发表机构
Harbin Institute of Technology; Jilin University（哈尔滨工业大学; 吉林大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种基于谱滤波器参数化的神经算子框架，通过精确分解能量误差为可观测分量，实现分布外泛化的可预测与可审计诊断。
AI 中文摘要
神经算子已成为求解偏微分方程（PDEs）的强大替代模型，然而它们在分布偏移下的可靠性仍是部署的关键障碍。现有针对算子学习中的分布外（OOD）泛化方法大多是经验性的且为黑箱：它们报告聚合误差指标，却不解释误差为何产生或何时增长。我们提出了一种保持结构的框架，使OOD泛化变得可预测且可审计。我们的关键思想是将学习到的解算子参数化为作用于底层椭圆算子特征值上的谱滤波器$h_\theta(\lambda)$，通过切比雪夫多项式展开实现，并使用弱形式目标进行训练。该参数化允许将能量范数误差精确分解为两个可观测分量：一个依赖于模型的谱逼近项和一个由输入引起的依赖于分布的谱加权项。由此分解我们推导出三个诊断指标：保守的带内上确界$\vareps_{\mathrm{sup}}$、全局RMS代理$\vareps_{\mathrm{rms}}$以及样本依赖的有效度量$\vareps_{\mathrm{eff}}(f)$。这些诊断指标无需访问真实解即可计算。通过四个受控实验，我们表明$\vareps_{\mathrm{eff}}(f)\\|f\\|$在分布内、带内谱偏移、带外尾部及复合偏移下始终能预测能量误差，而全局指标可能系统性误导。我们的框架将神经算子的OOD评估从黑箱基准测试转向算子结构诊断，为可审计的科学机器学习提供了实用途径。
英文摘要
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.
