🤖 AI Summary
This work addresses the challenges faced by physics-informed neural networks (PINNs) in solving differential equations—particularly spectral bias, stiffness, and insufficient accuracy for multiscale solutions. From the perspective of the neural tangent kernel (NTK), the study establishes the first theoretical framework to systematically analyze how the discriminator in adversarial training influences PINN dynamics, clarifies the conditions under which such approaches are effective, and provides a unified interpretation of various GAN variants within the PINN context. Building on this theory, the authors propose an efficient training algorithm that substantially mitigates ill-conditioning during optimization, achieving accuracy improvements of several orders of magnitude over existing methods across multiple benchmark problems.
📝 Abstract
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.