# StablePDENet: Enhancing Neural Operator Stability through Physics-Informed Residual-Sensitivity Regularization

arXiv: https://arxiv.org/abs/2601.06472 · HTML: https://mlsanity.com/paper/2601.06472/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 26 references that are in the map.
- First posted on arXiv 10 Jan 2026.

## Builds on (most cited first, 11 of 11)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [An Analysis of Adversarial Attacks and Defenses on Autonomous Driving Models](https://mlsanity.com/paper/2002.02175/) (2020)
- [RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations](https://mlsanity.com/paper/2509.01234/) (2025)

## Abstract (arXiv)

Learning solution operators for differential equations with neural networks has shown great potential in scientific computing, but ensuring their stability under input perturbations remains a critical challenge. We introduce the StablePDENet, a physics-informed adversarial training method that regularizes the residual sensitivity with respect to an input perturbation. The operator learning task is formulated as a min--max optimization problem, where the inner model searches admissible input perturbations by physics-based projected-gradient adversary, while the outer problem combines the attacked physics loss with a normalized residual-sensitivity penalty. Moreover, residual-sensitivity regularization is included to ensure that the local Lipschitz constant of the learned operator is a more accurate approximation to that of the exact operator. We evaluate the StablePDENet on several benchmark problems. Compared with PI-DeepONet and its adversarially trained variant, StablePDENet achieves higher accuracy under adversarial input perturbations while maintaining competitive accuracy on clean inputs. The numerical results also demonstrate that the StablePDENet can effectively improve the generalization accuracy for operator learning. The Helmholtz study further distinguishes learned-model sensitivity from amplification intrinsic to an ill-conditioned solution operator. The results support residual-sensitivity regularization as a practical route to more stable and physically consistent neural PDE operators.