基于自编码器的降阶神经常微分方程的误差传播分析
Analysis of Error Propagation in Autoencoder-Based Reduced-Order Neural Ordinary Differential Equations
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中文总结 AI 辅助
本文针对基于自编码器的降阶神经常微分方程，提出路径积分恒等式分析其误差传播，通过多步雅可比范数区分传播机制，实验揭示了Burgers与Gray-Scott系统的不同误差演化模式。
中文摘要 AI 辅助
神经常微分方程（Neural ODE）降阶模型常能达到相当的局部预测精度，但其长时程外推行为却可能存在显著差异。为分析该差异，本文提出一种路径积分恒等式，将学习到的潜动态中的局部误差注入与误差放大分离开来；相关的多步雅可比范数量化了传输灵敏度，并区分了不同的传播机制。针对Burgers系统和Gray-Scott系统的实验展现了两种不同的误差演化模式：在Burgers系统中，预测误差保持有界，主要与持续存在的局部误差相关；而在Gray-Scott系统的外推过程中，误差会出现明显放大，此时雅可比范数可作为灵敏度诊断指标，而非物理预测精度的直接指标。
英文摘要
Neural ODE reduced-order models often achieve comparable local prediction accuracy, yet their long-horizon extrapolation behavior can differ substantially. To analyze this discrepancy, we develop a path-integral identity that separates local discrepancy injection from amplification in the learned latent dynamics. The associated multi-step Jacobian norms quantify transport sensitivity and distinguish different propagation regimes. Experiments on the Burgers and Gray--Scott systems exhibit two distinct patterns of error evolution. In Burgers systems, prediction errors remain bounded and are primarily associated with persistent local discrepancies. In contrast, Gray--Scott systems exhibit pronounced amplification during extrapolation, where Jacobian norms serve as sensitivity diagnostics rather than direct indicators of physical prediction accuracy.
