发表机构
IMT Atlantique（IMT大西洋）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究经典代理模型预测粘性Burgers方程时在极值附近的系统误差，发现其源于粘性平滑不足，并提出了仅用预测量的修正方法。
AI 中文摘要
我们研究了经典机器学习代理模型的局部误差，这些模型用于近似一维粘性Burgers方程的时间演化。在相同的预测任务上，直接使用空间网格值比较了四种模型：径向基函数（RBF）核岭回归（KRR）、线性Ridge、ExtraTrees和随机森林。在所有四种模型中，单步残差（此处定义为每个网格点上真实值减去预测值）在预测的最大值和最小值附近形成清晰的弯曲分支。对KRR的更详细分析表明，这些误差与二阶空间导数（衡量局部曲率）的关系远强于与一阶空间导数的关系。在光滑极值附近，预测值和曲率形成局部两分支折叠。在我们基于局部曲率的残差模型下，这种折叠预测了预测值与残差之间的前导阶近抛物线关系。这一几何结果促使我们直接检验Burgers方程的对流（输运）项和扩散（平滑）项。对于KRR和Ridge，在留出轨迹上的回归测试、打破扩散项空间对齐的对照实验以及高频内容的谱测试均一致表明，在中高粘度下存在粘性平滑不足。在这种情况下，代理模型保留了比真实未来状态更多的小尺度结构。对于树模型，相同的物理解释要弱得多。最后，仅使用预测量的修正同时减少了单步误差和递归展开（每次预测用作下一次输入）期间的误差。
英文摘要
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.
Comments21 pages, 13 figures, 13 tables
