Artificial intelligence for partial differential equations in computational mechanics: A review

In recent years, Artificial intelligence (AI) has become ubiquitous, empowering various fields, especially integrating artificial intelligence and traditional science (AI for Science: Artificial intelligence for science), which has attracted widespread attention. In AI for Science, using artificial intelligence algorithms to solve partial differential equations (AI for PDEs: Artificial intelligence for partial differential equations) has become a focal point in computational mechanics. The core of AI for PDEs is the fusion of data and partial differential equations (PDEs), which can solve almost any PDEs. Therefore, this article provides a comprehensive review of the research on AI for PDEs, summarizing the existing algorithms and theories. The article discusses the applications of AI for PDEs in computational mechanics, including solid mechanics, fluid mechanics, and biomechanics. The existing AI for PDEs algorithms include those based on Physics-Informed Neural Networks (PINNs), Deep Energy Methods (DEM), Operator Learning, and Physics-Informed Neural Operator (PINO). AI for PDEs represents a new method of scientific simulation that provides approximate solutions to specific problems using large amounts of data, then fine-tuning according to specific physical laws, avoiding the need to compute from scratch like traditional algorithms. Thus, AI for PDEs is the prototype for future foundation models in computational mechanics, capable of significantly accelerating traditional numerical algorithms.

💡 Research Summary

This review paper surveys the rapidly emerging field of artificial intelligence for partial differential equations (AI for PDEs, also referred to as AI4PDEs) within the broader context of AI for Science (AI4Science). The authors begin by outlining how AI4Science has attracted widespread attention and how the solution of PDEs—central to computational mechanics—has become a focal research area. They argue that the core of AI4PDEs is the fusion of data with the governing equations, enabling the solution of virtually any PDE while reducing the reliance on large, high‑quality datasets.

The paper categorizes AI4PDE methods into three dominant paradigms: Physics‑Informed Neural Networks (PINNs), operator learning, and physics‑informed neural operators (PINO). Within PINNs, two formulations are distinguished: the strong‑form PINN, which directly penalizes PDE residuals in the loss function, and the energy‑form PINN (also called the Deep Energy Method, DEM), which minimizes the total potential energy of the system based on variational principles. The authors discuss the advantages of PINNs—mesh‑free representation, suitability for high‑dimensional problems via Monte‑Carlo sampling, and seamless handling of forward and inverse problems—as well as their drawbacks, such as non‑convex optimization, limited robustness, and often higher computational cost compared with classical numerical solvers.

Operator learning is presented next, with DeepONet and the Fourier Neural Operator (FNO) as representative examples. These approaches learn mappings from input functions to output functions, offering discretization‑invariance (i.e., the ability to evaluate on arbitrary grids without retraining) and strong performance when abundant data are available. However, purely data‑driven operator learning lacks physical constraints, which can lead to over‑fitting and poor extrapolation.

PINO combines the strengths of operator learning with physics‑based loss terms, thereby improving data efficiency and generalization while retaining the grid‑independent nature of neural operators. The review highlights that PINO can be viewed as a physics‑aware extension of DeepONet/FNO, bridging the gap between data‑rich and physics‑rich regimes.

Application domains are examined in depth. In solid mechanics, the authors discuss the use of PINNs and DEM for nonlinear elasticity, plasticity, fracture, and topology optimization. In fluid mechanics, they cover turbulence, shock‑wave propagation, multiphase flows, and moving‑boundary problems, emphasizing how operator learning can accelerate large‑scale simulations. In biomechanics, examples include soft‑tissue deformation, blood‑flow modeling, and material parameter identification in biological tissues.

The paper stresses that AI4PDEs are not intended to replace traditional numerical methods such as the finite element method (FEM) but rather to complement them. Hybrid strategies—surrogate modeling, accelerated multiscale simulations, and data‑assisted inverse analysis—are presented as practical pathways for integrating AI4PDEs into existing workflows.

Limitations are candidly addressed: data scarcity, lack of interpretability for purely data‑driven models, difficulties in assigning physical meaning to hyper‑parameters, and challenges in ensuring robustness and reliability across diverse problem settings. The authors argue that future progress will require tighter coupling of data‑driven learning with strong physical priors.

Finally, emerging research directions are outlined, including reinforcement‑learning‑based PDE control, generative surrogate modeling, and the use of large foundation models for scientific computing. While still nascent, these avenues suggest a trajectory toward AI‑augmented foundation models that could dramatically accelerate computational mechanics in the coming years.

Overall, the review provides a comprehensive taxonomy of AI‑based PDE solvers, critically evaluates their theoretical foundations and practical performance, and maps out both current applications and future opportunities in computational mechanics.