On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks
On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks
复制标题
DOI：
10.1016/j.cma.2021.113938
复制
发表时间：
2021-06-02
影响因子：
7.2
通讯作者：
Perdikaris, Paris
中科院分区：
文献类型：
--
作者：
Wang, Sifan;Wang, Hanwen;Perdikaris, Paris
物理信息神经网络（PINN）在将物理模型与有间隙和噪声的观测数据相结合方面表现出了显着的前景，但在要近似的目标函数表现出高频或多尺度特征的情况下，它们仍然很难实现。在这项工作中，我们通过神经正切核（NTK）理论的透镜研究这种限制，并阐明PINN如何偏向于学习函数沿着其限制NTK的主导特征方向。利用这一观察结果，我们构建了采用时空和多尺度随机傅立叶特征的新架构，并证明了这种坐标嵌入层如何能够产生鲁棒和准确的PINN模型。数值例子提出了几个具有挑战性的情况下，传统的PINN模型失败，包括波传播和反应扩散动力学，说明所提出的方法可以用来有效地解决正问题和反问题，涉及偏微分方程与多尺度行为。本手稿附带的所有代码和数据将在https://github.com/PredictiveIntelligenceLa/MultiscalePINNs上公开。(C)2021爱思唯尔有限公司版权所有。
Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. In this work we investigate this limitation through the lens of Neural Tangent Kernel (NTK) theory and elucidate how PINNs are biased towards learning functions along the dominant eigen-directions of their limiting NTK. Using this observation, we construct novel architectures that employ spatio-temporal and multi-scale random Fourier features, and justify how such coordinate embedding layers can lead to robust and accurate PINN models. Numerical examples are presented for several challenging cases where conventional PINN models fail, including wave propagation and reaction-diffusion dynamics, illustrating how the proposed methods can be used to effectively tackle both forward and inverse problems involving partial differential equations with multi-scale behavior. All code an data accompanying this manuscript will be made publicly available at https://github.com/PredictiveIntelligenceLa/MultiscalePINNs. (C) 2021 Elsevier B.V. All rights reserved.