Shikun Chen  Website  E-Mail: chenshikun@nbufe.edu.cn
Ningbo university of Finance and Economics
Interests: PDE, machine learning, PINN, optimization

Special Issue Information

This special issue focuses on AI-driven methodologies for the formulation, approximation, and numerical solution of differential equations, encompassing ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), and their coupled or hybrid variants. We seek contributions that develop, analyze, or systematically evaluate learning-based solvers and surrogate models for differential equations, with emphasis on theoretical grounding, rigorous error analysis, and computational scalability. Topics of interest span the full pipeline from problem formulation and network architecture design to training methodology, convergence analysis, and deployment in real-world scientific computing tasks. We particularly welcome works that address the mathematical foundations underpinning neural solvers—including approximation theory, generalization bounds, stability, and long-time integration accuracy—as well as contributions that demonstrate substantial empirical advances on challenging multi-scale, high-dimensional, or data-scarce benchmarks.

Dear Colleagues,

Differential equations constitute the fundamental mathematical language for modeling dynamical processes across virtually every branch of science and engineering—from fluid mechanics, solid mechanics, and electromagnetism to systems biology, financial mathematics, and climate science. Classical numerical methods, including finite difference, finite element, and spectral methods, have achieved remarkable success over decades; however, they often encounter severe computational bottlenecks when confronting high-dimensional problems, complex geometries, multi-scale phenomena, or real-time simulation requirements.

The rapid advancement of artificial intelligence and deep learning has opened transformative new avenues for tackling these longstanding challenges. Physics-informed neural networks (PINNs), neural operators (e.g., DeepONet, Fourier Neural Operator), diffusion-model-based solvers, and hybrid neural–numerical schemes have demonstrated impressive capabilities in approximating solutions to PDEs, ODEs, and SDEs, often achieving dramatic speedups over conventional solvers while maintaining competitive accuracy. Meanwhile, emerging paradigms such as operator learning, latent dynamics modeling, and score-based generative approaches are reshaping the theoretical and algorithmic landscape of scientific computing.

This special issue aims to bring together researchers working at the intersection of artificial intelligence and differential equations to present recent advances, identify open problems, and foster cross-disciplinary dialogue. We invite original research articles, comprehensive reviews, and benchmark studies that address—but are not limited to—the following topics:

• Physics-informed neural networks and their variants for forward and inverse PDE/ODE/SDE problems, including convergence analysis, adaptive training strategies, and failure mode characterization;
• Neural operator frameworks for learning solution maps across parametric families of differential equations, with attention to universality, resolution invariance, and out-of-distribution generalization;
• Deep learning approaches for high-dimensional PDEs arising in stochastic control, mean-field games, and computational finance, where classical mesh-based methods suffer from the curse of dimensionality;
• AI-augmented numerical methods that integrate learned components into established discretization schemes (e.g., learned preconditioners, neural-network-enhanced multigrid, data-driven closure models for turbulence);
• Generative and diffusion-model-based solvers for stochastic and uncertain differential systems, including uncertainty quantification and probabilistic solution characterization;
• Theoretical foundations of neural PDE solvers, including approximation theoretic results, sample complexity, training dynamics, and error bound analysis;
• Multi-scale and multi-physics modeling using AI, addressing the challenge of bridging disparate spatial and temporal scales in coupled differential systems;
• Transfer learning, meta-learning, and foundation models for differential equations, aiming to build general-purpose solvers that adapt efficiently to new equations or domains;
• Benchmark datasets, reproducibility frameworks, and systematic comparative studies that advance rigorous evaluation standards for AI-based differential equation solvers.

We believe that this special issue will serve as a valuable resource for researchers in applied mathematics, computational science, and machine learning, fostering deeper understanding of the capabilities, limitations, and future directions of AI methods in the numerical solution of differential equations.

We look forward to receiving your contributions.

Dr. Shikun Chen

Keywords

• Physics-informed neural networks
• Neural operators
• Partial differential equations
• Scientific machine learning
• Deep learning for differential equations
• Stochastic differential equations
• Operator learning
• High-dimensional PDEs
• Neural PDE solvers
• Surrogate modeling