Structured Extrema Errors in Classical Surrogates for Viscous Burgers: A Physics-Consistent Interpretation

• URL: http://arxiv.org/abs/2609.07952v1
• Date: Mon, 07 Sep 2026 20:17:58 GMT
• Title: Structured Extrema Errors in Classical Surrogates for Viscous Burgers: A Physics-Consistent Interpretation
• Abstract summary: We study the local errors of machine-learning surrogate models, which approximate the time evolution of the viscous Burgers equation.<n>Across all four models, the one-step residual, defined here as the true value minus the predicted value at each grid point, forms clear curved branches near predicted maxima and minima.<n>A more detailed analysis of KRR shows that these errors are much more strongly related to the second spatial derivative, which measures local curvature, than to the first spatial derivative.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the local errors of classical machine-learning surrogate models, which approximate the time evolution of the one-dimensional viscous Burgers equation. Four models are compared on the same prediction task, using the spatial grid values directly: radial basis function (RBF) kernel ridge regression (KRR), linear Ridge, ExtraTrees, and Random Forests. Across all four models, the one-step residual, defined here as the true value minus the predicted value at each grid point, forms clear curved branches near predicted maxima and minima. A more detailed analysis of KRR shows that these errors are much more strongly related to the second spatial derivative, which measures local curvature, than to the first spatial derivative. Near a smooth extremum, predicted value and curvature form a local two-branch fold. Under our local curvature-based model of the residual, this fold predicts a leading-order near-parabolic relation between predicted value and residual. This geometric result motivates a direct test of the Burgers advection (transport) and diffusion (smoothing) terms. For KRR and Ridge, regression tests on held-out trajectories, a control that breaks the spatial alignment of the diffusion term, and a spectral test of high-frequency content are consistent with insufficient viscous smoothing at moderate and high viscosity. In this case, the surrogate retains more small-scale structure than the true future state. The same physical explanation is much weaker for the tree models. Finally, a correction that uses only predicted quantities reduces both one-step error and error during recursive rollout, where each prediction is used as the next input.

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