Towards Universal Solvers: Using PGD Attack in Active Learning to Increase Generalizability of Neural Operators as Knowledge Distillation from Numerical PDE Solvers

• URL: http://arxiv.org/abs/2510.18989v1
• Date: Tue, 21 Oct 2025 18:13:05 GMT
• Title: Towards Universal Solvers: Using PGD Attack in Active Learning to Increase Generalizability of Neural Operators as Knowledge Distillation from Numerical PDE Solvers
• Authors: Yifei Sun,
• Abstract summary: PDE solvers require fine space-time discretizations and local linearizations, leading to high memory cost and slow runtimes.<n>We propose an adversarial teacher-student distillation framework in which a differentiable numerical solver supervises a compact neural operator.<n>Experiments on Burgers and Navier-Stokes systems demonstrate that adversarial distillation substantially improves OOD while preserving the low parameter cost and fast inference of neural operators.
• Score: 3.780792537808271
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Nonlinear PDE solvers require fine space-time discretizations and local linearizations, leading to high memory cost and slow runtimes. Neural operators such as FNOs and DeepONets offer fast single-shot inference by learning function-to-function mappings and truncating high-frequency components, but they suffer from poor out-of-distribution (OOD) generalization, often failing on inputs outside the training distribution. We propose an adversarial teacher-student distillation framework in which a differentiable numerical solver supervises a compact neural operator while a PGD-style active sampling loop searches for worst-case inputs under smoothness and energy constraints to expand the training set. Using differentiable spectral solvers enables gradient-based adversarial search and stabilizes sample mining. Experiments on Burgers and Navier-Stokes systems demonstrate that adversarial distillation substantially improves OOD robustness while preserving the low parameter cost and fast inference of neural operators.

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