用傅里叶特征嵌入硬约束神经网络的诺依曼边界条件
Hard-constraining Neumann boundary conditions in physics-informed neural networks via Fourier feature embeddings
- 用傅里叶特征嵌入将诺依曼条件直接融入网络结构
- 在多尺度和高频场景下优于现有硬约束方法和经典PINN
- 适合需要高精度边界控制的物理模拟任务
我们提出一种新方法，通过傅里叶特征嵌入在物理信息神经网络（PINNs）中硬约束诺依曼边界条件。诺依曼条件广泛用于描述各类关键物理过程，但在PINNs中比狄利克雷条件更难实现硬约束。本方法利用特定傅里叶特征嵌入，将诺依曼条件直接嵌入神经网络架构，而非让网络学习该条件。嵌入可自然扩展至高频模式，以更好捕捉高频现象。实验基于扩散问题验证了该方法的有效性，在多尺度和高频场景下，性能超越现有硬约束方法及经典PINN。
原文摘要 · Abstract (English)
We present a novel approach to hard-constrain Neumann boundary conditions in physics-informed neural networks (PINNs) using Fourier feature embeddings. Neumann boundary conditions are used to described critical processes in various application, yet they are more challenging to hard-constrain in PINNs than Dirichlet conditions. Our method employs specific Fourier feature embeddings to directly incorporate Neumann boundary conditions into the neural network's architecture instead of learning them. The embedding can be naturally extended by high frequency modes to better capture high frequency phenomena. We demonstrate the efficacy of our approach through experiments on a diffusion problem, for which our method outperforms existing hard-constraining methods and classical PINNs, particularly in multiscale and high frequency scenarios.
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