}\left(i\right)}{M}_{ij}^{(𝓁)}\right),$$

(21)

where ${\mathsf{\Phi }}_e,{\mathsf{\Phi }}_h$ are implemented through modules that combine MLPs on scalar invariants with geometric operations on equivariant terms. Examples include multiplication by $\Delta x_{ij}$, wedge products such as $\Delta x_{ij}\wedge \Delta x_{ik}$, or contractions with graded components of $H_j$ [10,32,54]. A close analogy with EGNN arises when dependencies are restricted to scalar invariants and $\Delta x_{ij}$ is used as the direction-carrying equivariant term [29,32,54].

For problems where coordinates are also updated, an equivariant update can be written as

$$x_i^{(𝓁+1)} = x_i^{(𝓁)} + \sum _{j\in \mathcal{N}\left(i\right)}{\psi }_x^{(𝓁)} \left({\langle M_{ij}^{(𝓁)}\rangle }_0,r_{ij}^2\right) \Delta x_{ij},$$

(22)

where ${\psi }_x$ returns a scalar. The correction is then a vector, which supports $E\left(n\right)$-equivariance and is compatible with the Clifford action when extended to $\mathrm{Pin}(p,q)$ [5,29,32]. This construction separates invariants that parameterize magnitudes from equivariant carriers that transport orientation. The same separation principle is used in general results on equivariant networks for compact groups and continuous domains [28,59,60].

Attention-based operators have also been extended to multivectors. The Geometric Algebra Transformer defines multivector queries, keys, and values and computes attention weights using invariants such as inner products or grade-wise norms. Rotational equivariance is preserved while benefiting from attention scaling [3,5,33].

The main computational concern is the $2^n$ growth in multivector components. Recent designs reduce cost through grade factorization, parameter sharing, and lightweight parameterizations. Comparable or improved accuracy has been reported with fewer effective parameters than tensor-based alternatives [33,58,61]. Figure 6 illustrates the graded hierarchy and the role of the geometric product as a compositional operator that couples oriented subspaces. This mechanism supports higher-order geometric interactions under controlled computational budgets [5,32,54].

4.1.3. Clifford and Geometric Algebra Applications in Mining

Modern mining involves systems with complex geometries, moving frames, uncertainty, and requirements for robust control and diagnosis. In this setting, the combination of equivariance and native geometric representation becomes practically relevant [5,19,23]. Three application axes are commonly identified.

Underground perception and robotics require consistency under rotations and frame changes in tasks such as visual tracking, pose estimation, and navigation. Hypercomplex and geometric algebra representations can encode orientation and spatial correlations without collapsing information into purely scalar descriptors [33,58,62]. For manipulators and autonomous equipment, these representations can be combined with $SE\left(3\right)$-equivariant modules to support more stable control and planning under perturbations and occlusions [5,29,31].

Particle systems, materials, and computational geoscience are also relevant. Several mining processes, including comminution, slurry transport, and sedimentation, can be modeled as interacting particle systems where rotational symmetry and conservation of physical quantities are central. In materials science, equivariant GNNs have enabled accurate and data-efficient interatomic potentials [63,64,65]. Geometric algebra provides an additional mechanism for encoding internal orientations, such as local planes and tensors, in a compact form. This is aligned with the broader trend of injecting explicit geometric quantities into message passing [5,54,66].

Structural geomechanics provides a third axis. In excavation stability, the orientation of discontinuity sets and their interactions govern failure kinematics. Bivectors in ${\mathsf{\Lambda }}^{2}V$ represent oriented planes directly, and their composition through the geometric product supports geometric coupling descriptions between discontinuities, including intersections and effective orientation changes, without ad hoc parameterizations. This direction is consistent with the growing use of GNNs in mechanics and structural engineering, where explicit geometric representation has been associated with improved generalization [54,67,68].

In summary, Clifford GNNs support a clear conceptual result. Geometry is not only inferred from data but is encoded as algebraic structure that enforces equivariance and supports higher-grade operators. This is consistent with the trajectory of modern geometric learning [4,5,32].

4.2. Generative Flow Networks on Graphs (GFlowNets)

While the previous sections focused on representation learning over defined geometric domains, engineering design often requires generating feasible discrete configurations from scratch. This section introduces generative flow networks (GFlowNets) as a framework for the sequential construction of graphs. We establish the probabilistic foundations required to sample diverse candidates under constraints, shifting the focus from static representation to the generative process itself.

4.2.1. Foundations: Beyond Prediction Toward Generation

In engineering, prediction is often not sufficient. Feasible high-performing discrete configurations must also be generated, including mine designs, ventilation layouts, extraction sequences, and molecules or materials. Classical generative models such as GANs and VAEs have been influential, but discrete combinatorial domains such as graphs remain challenging due to large action spaces, hard constraints, and expensive reward evaluation [69,70,71]. The limitation is particularly visible in molecular design and structural optimization, where solution diversity is required for exploration in addition to identifying a single optimum for exploitation [72,73,74].

