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

TL;DR

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.

Abstract

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.

Figures (30)

• Figure 1: The role of AI4PDEs in AI4Science, along with an introduction to AI4PDEs in computational mechanics, including solid, fluid, and biomechanics..
• Figure 2: Main methods of AI4PDEs: Physics-informed neural networks PINN_original_paperloss_is_minimum_potential_energydeep_ritz, operator learning DeepOnetli2020fourierkovachki2023neural, and physics-informed neural operators li2021physicschakraborty2021transfer.
• Figure 3: AI for PDEs method: Schematic of PINNs strong form PINN_original_paper.
• Figure 4: AI for PDEs method: Schematic of (a) DEM based on the principle of minimum potential energy loss_is_minimum_potential_energy, (b) DCEM based on the principle of minimum complementary energy wang2023dcm.
• Figure 5: AI for PDEs Method: Schematic of Operator Learning. (a) Neural Operator Layer Structure: Graph Neural Operator (GNO) li2020neural, Local Neural Operator (LNO), and Fourier Neural Operator (FNO) li2020fourier can serve as the core of the neural operator architecture kovachki2023neural. (b) Details of DeepONet DeepOnet. (c) Details of FNO li2020fourier.