[论文解读] Machine Learning for Partial Differential Equations

Steven L. Brunton, J. Nathan Kutz|arXiv (Cornell University)|Mar 30, 2023

Model Reduction and Neural Networks被引用 12

一句话总结

这是对机器学习在PDE研究中的进展的全面评审，包括发现控制方程、学习有效坐标和降阶模型，以及学习解算算子以改进数值PDE求解器。

ABSTRACT

Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This review will examine several promising avenues of PDE research that are being advanced by machine learning, including: 1) the discovery of new governing PDEs and coarse-grained approximations for complex natural and engineered systems, 2) learning effective coordinate systems and reduced-order models to make PDEs more amenable to analysis, and 3) representing solution operators and improving traditional numerical algorithms. In each of these fields, we summarize key advances, ongoing challenges, and opportunities for further development.

研究动机与目标

• Motivate the use of ML to learn new PDEs and coarse-grained closures for complex systems.
• Explore learning effective coordinate systems and reduced-order representations for PDEs.
• Examine representation of solution operators and improvements to traditional numerical PDE methods.

提出的方法

• Sparse and symbolic regression for PDE discovery (e.g., PDE-FIND) to identify governing terms from data.
• Weak form and ensemble approaches to improve noise robustness in PDE discovery.
• Koopman operator-based methods and neural network embeddings to find linearizing coordinates.
• Reduced-order modeling with POD, autoencoders, and data-driven dynamics (SINDy, DMD, Galerkin regression).
• Neural operators and DeepOnet to learn inverse operators and mesh-free solution mappings.
• Strategies to accelerate numerical PDE solvers via learned derivatives, improved stencils, and data-driven coarsening.

实验结果

研究问题

• RQ1 Can PDEs be learned directly from data to discover new physics or coarse-grained closures?
• RQ2 What coordinates or representations enable nonlinear PDEs to be treated with linear or simplified dynamics?
• RQ3 How can we learn operators that map function spaces to function spaces, enabling mesh-free solution propagation?
• RQ4 What are effective data-driven reduced-order models that balance accuracy and computational cost?
• RQ5 How can physics-informed learning improve traditional numerical PDE workflows and uncertainty propagation?

主要发现

• PDE-FIND and its PDE-specific sparse regression enable rediscovery of classical PDEs and discovery of new models and closures.
• Weak form formulations improve noise robustness and data efficiency in PDE discovery, with ensembling further enhancing reliability.
• Koopman-based coordinates and deep learning embeddings enable linearized or simplified representations of nonlinear PDE dynamics.
• Reduced-order modeling combines nonlinear manifolds with data-driven dynamics to achieve tractable simulations.
• Neural operators and DeepOnet provide discretization-invariant, mesh-free mappings between function spaces for operator learning.
• Neural operators and related methods offer a framework for accelerating numerical PDE solutions and improving solver conditioning.