Advances in Digital Twins and Hybrid Physical–Digital Systems

Dear Colleagues,

Digital twins (DTs) have emerged as a transformative technology paradigm that creates living digital representations of physical systems, enabling real-time monitoring, prediction, and optimisation across diverse domains from manufacturing and healthcare to smart cities and climate modelling. This Special Issue focuses on the fundamental mathematical challenges and innovations required to develop, analyse, and deploy digital twins and hybrid physical–digital systems at scale.

The mathematical foundations of digital twins span multiple disciplines, requiring novel theoretical frameworks that seamlessly integrate differential equations, stochastic processes, optimisation theory, machine learning, and control theory. The core challenge lies in developing mathematical methods that can handle the inherent complexity of coupling continuous physical dynamics with discrete digital computations while managing uncertainty, ensuring real-time performance, and providing theoretical guarantees for system behaviour.

This Special Issue seeks to advance the mathematical understanding of how physical and digital realms can be rigorously coupled, synchronised, and co-optimised. We invite contributions that address fundamental mathematical problems including the well-posedness of hybrid physical–digital systems, convergence analysis of data assimilation schemes, mathematical frameworks for multi-fidelity modelling, and the development of structure-preserving numerical methods that maintain physical invariants across the physical–digital interface.

Scope and Topics

Core Mathematical Themes:

• Hybrid Dynamical Systems Theory: Mathematical formulations for coupled continuous-discrete systems, switched systems, and hybrid automata in the context of digital twins
• Data Assimilation and State Estimation: Variational methods (4D-Var), ensemble Kalman filtering, particle filters, and novel Bayesian approaches for maintaining synchronisation between physical assets and digital representations
• Multi-Fidelity and Multi-Scale Modelling: Mathematical frameworks for combining models of varying accuracy and computational cost, including surrogate modelling, reduced-order modelling (ROM), and hierarchical multiscale methods
• Physics-Informed Machine Learning: Integration of physical laws into neural network architectures (PINNs), neural ODEs/PDEs, and Hamiltonian neural networks for digital twin applications
• Uncertainty Quantification and Propagation: Polynomial chaos expansions, stochastic Galerkin methods, and probabilistic frameworks for handling aleatory and epistemic uncertainties
• Optimisation and Optimal Control: Real-time optimisation algorithms, model predictive control (MPC), reinforcement learning, and inverse problems for digital twin-based decision making
• Graph Theory and Network Science: Mathematical models for interconnected digital twins, including synchronisation theory, consensus algorithms, and distributed estimation
• Topological and Geometric Methods: Persistent homology, manifold learning, and differential geometry for understanding the shape and structure of data spaces in digital twins
• Formal Methods and Verification: Mathematical logic, temporal logics, and model checking for ensuring correctness and safety properties of digital twin systems
• Information-Theoretic Approaches: Entropy methods, mutual information, and rate-distortion theory for optimal sensor placement and data compression in digital twins

Scientific Impact: This Special Issue will establish rigorous mathematical foundations for digital twin technology, moving beyond ad hoc implementations toward theoretically grounded methodologies. Expected contributions include the following:

• New existence and uniqueness theorems for hybrid physical–digital coupled systems;
• Convergence proofs for novel data assimilation algorithms in high-dimensional spaces;
• Mathematical frameworks for quantifying and controlling the fidelity gap between physical systems and their digital representations;
• Theoretical bounds on prediction accuracy and uncertainty in digital twin forecasts.

Technological Impact

The mathematical advances will directly enable the following:

• More accurate and reliable digital twins for critical infrastructure and safety-critical systems;
• Reduced computational costs through mathematically optimal model reduction techniques;
• Enhanced predictive capabilities for preventive maintenance and anomaly detection;
• Scalable algorithms for managing networks of thousands of interconnected digital twins.

Societal and Economic Impact:

• Industry 4.0/5.0: Mathematical foundations for smart manufacturing, reducing downtime by 30–50% through predictive maintenance;
• Healthcare: Patient-specific digital twins for personalised medicine and surgical planning;
• Climate and environment: High-fidelity Earth system digital twins for climate prediction and disaster response;
• Smart cities: Urban digital twins for traffic optimisation, energy management, and infrastructure planning;
• Sustainability: Optimisation of resource consumption and carbon footprint through digital twin-based lifecycle management.

Cross-Disciplinary Impact

This Special Issue will foster collaboration between the following:

• Pure and applied mathematicians;
• Computer scientists and data scientists;
• Engineers across multiple domains;
• Physicists and computational scientists;
• Industry practitioners and technology developers.

Target Audience

• Researchers in applied mathematics, computational mathematics, and mathematical physics;
• Engineers working on cyber–physical systems, IoT, and Industry 4.0;
• Data scientists and machine learning researchers interested in physics-informed methods;
• Industrial practitioners developing digital twin solutions;
• Graduate students and postdocs entering this emerging field.

Prof. Dr. Bogdan Mocan
Dr. Sergiu Dan Stan
Prof. Dr. Milos Manic
Dr. Milos Simonovic
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

• digital twins
• hybrid systems
• data assimilation
• physics-informed machine learning
• multi-fidelity modelling
• uncertainty quantification
• model reduction
• real-time optimisation
• cyber–physical systems
• inverse problems
• state estimation
• surrogate modelling
• structure-preserving algorithms
• multi-scale mathematics
• system identification