Generative flow networks introduce an alternative paradigm. A stochastic policy is learned to construct an object x sequentially such that the induced distribution over complete objects is proportional to a nonnegative reward $R\left(x\right)$ [47,48,49]. A DAG of states $\mathcal{S}$ is defined with an initial state $s_0$ and terminal states $\mathcal{X}\subset \mathcal{S}$ representing complete objects. A trajectory $\tau =(s_0 \to s_1 \to \dots \to x)$ is sampled by a forward policy $P_F\left(a_t\right|s_t)$, and a backward policy $P_B\left(s_t\right|s_{t+1})$ assigns probability to reversing steps. Learning is formulated through state flows $F\left(s\right)$ and edge flows $F(s\to s^{\prime })$ that satisfy flow conservation at intermediate states:

$$\sum _{s^{\prime }: s\to s^{\prime }}F(s\to s^{\prime }) = \sum _{s^{″}: s^{″}\to s}F(s^{″}\to s), \forall s\in \mathcal{S}\setminus (\left\{s_0\right\}\cup \mathcal{X}),$$

(23)

and a terminal condition

$$F\left(x\right)=R\left(x\right), \forall x\in \mathcal{X}.$$

(24)

Under these constraints, the induced terminal distribution satisfies

$$P\left(x\right) = {\displaystyle \frac{F\left(x\right)}{Z}} = {\displaystyle \frac{R\left(x\right)}{{\sum }_{x^{\prime }\in \mathcal{X}}R\left(x^{\prime }\right)}},$$

(25)

so sampling concentrates on high-reward objects while retaining diversity [48,49,75].

A widely used training objective for discrete graphs is trajectory balance, which enforces trajectory-level flow consistency [48,49,76]. For a trajectory $\tau$ ending in x, the trajectory balance condition is  

$$Z \prod _{t=0}^{T-1}{P}_F\left(a_t\right|s_t) = R\left(x\right) \prod _{t=0}^{T-1}{P}_B\left(s_t\right|s_{t+1}),$$

(26)

with a learned normalizer Z. A common squared-error loss in log space is

$${\mathcal{L}}_{\mathrm{TB}}\left(\tau \right)={\left(logZ+\sum _{t=0}^{T-1}log{P}_F\left(a_t\right|s_t)-logR\left(x\right)-\sum _{t=0}^{T-1}log{P}_B\left(s_t\right|s_{t+1})\right)}^2,$$

(27)

which improves credit assignment in deep combinatorial construction processes [48,73,76].

Figure 7 illustrates the operational pipeline. An encoder, often implemented as a GNN, summarizes the partial graph, and a policy network proposes stochastic expansion or termination actions. Training is performed using trajectory balance objectives [13,48,49].

4.2.2. GFlowNet Architectures for Graphs

For graphs, the state $s_t$ is a partial graph $G_t=(V_t,E_t)$ and the action set $\mathcal{A}\left(G_t\right)$ typically includes (i) adding a node, (ii) adding an edge $(u,v)$, (iii) assigning attributes, and (iv) stopping. The main difficulty is the combinatorics of the action space. Selecting an edge can be $O\left(\right|V_t{|}^2)$. A common approach is action factorization, where u is selected first and then v, or actions are parameterized through templates. This reduces complexity and improves training stability [50,77,78].

In a typical implementation, a GNN encoder is used to compute node embeddings and a global embedding $g_t$:

$${\left\{h_v^{\left(t\right)}\right\}}_{v\in V_t}, g_t = {\mathrm{GNN}}_{\theta }\left(G_t\right),$$

(28)

using standard message passing or attention-based variants [9,10,13]. A forward policy is then defined as a distribution over actions parameterized by $g_t$ and/or $\left\{h_v^{\left(t\right)}\right\}$:

$$P_F\left(a\right|G_t)=\mathrm{softmax}\left({\pi }_{\theta }(a;g_t,\left\{h_v^{\left(t\right)}\right\})\right),$$

(29)

where ${\pi }_{\theta }$ can incorporate attention mechanisms to select nodes or pairs of nodes efficiently [3,50,74]. The backward policy $P_B$ can be defined symmetrically by reversing the last action, and it is used in trajectory balance objectives to close the probabilistic consistency condition [48,49,76].

A practically relevant result is that GFlowNets can address combinatorial optimization over graphs by sampling high-reward solutions with structural diversity. This contrasts with classical reinforcement learning and deterministic methods that often collapse to a single mode [50,73,77]. Variants have also been used for Bayesian structure learning, such as causal graph discovery, where the generated object is a discrete structure with a probabilistic score [48,49,79].

4.2.3. Generative Flow Networks for Mining and Molecular Design

GFlowNets are suitable for design problems in mining for three reasons. The domain is inherently discrete and combinatorial, rewards are multiobjective and constrained, and plan diversity is needed for scenario analysis [19,23,73].

Generative design of mine plans and mine topologies can be formulated by representing a design as a graph of galleries, ramps, and stopes with topological and geotechnical constraints. A reward can be defined as

$$R\left(G\right)=exp \left(\alpha \mathrm{VAN}\left(G\right)-\beta {\mathcal{R}}_{\mathrm{seg}}\left(G\right)-\gamma {\mathcal{C}}_{\mathrm{op}}\left(G\right)\right),$$

(30)

where $\mathrm{VAN}$ denotes net present value, ${\mathcal{R}}_{\mathrm{seg}}$ penalizes safety risks such as rockburst susceptibility, and ${\mathcal{C}}_{\mathrm{op}}$ denotes operational costs. The exponential form ensures $R\left(G\right)\ge 0$ and compatibility with (24). When ${\mathcal{R}}_{\mathrm{seg}}$ is evaluated by learned models, including physics- or topology-aware GNNs, the GFlowNet can act as a generative engine that proposes feasible alternatives for planning-oriented digital twins [21,23,24].

Materials and molecular discovery for mining processes can also be treated as graph generation. Reactants, flocculants, and collectors can be modeled as chemical graphs. GFlowNets have shown competitive performance in biological sequence design and other complex discrete objectives [48,49,80]. Rewards can be defined through graph-based predictors such as MPNNs or equivariant GNNs trained on chemical benchmarks [63,81,82], and they can be coupled to high-fidelity equivariant potentials when needed [64,65,66].

