Predicting Out-of-Distribution Generalization of Neural Operators via Observable Spectral Error Decomposition

2026-09-20

AI Summary

本文提出了一种通过谱误差分解来预测神经算子在分布外泛化性能的方法，使用Chebyshev多项式展开和弱形式目标训练，并提供可计算的诊断工具以提高模型的可审计性。

Abstract

Neural operators have emerged as powerful surrogates for solving partial differential equations (PDEs), yet their reliability under distribution shift remains a critical barrier to deployment. Existing approaches to out-of-distribution (OOD) generalization in operator learning are largely empirical and black-box: they report aggregate error metrics without explaining why errors arise or when they will grow. We propose a structure-preserving framework that makes OOD generalization predictable and auditable. Our key idea is to parameterize the learned solution operator as a spectral filter $h_θ(λ)$ acting on the eigenvalues of the underlying elliptic operator, implemented via Chebyshev polynomial expansions and trained with a weak-form objective. This parameterization admits an exact decomposition of the energy-norm error into two observable components: a model-dependent spectral approximation term and a distribution-dependent spectral weighting term induced by the input. From this decomposition we derive three diagnostics: a conservative in-band supremum $\vareps_{\mathrm{sup}}$, a global RMS proxy $\vareps_{\mathrm{rms}}$, and a sample-dependent effective metric $\vareps_{\mathrm{eff}}(f)$. These diagnostics can be computed without access to ground-truth solutions. Through four controlled experiments, we show that $\vareps_{\mathrm{eff}}(f)\|f\|$ consistently predicts energy error under in-distribution, in-band spectral shift, out-of-band tail, and compound shifts, whereas global metrics can be systematically misleading. Our framework shifts OOD assessment of neural operators from black-box benchmarking to operator-structure diagnostics, providing a practical route to auditable scientific machine learning.

Problem

• Neural Operators
• Out-of-Distribution Generalization
• Spectral Error Decomposition
• Distribution Shift
• Operator Learning

Innovation

• Spectral Error Decomposition
• Observable Diagnostics
• Structure-Preserving Framework
• Neural Operators
• Out-of-Distribution Generalization