In the recent years, Physics Informed Neural Networks (PINNs) have received strong interest as a method to solve PDE driven systems, in particular for data assimilation purpose. This method is still in its infancy, with many shortcomings and failures that remain not properly understood. In this paper we propose a natural gradient approach to PINNs which contributes to speed-up and improve the accuracy of the training. Based on an in depth analysis of the differential geometric structures of the problem, we come up with two distinct contributions: (i) a new natural gradient algorithm that scales as $\min(P^2S, S^2P)$, where $P$ is the number of parameters, and $S$ the batch size; (ii) a mathematically principled reformulation of the PINNs problem that allows the extension of natural gradient to it, with proved connections to Green's function theory.

翻译：近年来，物理信息神经网络（PINNs）作为一种求解偏微分方程驱动系统的方法，特别是在数据同化方面，受到了广泛关注。该方法仍处于起步阶段，存在许多尚未被充分理解的缺陷和失败案例。本文提出了一种针对PINNs的自然梯度方法，有助于加速训练并提高其准确性。基于对该问题微分几何结构的深入分析，我们提出了两个独立的贡献：（i）一种新的自然梯度算法，其计算复杂度为$\min(P^2S, S^2P)$，其中$P$为参数数量，$S$为批次大小；（ii）对PINNs问题进行了数学原理上的重构，使得自然梯度方法得以扩展至该问题，并证明了其与格林函数理论的联系。