Sequencing and logistics problems arise in operational scheduling, dispatch, and maintenance planning, where sequential decisions must be optimized under uncertainty. GFlowNets can sample multiple high-reward sequences and can reduce dependence on a single fragile solution. This perspective complements neural combinatorial optimization and reinforcement learning approaches for discrete optimization [74,83,84].

In summary, GFlowNets shift the focus from prediction to design by learning distributions over high-quality solutions with controlled diversity, a capability particularly relevant in mining engineering and materials science where exploration under constraints is essential [48,50,78]. Building on this reward-guided generation of discrete graph structures, the focus now shifts toward higher-order representational formalisms—specifically simplicial and cellular complexes—which are required when non-pairwise interactions and topological features, such as cycles and cavities, are fundamental components of the system state.

4.3. Simplicial and Cell Complex Neural Networks (Topological Deep Learning)

Shifting the focus from the sequential construction of graphs (Section 4.2) toward the representation of complex interactions, we now examine simplicial and cell complex neural networks. By extending the relational viewpoint beyond pairwise edges, this framework introduces dimension-aware operators—such as the boundary map and Hodge Laplacian—to capture collective system dynamics that simple graphs cannot resolve.

4.3.1. Foundations: Beyond Graphs Toward Topology

Standard GNNs model binary interactions through edges. Many real systems exhibit higher-order interactions, including triads, cycles, cavities, collective constraints, and couplings that are not reducible to pairwise terms [18,85,86]. From a relational viewpoint, representing multi-agent phenomena as sums of pairwise interactions can introduce structural bias by losing information about closure and global coherence [11,36,86]. Hypergraphs provide an intermediate representation in which a hyperedge connects k nodes, and neural networks on hypergraphs have increased expressive reach relative to simple graphs [86,87,88]. For orientation, boundary structure, and dimension-aware neighborhoods, algebraic topology provides a richer framework through simplicial and cellular complexes [26,36,89].

A simplicial complex K is a collection of simplices closed under inclusion. If $\sigma \in K$, then every face $\tau \subset \sigma$ also belongs to K. A k-simplex $\sigma =[v_0,\dots ,v_k]$ represents a k-dimensional entity, including vertices, edges, triangles, tetrahedra, and higher-dimensional analogues. A cellular (CW) complex generalizes this construction by gluing cells of different dimensions without restricting to simplices, for example, by allowing arbitrary polygons as 2-cells [26,35,36].

The algebraic formalization is expressed through chain groups $C_k\left(K\right)$ and boundary operators ${\partial }_k:C_k\to C_{k-1}$ satisfying ${\partial }_{k-1}{\partial }_k=0$. This relation encodes topological consistency and enables invariants such as homology and persistent homology to characterize cycles and cavities [89,90,91]. The conceptual shift is that information propagation is no longer restricted to edges (1-cells) but can also occur through faces (2-cells) and volumes (3-cells), which supports collective couplings and long-range patterns [17,36,92].

4.3.2. Architectures of Topological Neural Networks

A key generalization of the graph Laplacian is the dimension-specific Hodge Laplacian. Let $B_k$ be the oriented incidence matrix representing the boundary operator ${\partial }_k$ in a discrete basis. The k-Hodge Laplacian is defined as

$$L_k = B_k^{\top }{B}_k + B_{k+1}{B}_{k+1}^{\top },$$

(31)

where $B_k^{\top }{B}_k$ is the lower term (interaction through $(k-1)$-boundaries) and $B_{k+1}{B}_{k+1}^{\top }$ is the upper term (interaction through $(k+1)$-co-boundaries) [37,38,92]. This decomposition makes explicit three sources of neighborhood for a k-cell: lower adjacency (shared boundary), upper adjacency (co-incidence within a higher-dimensional cell), and same-order adjacency induced by the combination of lower and upper relations.

Simplicial neural networks and simplicial complex neural networks implement spectral or spatial filters on $L_k$ for each dimension k and update signals $x^{\left(k\right)}$ defined on k-cells:

$$x_{𝓁+1}^{\left(k\right)} = \sigma \left(W_0^{\left(k\right)}{x}_{𝓁}^{\left(k\right)} + W_L^{\left(k\right)}{B}_k^{\top }{x}_{𝓁}^{(k-1)} + W_U^{\left(k\right)}{B}_{k+1}{x}_{𝓁}^{(k+1)}\right),$$

(32)

where $\sigma$ is a nonlinearity. The terms $B_k^{\top }{x}^{(k-1)}$ and $B_{k+1}{x}^{(k+1)}$ implement downward and upward information flow across dimensions [17,93,94]. CW networks extend this construction to cellular complexes while preserving the more general combinatorial structure of CW complexes [35,36,92].

More recent work has proposed topological message passing mechanisms that explicitly emphasize higher-order interactions and long-range dependencies, for example, by using operators in which cycles and cavities act as carriers of global information [18,26,92]. This direction is consistent with the observation that many critical patterns, such as recirculation, redundant routes, bottlenecks, and cavities, are topological invariants that are not captured by purely metric descriptors [89,90,91]. There is also evidence of links between topology and the capacity and optimization behavior of deep networks, which supports the use of topology as an inductive bias [26,36,95].

4.3.3. Topological and Cellular Complex Applications in Mining

Underground mine ventilation and contaminant transport are intrinsically volumetric. Air moves along tunnels that can be treated as 1-cells, but mixing and effective distribution depend on chambers, intersections, and regions that require 2- and 3-cells. Representing a mine as a cellular complex allows higher-order connectivity to be encoded and supports the integration of conservation constraints for mass and energy. In this setting, cellular complex neural networks can be coupled with physics-informed learning to regularize transport and flow equations and reduce out-of-regime error [43,44,96]. For turbulence closures and reduced-order models, energy-conserving parameterizations are consistent with recent progress on networks that preserve energy in fluid mechanics [96,97,98].

