StablePDENet: Enhancing Stability of Operator Learning for Solving Differential Equations

2026-01-10

arXiv.org

AI Summary

This work addresses the sensitivity of neural operators to input perturbations, which undermines their reliability in real-world scenarios involving noise or uncertainty. To enhance robustness, the study introduces adversarial training into neural operator learning for the first time, proposing a self-supervised min–max optimization framework that formulates operator robustness as a defense against worst-case perturbations. The method significantly improves model stability and fidelity under both standard and adversarially perturbed inputs, without compromising accuracy. By explicitly accounting for input uncertainties through a principled adversarial lens, this approach establishes a new paradigm for developing highly reliable, data-driven solvers for partial differential equations.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness
• Search and Optimization: Adversarial Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

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. This paper presents a robust self-supervised neural operator framework that enhances stability through adversarial training while preserving accuracy. We formulate operator learning as a min-max optimization problem, where the model is trained against worst-case input perturbations to achieve consistent performance under both normal and adversarial conditions. We demonstrate that our method not only achieves good performance on standard inputs, but also maintains high fidelity under adversarial perturbed inputs. The results highlight the importance of stability-aware training in operator learning and provide a foundation for developing reliable neural PDE solvers in real-world applications, where input noise and uncertainties are inevitable.

Problem

• operator learning
• stability
• differential equations
• input perturbations
• neural operators

Innovation

• neural operator
• adversarial training
• stability
• min-max optimization
• PDE solver