On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networks

作     者：Rohrhofer, Franz M. Posch, Stefan Gößnitzer, Clemens Geiger, Bernhard C. 

作者机构：Know-Center GmbH Research Center for Data-Driven Business & Big Data Analytics Sandgasse 36/4 Graz8010 Austria LEC GmbH Large Engines Competence Center Inffeldgasse 19 Graz8010 Austria 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2022年

核心收录：

主　　题：Dynamical systems 

摘      要：This paper empirically studies commonly observed training difficulties of Physics-Informed Neural Networks (PINNs) on dynamical systems. Our results indicate that fixed points which are inherent to these systems play a key role in the optimization of the in PINNs embedded physics loss function. We observe that the loss landscape exhibits local optima that are shaped by the presence of fixed points. We find that these local optima contribute to the complexity of the physics loss optimization which can explain common training difficulties and resulting nonphysical predictions. Under certain settings, e.g., initial conditions close to fixed points or long simulations times, we show that those optima can even become better than that of the desired solution. Copyright © 2022, The Authors. All rights reserved.