从神经正切核视角揭示对抗训练如何提升物理信息神经网络
When and Why Adversarial Training Improves PINNs: A Neural Tangent Kernel Perspective
- 用神经正切核分析对抗训练中判别器对PINN训练动态的影响
- 实测新方法使模型精度提升数个数量级，显著改善训练病态性
- 适合关注微分方程求解与神经网络训练机制的研究者
物理信息神经网络（PINNs）虽是微分方程的强大代理模型，但因频谱偏差、刚性及高频或多尺度解的低精度而难训练。基于生成对抗网络（GANs）的对抗训练近期表现出惊人效果，但其内在机制尚不明确。本文提出一种新分析框架，关键观察在于判别器如何影响PINN的训练动态。该框架首次为对抗训练在PINNs中的有效性提供理论依据，并统一分析多种GAN变体，最终导出一种新的、高效实用的PINN训练算法。实验表明，该方法能显著缓解训练病态性，获得性能更优的模型，精度常比现有方法高出数个数量级。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) are powerful surrogates for differential equations but are notoriously difficult to train due to spectral bias, stiffness, and poor accuracy on high-frequency or multiscale solutions. Adversarial training based on generative adversarial networks (GANs) has recently gained surprisingly strong empirical results in improving training, but the underlying mechanisms remain elusive. To this end, we propose a new analysis framework for adversarially trained PINNs, based on the key observation of how the discriminator in GANs can influence the training dynamics of PINNs. The framework first provides a much needed theoretical grounding to why and when adversarial training is effective in PINNs, then presents a unified analysis of GANs variants in such training, and finally leads to a new, practical, efficient training algorithm for PINNs. Empirical results demonstrate that our method can significantly reduce the pathology of PINNs training, thereby providing better models with superior performances, often several magnitudes more accurate than alternative methods.
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