自适应分解域的神经网络，让求解复杂方程更高效
AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decomposition
- 根据残差大小动态调整子域划分，重点强化难解区域
- 在多尺度问题上表现优异，解决传统方法收敛难问题
- 适合处理复杂偏微分方程，减少调参需求
我们提出自适应基物理信息神经网络（AB-PINNs），一种新型域分解方法，用于训练物理信息神经网络。该方法使已有子域在训练过程中动态适应未知解的内在特征。受经典网格细化技术启发，我们在残差损失高的区域实时引入新子域，从而在解难以表示的区域提升表达能力。该灵活的域分解策略特别适用于多尺度问题，不同子域可分别学习捕捉不同尺度的解。此外，训练中动态添加子域有助于避免陷入不良局部极小值，相比静态域分解方法，显著降低对超参数调优的需求。通过大量数值实验，验证了AB-PINNs在求解多种复杂多尺度偏微分方程上的有效性。
原文摘要 · Abstract (English)
We introduce adaptive-basis physics-informed neural networks (AB-PINNs), a novel approach to domain decomposition for training PINNs in which existing subdomains dynamically adapt to the intrinsic features of the unknown solution. Drawing inspiration from classical mesh refinement techniques, we also modify the domain decomposition on-the-fly throughout training by introducing new subdomains in regions of high residual loss, thereby providing additional expressive power where the solution of the differential equation is challenging to represent. Our flexible approach to domain decomposition is well-suited for multiscale problems, as different subdomains can learn to capture different scales of the underlying solution. Moreover, the ability to introduce new subdomains during training helps prevent convergence to unwanted local minima and can reduce the need for extensive hyperparameter tuning compared to static domain decomposition approaches. Throughout, we present comprehensive numerical results which demonstrate the effectiveness of AB-PINNs at solving a variety of complex multiscale partial differential equations.
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