Advances in Scientific Machine Learning and Mathematical Modeling

Dear Colleagues,

Scientific Machine Learning (SciML) has matured significantly, driven by rigorous mathematical foundations, scalable architectures, and immense computational power. Modern approaches—spanning physics-informed neural networks (PINNs), operator learning models (e.g., Fourier Neural Operators), foundation models tailored to PDEs, diffusion-based generative systems, and hybrid physics-ML frameworks—enable high-fidelity representation of complex dynamical systems, multi-physics phenomena, and data assimilation far beyond traditional numerical methods.

These developments are transforming scientific discovery and engineering practice across disciplines. Deep learning now serves as efficient surrogates for costly PDE simulations, accelerates hypothesis generation, enables inverse problems and uncertainty quantification, and unifies heterogeneous data sources in fields like physics, biology, chemistry, materials science, and climate modeling.

Accordingly, the Special Issue of Mathematics entitled “Advances in Scientific Machine Learning and Mathematical Modeling” invites contributions bridging mathematical theory, innovative neural architectures, and impactful AI applications in the sciences. We especially welcome rigorous theoretical work, novel algorithmic contributions, empirically validated methods on challenging real-world problems, and studies that tightly connect theory to practice or provide mathematically grounded insights into SciML systems.

Topics of interest include, but are not limited to, the following:

• Mathematical foundations of neural networks and SciML, including approximation theory, operator learning guarantees, optimization landscapes, generalization bounds, and information-theoretic perspectives;
• Theoretical and empirical analysis of training dynamics, expressivity, robustness, spectral properties, and convergence in physics-informed and data-driven models;
• Advances in neural architectures for scientific computing (e.g., transformers for sequences in time/space, graph neural networks, physics-informed networks, Fourier/DeepONet-style operator learners, Mamba-based state-space models, and emerging foundation models for PDEs across domains);
• AI for Science in biology and medicine (e.g., protein folding and interaction prediction, genomics, systems biology modeling, medical image analysis, and biomechanical simulations);
• AI for Science in physics and engineering (e.g., PDE surrogates and solvers, turbulence closure modeling, plasma physics, fluid dynamics, astrophysics, dynamical systems identification, and multi-scale/multi-physics problems);
• AI for Science in chemistry and materials science (e.g., molecular generation and inverse design, reaction pathway prediction, property forecasting, high-throughput screening, and quantum chemistry approximations);
• Data-driven and hybrid modeling approaches (e.g., physics-informed neural networks, neural differential equations, data assimilation, equation discovery, and mechanistic-ML integration);
• Uncertainty quantification, probabilistic modeling, reliability, safety, and calibration in neural-network-based scientific surrogates;
• Explainable, interpretable, and physics-constrained AI for model validation, scientific insight, and trust in discovery pipelines;
• Domain adaptation, transfer learning, multi-fidelity modeling, few-shot learning, and cross-domain generalization in scientific and industrial contexts;
• Reinforcement learning, optimal control, decision-making, and active learning for physical, biological, or engineered systems governed by PDEs;
• Novel numerical algorithms, scalable training strategies, efficient inference, adaptive sampling, and high-performance computing techniques for large-scale SciML in PDE contexts.

Dr. Qingguang Guan
Guest Editor

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 250 words) can be sent to the Editorial Office for assessment.

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-anonymized 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.