StablePDENet: Enhancing Stability of Operator Learning for Solving Differential Equations

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.