Structure-Preserving Physics-Informed Neural Networks With Energy or
  Lyapunov Structure
Haoyu Chu Yuto Miyatake Wenjun Cui Shikui Wei Daisuke Furihata
Recently, there has been growing interest in using physics-informed neural
networks (PINNs) to solve differential equations. However, the preservation of
structure, such as energy and stability, in a suitable manner has yet to be
established. This limitation could be a potential reason why the learning
process for PINNs is not always efficient and the numerical results may suggest
nonphysical behavior. Besides, there is little research on their applications
on downstream tasks. To address these issues, we propose structure-preserving
PINNs to improve their performance and broaden their applications for
downstream tasks. Firstly, by leveraging prior knowledge about the physical
system, a structure-preserving loss function is designed to assist the PINN in
learning the underlying structure. Secondly, a framework that utilizes
structure-preserving PINN for robust image recognition is proposed. Here,
preserving the Lyapunov structure of the underlying system ensures the
stability of the system. Experimental results demonstrate that the proposed
method improves the numerical accuracy of PINNs for partial differential
equations. Furthermore, the robustness of the model against adversarial
perturbations in image data is enhanced.