PINNs失败源于过拟合，正则化可解决，新方法用更少采样点达顶尖性能。
PINNs Failure Modes are Overfitting
- 通过可视化残差发现失败是因过拟合，损失仅在采样点上最小化
- 加入正则化后失败模式消失，验证了过拟合本质
- 用全残差双反向传播，采样点减少23倍仍保持最优表现
物理信息神经网络（PINNs）是一类基于机器学习的偏微分方程求解器，通过最小化编码方程的残差损失来训练网络以表示解。尽管取得成功，它们在某些简单方程上仍会失败，表现为收敛到错误解但损失值很低。过去几年中，这类失败模式引发广泛关注，催生了多种架构与优化改进方案。本文通过直接可视化残差，证明这些失败是过拟合所致：损失仅在采样点处被最小化，而不在全局范围内。引入正则化后，失败模式完全消失。此外，我们扩展了对全部残差的双反向传播方法，在四个标准失败方程上实现了当前最优性能，使用最多减少23倍的采样点，且仅依赖基础架构。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) are a common class of machine learning-based partial differential equation (PDE) solvers which train a network to represent a solution by minimizing a residual loss that encodes the PDE. Despite their successes, they are known to fail on certain simple equations, converging to an incorrect solution despite low loss. These failure modes have garnered significant attention in the literature over the past several years, motivating both architectural and optimization based solutions. By directly visualizing the residual, we show that failure modes are the result of overfitting: the loss is minimized on the collocation points, but not elsewhere. Applying regularization causes the failure modes to vanish. Finally, we extend double backpropagation over the full set of residuals, and use it to achieve state-of-the-art performance on four standard failure mode equations with up to $23\times$ fewer collocation points and a vanilla architecture.
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