StablePDENet: Enhancing Stability of Operator Learning for Solving Differential Equations
Chutian Huang, Chang Ma, Kaibo Wang, Yang Xiang
TL;DR
Neural operators for PDEs often lack stability guarantees under input perturbations. This paper introduces StablePDENet, a self-supervised framework that couples adversarial training with operator learning to enforce stability, formulating a min–max objective that bounds the sensitivity of the learned operator, effectively controlling the Fréchet derivative $D G_\theta(f)$. The method uses a physics-informed loss and PGD-based attacks to train against worst-case perturbations, achieving high accuracy on nominal inputs while remaining robust to perturbations across a broad family of PDEs, including Poisson, elliptic with Neumann boundary, heat, diffusion–reaction, and Stokes equations. Empirical results show that StablePDENet reduces the spectral norm of the Jacobian and maintains low relative $L^2$ errors under adversarial inputs, supporting the viability of reliable neural PDE solvers in noisy real-world settings.
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.