通过物理约束提升神经微分方程算子的稳定性

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.