Structure-Preserving Physics-Informed Neural Networks With Energy or Lyapunov Structure

TL;DR

The paper tackles the issue that standard physics-informed neural networks (PINNs) often fail to preserve the intrinsic energy or stability structure of physical systems, which can lead to nonphysical behavior and limited usefulness for downstream tasks. It introduces structure-preserving PINNs (SP-PINNs) with two branches: a PDE-focused model that enforces energy dissipation via a discrete global energy $J(u)$ and a structure loss, and an image-recognition framework that enforces Lyapunov stability by projecting learned dynamics onto a stability-constrained space. Experiments on the Allen–Cahn equation show SP-PINN improves numerical accuracy and runs faster than traditional structure-preserving schemes, while image-recognition benchmarks (MNIST, SVHN, CIFAR10/100) reveal enhanced robustness against white-box adversarial attacks, further boosted when combined with adversarial training. Overall, the work demonstrates that embedding structural priors into PINNs extends their applicability to downstream tasks and improves robustness, offering a principled path to energy- and stability-aware neural computation.

Abstract

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.

Figures (6)

• Figure 1: SP-PINN for Allen–Cahn equation. The SP-PINN is defined as a $N$-layered FCN with inputs $x,t$ and output $\hat{u}(x,t)$.
• Figure 2: SP-PINN for robust image recognition. The SP-PINN is defined as a $N$-layer residual network. The orange line denotes the process of solving the inverse problem, in which a data-driven way is resorted to obtain the unknown parameter $\boldsymbol{\alpha}$ of the underlying system. For solving the forward problem, the output is the approximate solution $\boldsymbol{\hat{u}}$. The blue arrow represents a state that satisfies the Lyapunov exponential stability condition. The FC layer predicts the category of the image. When calculating the loss regarding the initial condition, the time $t$ is set to 0. When performing classification, the time $t$ is set to 1.
• Figure 3: Numerical solutions of the Allen–Cahn equation. The orange line is obtained by DVDM. The blue line is obtained by our method.
• Figure 4: (left) The error between the results obtained by the proposed model and DVDM. (right) The error between the results obtained by the baseline model and DVDM.
• Figure 5: The comparison of the time consuming on same experimental settings. The ODE solver used in Neural ODEs is Dopri5.