Rock mass stability and failure in fractured media are emergent phenomena where interactions among multiple discontinuities can produce kinematic mechanisms that are not reducible to pairwise effects. Representing blocks and contacts as cells and co-boundaries can expose higher-order couplings and can support the prediction of critical zones, including rockburst-prone areas, using topological inductive biases [21,85,86]. This perspective is consistent with the growing use of graph-based learning in structural analysis and applied mechanics, where discretizations such as meshes and assemblies are treated as relational objects with nontrivial structure [67,68,99].

Sensor networks for seismic, deformation, and thermal monitoring produce multiscale signals. Computational topology tools such as persistent homology can extract robust descriptors of wavefronts, cavities, and effective connectivity. These descriptors can be integrated as features or losses in topological learning models [89,90,91]. In a digital twin setting, such topological signatures can act as global observables for validation and diagnosis [23,24,25].

In conclusion, topological deep learning formalizes a practical principle. When physical and engineering behavior depends on collective interactions and on the structure of cycles and cavities, it is methodologically preferable to lift the representation from graphs to complexes, because the associated operators, including Hodge Laplacians and boundary maps, align the architecture with the underlying geometry of the phenomenon [36,37,92]. To synthesize the frameworks discussed across these sections, Figure 8 provides a consolidated overview of these advanced GNN architectures and their targeted applications in the mining industry.

4.4. Symplectic/Hamiltonian Graph Neural Networks (SympGNN)

4.4.1. Foundations: Why Physical Conservation Is Needed

Reliable simulation of physical systems requires structural invariants to be respected, including energy, momentum, and volume preservation in phase space. Generic neural predictors that learn $z_{t+\Delta t}=f_{\theta }\left(z_t\right)$ often accumulate error and can diverge over long horizons, even when short-term error is small [39,42,44]. In Hamiltonian mechanics, the state $z=(q,p)\in {\mathbb{R}}^{2d}$ evolves according to

$$\dot{z}=J\nabla H\left(z\right), J=\left[\begin{array}{cc}0& I\\ -I& 0\end{array}\right],$$

(33)

where $H(q,p)$ is the Hamiltonian. The exact flow ${\phi }_t$ is symplectic, meaning that

$${\left(D{\phi }_t\left(z\right)\right)}^{\top }J\left(D{\phi }_t\left(z\right)\right)=J, \forall t,$$

(34)

and therefore preserves the symplectic form and qualitative properties of the dynamics. The literature on structure-preserving neural networks stresses that enforcing (34), or approximating it in a controlled way, is essential for long-term stability [40,42,100]. This is consistent with physics-informed learning, where governing equations and constraints provide inductive bias that reduces erroneous extrapolation [43,44,98].

Figure 9 summarizes the design principle. The model is not only a state regressor. It is an explicit or implicit learned integrator that preserves Hamiltonian or symplectic structure and, when needed, is combined with equivariant modules so that spatial symmetries are respected [39,40,54].

4.4.2. Architectures of Physics-Informed Neural Networks

Hamiltonian neural networks parameterize a scalar Hamiltonian $H_{\theta }(q,p)$ and obtain dynamics through (33) by replacing $\nabla H$ with $\nabla H_{\theta }$:

$$\dot{z}\approx J\nabla H_{\theta }\left(z\right).$$

(35)

This yields energy conservation in the continuous-time model under exact integration. In practice, numerical integration must be chosen carefully to control energy drift over long horizons [39,42,100]. Lagrangian neural networks learn a Lagrangian and derive Euler–Lagrange equations, which can make it easier to incorporate holonomic constraints and symmetries [42,98,101].

SympNets learn a discrete-time map $z_{t+\Delta t}={\mathsf{\Psi }}_{\theta }\left(z_t\right)$ that is symplectic by construction, typically by composing simple transformations that preserve (34) [40,42,100]. Related models such as symplectic ODE-Net incorporate control signals and Hamiltonian structure into learned dynamics [40,100,102]. For sequential prediction, symplectic recurrent neural networks have been proposed to maintain temporal stability [41,42,102]. Hamiltonian generative networks connect probabilistic modeling with physical dynamics in latent spaces and are relevant for inverse problems and conditional generation [39,98,103].

A classical characterization of symplectic maps uses generating functions. For instance, a type-II generating function $S(q,p^{\prime })$ defines the map

$$p={\displaystyle \frac{\partial S}{\partial q}}, q^{\prime }={\displaystyle \frac{\partial S}{\partial p^{\prime }}},$$

(36)

which is symplectic under mild regularity assumptions. In learning-based settings, one parameterizes $S_{\theta }$ with a neural network and obtains a structure-preserving integrator [40,42,100]. For constrained systems, such as those with kinematic constraints or contacts, the Hamilton–Dirac framework extends this idea. Neural formulations have been explored to handle constraints in a geometrically consistent way [42,98,104].

For many-particle systems, the Hamiltonian often decomposes as

$$H(q,p)=\sum _{i=1}^{N}T\left(p_i\right) + \sum _{(i,j)\in E}U(q_i,q_j),$$

(37)

where E represents interactions such as contacts or pairwise forces. Graph networks have been effective for learning simulators from data when interactions are represented as edges [11,99,105]. A SympGNN combines (37) with message passing modules to approximate the interaction term U or a generating function $S_{\theta }$ and then updates $(q,p)$ using a symplectic map, for example, via (36). This reduces energy drift over long horizons [40,99,100]. If spatial equivariance to $E\left(3\right)$ is required, equivariant messages based on explicit geometric quantities can be used, following modern equivariant architectures [5,29,54].

