ELADO: Elliptic PDE Assessment Datasets for Operator Learning

2026-06-18

AI Summary

This work addresses a critical limitation in the evaluation of neural operators for elliptic partial differential equations (PDEs), where performance deficiencies are often obscured by standard datasets and average error metrics. The authors propose ELADO—a systematic benchmark suite centered on variable-coefficient Poisson and Helmholtz equations—that precisely characterizes five key challenges: heavy-tailed solution distributions, input spectral shifts, heavy-tailed behavior in the solution frequency domain, operator sensitivity to input perturbations, and input complexity. These challenges are engineered through controlled random coefficient fields, spectral analysis, and local Lipschitz estimates. Experiments demonstrate that these factors substantially degrade model accuracy, yet remain undetected under conventional evaluation protocols. By moving beyond aggregate performance measures, this study establishes a new paradigm for systematically identifying and isolating the fundamental sources of difficulty in learning elliptic PDE operators.

Abstract

We introduce ELADO (Elliptic PDE Assessment Datasets for Operator Learning), a systematic benchmark suite constructed to show and quantify failure modes of neural operator architectures when learning solution operators of elliptic PDEs. While the benchmarks of existing datasets focus on average case performance, the ELADO datasets are constructed to highlight challenges that arise naturally in elliptic PDE problems. In particular, we construct several datasets built around Poisson's equation and the Helmholtz equation, each with non-constant coefficients. We define a controllable data-generating process to create datasets, that are designed to isolate a distinct source of difficulty. Specifically, these are (1) heavy-tailed solution distributions arising from light-tailed coefficient field distributions, (2) spectral distribution shift of the input data, (3) heavy-tailed distributions in the frequency domain of solutions, arising from light-tailed coefficient field distributions, (4) input sensitivity of learned operators, quantified by an empirical local Lipschitz analysis, and (5) the effect of input signal complexity on prediction accuracy under controlled amplitude normalization. We evaluate several neural operator architectures across all datasets and show that heavy-tailed targets, spectral shift, and input sensitivity each cause substantial degradation of the prediction accuracy that standard datasets and metrics (e.g., the mean relative $L^2$ error) may obscure.

Problem

• elliptic PDEs
• neural operators
• failure modes
• heavy-tailed distributions
• spectral shift

Innovation

• neural operators
• elliptic PDEs
• benchmark dataset
• heavy-tailed distributions
• spectral distribution shift