[论文解读] Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next

Salvatore Cuomo, Vincenzo Schiano Di Cola+4 more|arXiv (Cornell University)|Jan 14, 2022

Model Reduction and Neural Networks被引用 83

一句话总结

对物理信息神经网络（PINNs）的综合评述，其构建模块、变体、应用、优势、局限性和未来方向。

ABSTRACT

Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.

研究动机与目标

• 定义PINNs及其用于求解PDE及相关问题的动机。
• 总结PINNs中使用的构建模块和神经网络架构。
• 调查变体（PCNN、hp-VPINN、CPINN）及它们在将物理学与学习整合中的作用。
• 讨论学习理论、准确性、收敛性以及可用的工具链。
• 突出PINNs中的尚待解决的理论问题和未来研究方向。

提出的方法

• 将PDE残差以及边界/初始条件嵌入到神经网络损失中的PINN框架。
• 将一般PDE问题与PINN目标正式化为物理残差、边界条件和数据损失的加权和。
• 回顾神经网络架构（FFNN/MLP、CNN、RNN）及其对PINNs的适用性。
• 讨论影响训练与精度的激活函数与结构选择。
• 调研如hp-VPINN、CPINN、PCNN，以及DeepONet风格的多网络设置等变体。
• 总结相关方法的景观（如DRM、Deep Galerkin、hp-VPINN）及其与PINNs的关系。

实验结果

研究问题

• RQ1 什么是物理信息神经网络，它们旨在解决哪些问题？
• RQ2 PINNs的主要构建模块和结构选择有哪些？
• RQ3 除了普通的PINNs之外存在哪些变体，它们如何整合物理约束？
• RQ4 PINNs在理论和实践方面面临的挑战（准确性、收敛、训练）有哪些，提出了哪些未来方向？
• RQ5 在能力与局限性方面，PINNs与传统数值方法及其他机器学习方法相比如何？

主要发现

• PINNs是网格自由的神经网络，将PDE残差及边界/初始条件纳入训练损失。
• PINNs可以在同一优化框架内解决前向问题和逆问题。
• 各种神经网络架构（FFNNs、CNNs、RNNs）和激活函数会影响PINN的性能。
• 已提出若干变体（PCNN、hp-VPINN、CPINN）和多网络设计，以提升物理集成性和鲁棒性。
• PINNs在处理复杂几何和高维问题方面具有优势，但理论理解和收敛性保证仍未解决。
• 自2019年以来，PINNs文献迅速增长，广泛适用于微分方程、积分微分方程和随机偏微分方程。