4.4.3. Applications in Mining and Physical Systems

Long-term stability matters for mining assets with extended operating cycles, where prediction errors can translate into operational risk or economic losses. In this context, SympGNNs support dynamic digital twins with structural guarantees [22,23,25].

Massive structures and heavy equipment, including stackers, reclaimers, and overhead cranes, are subjected to cyclic loading and transient excitations. A surrogate model that preserves energy-related structure reduces numerical drift when simulating very long horizons, which can support cumulative damage estimation and remaining useful life assessment. This aligns with recent progress in mechanics-informed models and graph-based methods for structural analysis [67,68,106] and with digital twin paradigms for monitoring and maintenance [23,24,25].

Process dynamics in grinding, slurry transport, and vibration monitoring involve many-particle interactions and energy–momentum couplings. Graph networks have been used for learned physical simulation on meshes and complex systems [13,99,105]. Enforcing symplectic or energy-preserving structure is consistent with recent results on energy-preserving networks for turbulence closures and complex dynamics [42,97,100].

Intermittent contacts, operational bounds, and geometric constraints can be modeled through constrained formulations. The Hamilton–Dirac approach provides a systematic framework for constraints, and learning-based structure-preserving variants offer a path toward robust simulation under contacts and regime changes [42,98,104].

The central methodological message of SympGNNs is that long-term fidelity is not achieved only by more data or deeper networks. It also requires enforcing the correct phase-space geometry, namely, symplectic structure, and incorporating spatial symmetries in message passing when the problem demands it [5,40,54].

4.5. Challenges and Real-World Deployment Barriers

Synthesizing the evidence regarding the performance indicators defined in Section 3.4 (asymptotic complexity, scalability, and numerical stability), we identify specific computational and operational barriers for deployment in industrial mining environments:

• Clifford GNNs impose a significant memory footprint due to the multivector representation, where feature dimension scales as $2^n$ with the space dimension n. Although grade factorization reduces this cost [32], the multiply–accumulate (MAC) operations for geometric products remain heavier than standard scalar products, potentially limiting real-time inference on edge devices.
• Simplicial and Cellular Networks require the pre-computation and storage of incidence matrices and boundary operators for high-order structures (faces, volumes). For large-scale mine topologies, this leads to a combinatorial explosion in memory usage, which necessitates the use of sparse adjacency structures and dimension-restricted message passing [17].
• Symplectic and Hamiltonian GNNs, while stable for long-term simulation, are sensitive to the choice of time-stepping integrators. Stiff dynamics, common in rock mechanics, may require implicit solvers that are computationally expensive to differentiate through during training [101].
• Generative Flow Networks (GFlowNets) face a distinct bottleneck in sample efficiency. Unlike supervised learning, they require extensive exploration of the combinatorial action space to converge to the target distribution. Reward engineering and the calibration of temperature parameters are non-trivial and often require domain-specific heuristics to prevent mode collapse [48].

To mitigate these issues, future work is increasingly adopting hybrid strategies, such as using low-dimensional equivariant backbones for control loops while reserving high-fidelity Hamiltonian integration for offline planning [5,54].

5. Discussion

This discussion integrates the findings of the manuscript under a single methodological principle: improvements in generalization and stability are not obtained only by increasing parameter count but by enforcing mathematical structure (symmetry, topology, conservation, and probabilistic flow) as inductive constraints. This principle is consistent with the evolution from generic deep learning models toward geometric, equivariant, and physics-informed learning [1,4,44]. In the context of this review, the four families are best understood as complementary answers to the same limitation of standard message passing: purely data-driven aggregation is often insufficient when the target system is governed by rigid transformations, higher-order connectivity, or long-horizon invariants [5,36,42].

5.1. Comparative Synthesis of the Four Families

A unifying view is constrained learning:

$$\underset{\theta }{min} {\mathbb{E}}_{(x,y)\sim \mathcal{D}}\left[𝓁(f_{\theta }\left(x\right),y)\right] \mathrm{s}.\mathrm{t}. \mathcal{C}\left(f_{\theta };x\right)=0,$$

(38)

where $\mathcal{C}$ encodes a geometric, topological, physical, or flow-consistency constraint that reduces the effective hypothesis space. When $\mathcal{C}=0$ is satisfied by construction, the model typically exhibits improved sample efficiency and better out-of-distribution behavior, consistent with the role of equivariance and inductive bias [5,27,28]. When constraints are enforced through penalties, a common approach is to add ${\lambda \parallel \mathcal{C}\parallel }^2$ to the objective, and performance depends strongly on the calibration of $\lambda$ and on optimization stability [42,43,98].

From a practical standpoint, selecting a family for an engineering task can be organized as a sequence of checks:

1.

Symmetry check. Identify the relevant group G acting on the inputs and decide whether the model must be equivariant:

$$f_{\theta }\left({\rho }_{\mathrm{in}}\left(g\right)x\right)={\rho }_{\mathrm{out}}\left(g\right) f_{\theta }\left(x\right) \forall g\in G.$$

(39)

This criterion motivates equivariant GNNs and Clifford-based designs when rotations and frame changes are central [4,5,27].

2.

Interaction order check. Decide whether binary edges capture the essential interactions or whether higher-order structure is intrinsic. If higher-order relations matter, the base object becomes a complex K with boundary operators and Hodge Laplacians [17,36,37].

3.

