StablePDENet: Enhancing Stability of Operator Learning for Solving Differential Equations

TL;DR

StablePDENet introduces a stability-enhanced neural operator framework for solving differential equations, employing adversarial training to ensure robustness against input perturbations while maintaining accuracy.

Contribution

It proposes a novel adversarial training approach formulated as a min-max optimization to improve the stability of neural operators in differential equation solving.

Findings

Achieves high fidelity under adversarial input perturbations

Maintains accuracy on standard inputs

Provides a foundation for reliable neural PDE solvers

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.