论文标题
在傅立叶特征网络的特征向量偏置上：从回归到解决具有物理信息的神经网络的多尺度PDE
On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks
论文作者
论文摘要
物理知识的神经网络（PINN）在将物理模型与Gappy和嘈杂的观察数据集成在一起时表现出了巨大的希望，但是在近似目标函数表现出高频或多尺度特征的情况下，它们仍然在努力。在这项工作中，我们通过神经切线核（NTK）理论的镜头研究了这一局限性，并阐明了Pinn的偏向于学习函数的偏向于其限制NTK的主要特征方向。使用此观察结果，我们构建了采用时空和多尺度随机傅立叶特征的新型体系结构，并证明这种坐标嵌入层如何导致稳健而准确的PINN模型是合理的。给出了几种挑战性的示例，这些示例是传统的PINN模型失败（包括波传播和反应扩散动力学）的几个挑战性示例，说明了如何使用所提出的方法有效解决涉及与多尺度行为的部分区分方程的前进和反向问题。所有代码伴随本手稿的数据将在\ url {https://github.com/predictivectiveIntelligencelab/multiscalepinns}上公开提供。
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 \url{https://github.com/PredictiveIntelligenceLab/MultiscalePINNs}.