Long-horizon dynamics check. If the system is Hamiltonian or near-Hamiltonian, long-term fidelity depends on preserving symplectic structure. For a learned phase-space map ${\mathsf{\Phi }}_{\theta }$:

$${\left(D{\mathsf{\Phi }}_{\theta }\right)}^{\top }J\left(D{\mathsf{\Phi }}_{\theta }\right)=J,$$

(40)

a property implemented by symplectic networks and related approaches [40,42,100].

4.

Objective check: prediction versus design. If the goal is to produce diverse high-quality discrete solutions, it is natural to learn a distribution $P_{\theta }\left(x\right)\propto R\left(x\right)$ using GFlowNets, rather than a single point estimate [48,50,76].

Table 1. Comparative synthesis of mathematically grounded GNN families: dominant constraint, key mathematical object, and typical application domain.

Family	Main Limitation Addressed	Key Mathematical Object	Typical Applications
Clifford GNNs	Continuous symmetries and oriented geometric entities	Clifford algebra $\mathrm{Cl}(p,q)$, $\mathrm{Spin}/\mathrm{Pin}$ action, equivariance (39)	3D perception and robotics; physics and materials [32,33,58]
Simplicial/Cell NNs	Higher-order interactions, cycles, cavities, global consistency	Complexes K, boundaries ${\partial }_k$, Hodge Laplacians $L_k$ [35,37,38]	Networked flows; ventilation; volumetric geomechanics [26,36,92]
Symplectic/ Hamiltonian GNNs	Long-horizon stability and invariant preservation	Hamiltonian mechanics, symplectic maps (40) [39,40,42]	Stable surrogate simulation; prognostics; dynamical systems [97,100,104]
GFlowNets	Diverse generation of high-reward discrete structures	Flow conservation on a DAG; trajectory balance; $P_{\theta }\left(x\right)\propto R\left(x\right)$ [48,49,76]	Generative design; combinatorial optimization; exploration [50,77,78]

summarizes the families by dominant constraint, mathematical object, and application style.

To complement this synthesis, Figure 10 provides a multi-dimensional evaluation of the four advanced GNN families, illustrating the inherent trade-offs between geometric expressivity, computational complexity, and physical fidelity. The main trade-off is that structural guarantees usually increase mathematical and computational complexity. This makes scalability and verification central topics, not secondary implementation details [13,53,107].

5.2. Compositional Perspective

The families are not mutually exclusive. Combining them corresponds to enforcing multiple constraints $\mathcal{C}$ in (38). Two compositions are particularly natural for mining-scale systems.

First, symplectic and geometric representations can be combined when the state includes rigid-body orientation. Rotations can be represented by rotors in $\mathrm{Spin}\left(3\right)$ (or more generally elements of $\mathrm{Pin}(p,q)$), and a learned integrator can be constructed to preserve symplectic structure while remaining equivariant under frame changes. This aligns with the emphasis on explicit geometric quantities in message passing [29,32,54] and with the broader program of structure-preserving dynamics learning [40,42,100].

Second, topology can be integrated into generative design by defining rewards that include global connectivity targets such as redundancy, bottlenecks, and cavity structure. One example is

$$R\left(G\right)=exp \left(\alpha \mathcal{U}\left(G\right) - \beta \mathcal{R}\left(G\right) - \gamma \parallel \mathbf{b}\left(G\right)-{\mathbf{b}}^{\star }{\parallel }_1\right),$$

(41)

where $\mathbf{b}\left(G\right)$ collects topological invariants (e.g., Betti numbers or persistence-based descriptors). These descriptors can be computed with persistent homology [89,90,91] or approximated through topological message passing on complexes [26,36,92], while GFlowNets provide reward-proportional diverse sampling [48,49,76].

5.3. Implications for Mining and Digital Twins

Mining systems combine multi-physics processes, complex geometry, and discrete decision layers under strict safety constraints. Systematic evidence indicates growing adoption of machine learning in mining and geoscience, alongside documented failure modes when models remain purely correlational and ignore physical or geometric structure [19,20,21]. In parallel, digital twin programs emphasize predictive models that are mathematically grounded, traceable, and trustworthy for lifecycle decision support [22,23,24,25]. A minimal dynamic digital twin can be expressed as

$$\begin{array}{cc}\hfill z_{t+\Delta t}& ={\mathcal{F}}_{\theta }(z_t,u_t,{\xi }_t),\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill y_t& =\mathcal{H}\left(z_t\right)+{\epsilon }_t,\hfill \end{array}$$

(43)

where $z_t$ is the physical and operational state, $u_t$ denotes controls, and $y_t$ are sensor observations. The key implication of mathematically grounded graph learning is to construct ${\mathcal{F}}_{\theta }$ under domain-consistent constraints, reducing drift and improving extrapolation under distribution shifts [23,42,44].

For critical assets and processes, long-horizon stability is often decisive. Symplectic and Hamiltonian graph learning provides a route to stable surrogate simulation by controlling energy drift and preserving qualitative behavior [39,40,100]. When orientation and changing reference frames are dominant degrees of freedom, Clifford-based representations provide a direct way to encode rotations and oriented subspaces, complementing $SE\left(3\right)$-equivariant message passing [29,32,33]. For infrastructure and rock-mass structure, higher-order topology is relevant because failure mechanisms and flow patterns can depend on cycles, cavities, and collective interactions. Representing the domain as cellular or simplicial complexes enables Hodge-based operators that align the architecture with these constraints [36,37,92]. Finally, for planning, layouts, and sequencing, the objective is often to generate diverse feasible alternatives rather than a single optimum. GFlowNets support this by learning reward-proportional distributions over discrete designs [48,50,78].

