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

arXiv: https://arxiv.org/abs/2401.04986 · HTML: https://mlsanity.com/paper/2401.04986/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 53 references that are in the map.
- First posted on arXiv 10 Jan 2024.

## Builds on (most cited first, 24 of 24)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Adversarial examples in the physical world](https://mlsanity.com/paper/1607.02533/) (2016)
- [Adversarial Examples Are Not Bugs, They Are Features](https://mlsanity.com/paper/1905.02175/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Distillation as a Defense to Adversarial Perturbations against Deep Neural Networks](https://mlsanity.com/paper/1511.04508/) (2015)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [Adversarial Examples: Opportunities and Challenges](https://mlsanity.com/paper/1809.04790/) (2018)
- [Interpolated Adversarial Training: Achieving Robust Neural Networks without Sacrificing Too Much Accuracy](https://mlsanity.com/paper/1906.06784/) (2019)
- [Dissipative Hamiltonian Neural Networks: Learning Dissipative and Conservative Dynamics Separately](https://mlsanity.com/paper/2201.10085/) (2022)
- [Revealing hidden dynamics from time-series data by ODENet](https://mlsanity.com/paper/2005.04849/) (2020)
- [FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities](https://mlsanity.com/paper/2210.00272/) (2022)
- [Lyapunov-Stable Deep Equilibrium Models](https://mlsanity.com/paper/2304.12707/) (2023)

## Built on by (most cited first, 6 of 6)

- [Dual-Balancing for Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.11117/) (2025)
- [Non-Asymptotic Stability and Consistency Guarantees for Physics-Informed Neural Networks via Coercive Operator Analysis](https://mlsanity.com/paper/2506.13554/) (2025)
- [Learning Aligned Stability in Neural ODEs Reconciling Accuracy with Robustness](https://mlsanity.com/paper/2509.21879/) (2025)
- [Lyapunov Stable Graph Neural Flow](https://mlsanity.com/paper/2603.12557/) (2026)
- [A Comparative Investigation of Thermodynamic Structure-Informed Neural Networks](https://mlsanity.com/paper/2603.26803/) (2026)
- [PhysAttNet: Enhancing Predictive Performance in Industrial and Astrophysical Time Series via Physics-Informed Attention](https://mlsanity.com/paper/2608.07681/) (2026)

## Abstract (arXiv)

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.