Spectral Audit of In-Context Operator Networks

2026-06-01

AI Summary

Current evaluations of neural operators rely solely on prediction errors, which fail to reveal their fidelity to the local dynamical structure of partial differential equations. This work proposes a Jacobian-based spectral auditing method: by leveraging automatic differentiation, it computes the Jacobian of the output of context-aware operator networks with respect to query functions under fixed prompts, then projects this Jacobian into Fourier modal space to analyze frequency gain, phase structure, and modal coupling characteristics. For the first time, this approach assesses the mechanistic fidelity of neural operators from the perspective of local spectral properties, successfully uncovering defects—such as high-frequency degradation, erroneous phase recovery, and prompt-operator inconsistency—that are invisible to conventional prediction-error metrics. The findings demonstrate that predictive accuracy and local operator fidelity are largely independent.

Abstract

Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction--diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt--operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency.

Problem

• neural operators
• in-context learning
• spectral analysis
• Jacobian
• PDE operators

Innovation

• spectral audit
• in-context operator learning
• Jacobian analysis
• neural operators
• Fourier mode projection