Figure 8 can be interpreted as a modular composition map: topological modules for infrastructure and rock-mass structure [17,92,108]; geometric and equivariant modules for perception and robotics [30,31,32]; structure-preserving dynamics modules for stable simulation and forecasting [40,42,97]; and generative modules for design and planning under uncertainty [48,50,75]. This modular view aligns with common digital twin practice, where components are connected through explicit interfaces and validated against operational constraints [24,25,109].

Figure 11 provides a bibliometric summary of the backbone of the field: message passing foundations [7,10,16]; geometric deep learning and equivariance [4,5,27]; topological deep learning [26,36,37]; physics-informed and structure-preserving learning [42,43,44]; and flow-based generation [48,49,76]. A practical implication is that more consolidated subareas tend to provide stronger evaluation protocols, benchmarks, and software ecosystems, which matters for industrial transfer [6,13,53].

Figure 11 provides a bibliometric summary of the foundational publications that form the backbone of the field. This includes the establishment of neural message passing (MPNN) [10] and hypergraph extensions (HGNN) [87], the formalization of the geometric deep learning blueprint [4], and the introduction of $E\left(n\right)$-equivariant networks (EGNN) [29]. It also tracks the emergence of combinatorial generative frameworks (GFlowNets) [48], higher-order cellular topologies (CW Networks) [35], structure-preserving symplectic models (SympNets) [40], and recent geometric algebra architectures (Clifford GNN) [32]. A practical implication is that more consolidated subareas tend to provide stronger evaluation protocols, benchmarks, and software ecosystems, which matters for industrial transfer [6,13,53].

5.4. Challenges and Future Directions

Despite strong potential, industrial deployment requires progress on verification, scalability, and system integration.

One challenge is mathematical complexity and verification. Equivariant, topological, and structure-preserving models require careful validation that the intended properties hold in the implemented pipeline, including discretization and numerical effects [5,36,42]. A concrete direction is automated symmetry and constraint diagnostics, informed by work on symmetry discovery and Lie-theoretic analysis for neural networks [55,57,110].

A second challenge is computational scalability. Clifford-based models can incur overhead from multivector representations, and topological models scale with the number of higher-dimensional cells and with the cost of incidence operators. Learned simulators on meshes and particle systems are also heavy at mine scale [13,99,105]. Progress will likely require both parameter-efficient designs [32,58,61] and scalable kernels and parallelism strategies, as explored in large-scale equivariant potentials [64,65,107]. Hardware considerations remain relevant when moving from prototypes to operational systems.

A third challenge is end-to-end composition and data integration. The highest impact arises when generation, evaluation, simulation, and decision are connected into a closed loop. One view is a bi-level objective:

$$\begin{array}{cc}\hfill \underset{\theta }{max} & {\mathbb{E}}_{G\sim P_{\theta }}\left[R_{\psi }\left(G\right)\right]\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}. & R_{\psi }\left(G\right)=\mathcal{R}\left(G, {\mathcal{E}}_{\varphi }\left(G\right), {\mathcal{S}}_{\eta }\left(G\right)\right),\hfill \end{array}$$

(45)

where $P_{\theta }$ is a GFlowNet, ${\mathcal{E}}_{\varphi }$ is a geomechanical or topological evaluator, and ${\mathcal{S}}_{\eta }$ is a structure-preserving simulator. Stable training requires robust flow objectives [48,49,76] and reliable continuous-time learning tools such as neural ODEs and structure-preserving integrators [42,100,111]. The operational reality of reconfiguration and time-varying graphs also motivates dynamic graph learning components [13,112,113].

A fourth challenge is interpretability and operational trust. Practical routes include attention-based explanations in graph models [3,9,13], topological summaries as global explanations [89,90,91], and symmetry auditing to verify expected invariances [55,57,110]. In digital twin settings, traceability also requires explicit definition of scope, fidelity targets, update rules, and uncertainty handling [23,24,25].

Recent studies from early 2026 confirm the accelerating convergence of these paradigms. For instance, Sun et al. [114] argue that Riemannian geometry is becoming the unifying foundation for graph learning, while Ji [115] demonstrated that Clifford geometric algebra can completely replace standard interaction layers in vision and graph backbones. Addressing the stability of deep architectures, Su and You [116] established that respecting the underlying manifold geometry via dynamic scaling is critical for preventing representation collapse. Furthermore, the industrial applicability of symplectic priors was reinforced by Ma et al. [117], who integrated Hamiltonian energy conservation into few-shot learning frameworks, proving that physics-informed constraints effectively stabilize training in data-scarce regimes.

Overall, the accumulated evidence supports a shift from larger models to models that match the mathematics of the domain. In practical terms, this means using equivariance when symmetry is known [60,118,119], using topology when global connectivity dominates [26,36,92], using structure preservation when long horizons matter [40,97,100], and using flow-based generative models when the goal is to design and explore discrete solution spaces rather than only to predict [48,50,78].

6. Conclusions

This systematic review, conducted under the PRISMA 2020 standard, consolidates the available evidence (96 high-impact articles) on recent advances in graph neural networks. The main conclusion is consistent across the literature: the field is moving from models that primarily fit statistical correlations toward architectures that enforce geometric, topological, and physical structure by construction. From an engineering perspective, this shift replaces unconstrained universal approximators with learned operators that respect symmetries, higher-order connectivity, and conservation laws, improving stability, long-horizon behavior, and transfer to operational settings.

Modern mining can be formalized as a multi-scale cyber-physical system that combines (i) discrete design and sequencing decisions, (ii) continuous fields (stress, deformation, flow), and (iii) transient dynamics of critical assets. In compact form, the strategic problem can be expressed as constrained multiobjective optimization:

$$\underset{\pi \in \mathsf{\Pi }}{max} \mathbb{E} \left[\sum _{t=0}^T{\displaystyle \frac{C{F}_t\left(\pi \right)}{{(1+r)}^t}}\right] \mathrm{subject} \mathrm{to} g_{\mathrm{geo}}\left(\pi \right)\le 0, g_{\mathrm{vent}}\left(\pi \right)\le 0, g_{\mathrm{ope}}\left(\pi \right)\le 0, g_{\mathrm{seg}}\left(\pi \right)\le 0,$$

(46)

where $\pi$ denotes the plan (geometric design, sequence, logistics), $C{F}_t$ is cash flow, r is the discount rate, and $\left\{g_{·}\right\}$ encode geotechnical, ventilation, operational, and safety constraints. The reviewed methods support practical approximations of the operators required to evaluate and optimize (46) at industrial scale while maintaining physically meaningful structure.

The findings can be translated into an operational guide for mining deployment, consistent with the comparative synthesis in Table 1, the multi-criteria analysis in Figure 10, and the application map in Figure 8:

1.

When geometry and frame changes dominate: Clifford GNNs. In underground robotics, inertial instrumentation, 3D perception, and equipment kinematics, performance depends on consistency under rotations and rigid transformations. Clifford GNNs replace purely vectorial features with multivectors and use geometric algebra operations (Figure 5 and Figure 6), so that rotations are represented by rotors,

$$x^{\prime }=R x \tilde{R}, R\tilde{R}=1,$$

(47)

avoiding fragile parameterizations and enabling coherent state estimation and control in environments with strong geometric uncertainty.

2.

When collective interactions and volumetric structure dominate: simplicial and cell complex neural networks. Rock-mass stability, ventilation, and underground hydrology are not purely pairwise phenomena; they depend on couplings across surfaces and volumes. Topological deep learning lifts the domain from graphs to complexes (Figure 8), where information propagation is organized by dimension through boundary operators and Hodge Laplacians:

$$L_k=B_k^{\top }{B}_k+B_{k+1}{B}_{k+1}^{\top }.$$

(48)

This supports models that respect the edge–face–volume hierarchy and capture emergent mechanisms such as stress propagation through contact faces, improving fidelity for risk evaluation and control design.

3.

When long-horizon dynamics dominate: symplectic and Hamiltonian graph learning. For critical assets (mills, conveyors, pumps), useful prediction requires long time horizons; without structural preservation, small errors accumulate. Symplectic GNNs (Figure 9) incorporate Hamiltonian or symplectic structure as an architectural constraint. For a learned phase-space map ${\mathsf{\Phi }}_{\theta }$ with $z=(q,p)$, the key condition is

$${\left(D{\mathsf{\Phi }}_{\theta }\right)}^{\top }J\left(D{\mathsf{\Phi }}_{\theta }\right)=J,$$

(49)

which stabilizes surrogate simulation and supports operational digital twins for predictive maintenance and long-term degradation forecasting under cyclic loads.

4.

When the objective is to design and explore, not only to predict: GFlowNets. Mine planning lives in discrete combinatorial spaces. GFlowNets (Figure 7) shift the objective toward sampling diverse high-quality solutions G proportional to a positive reward $R\left(G\right)$:

$$P\left(G\right)\propto R\left(G\right), R\left(G\right)=exp\left(\alpha \mathrm{VAN}\left(G\right)-\beta {\mathcal{R}}_{\mathrm{seg}}\left(G\right)-\gamma {\mathcal{C}}_{\mathrm{amb}}\left(G\right)\right),$$

(50)

where ${\mathcal{R}}_{\mathrm{seg}}$ penalizes geotechnical and operational risk and ${\mathcal{C}}_{\mathrm{amb}}$ accounts for environmental impact. This enables the generation of portfolios of feasible alternatives for scenario analysis rather than a single potentially brittle plan.

A practical implication for the mining industry is that these architectural families can be composed into a unified, end-to-end system where specialized operators are connected through physically meaningful interfaces—such as state variables, constraints, and rewards—and validated against real-world sensor measurements. This approach provides a principled checklist for designing learning-enabled digital twins with improved reliability, shifting the focus from purely predictive modeling toward the constraint-aware exploration of discrete design spaces. However, successful industrial adoption still requires addressing concrete engineering challenges, including validation and traceability beyond average accuracy, computational scalability for real-time latency and the integration of heterogeneous sensor data into consistent topological representations.

Implications and Practical Applications: The identified inductive biases—E(n)-equivariance, higher-order topological operators, and symplectic structure—provide a principled checklist for designing learning-enabled digital twins with improved reliability. These conclusions motivate the use of generative frameworks like GFlowNets for constraint-aware exploration of discrete design spaces (e.g., layouts and schedules) rather than purely predictive modeling. Furthermore, this synthesis supports the implementation of deployment-oriented governance criteria, including stability under rollout, robustness to distribution shift, explicit constraint satisfaction, and decision-oriented interpretability, all of which are essential for safety-critical industrial adoption.

Future Research Directions: Building on this synthesis, several extensions are particularly impactful for smart mining deployment: (i) the creation of mining-specific benchmark suites with open evaluation protocols; (ii) the development of compositional pipelines that couple GFlowNets for discrete design exploration with topological evaluators and symplectic simulators; (iii) the advancement of verification and uncertainty quantification (e.g., constraint certification and calibrated risk) to support safety-critical decisions; and (iv) the engineering of scalable implementations for dynamic, sensor-driven digital twins operating under real-time constraints.