用半群一致性诊断物理模拟器的长期误差，发现能有效预判滚动预测失败。
Semigroup Consistency as a Diagnostic for Learned Physics Simulators
- 通过比较直接演化与分步演化的结果差异，检测模型时间累积错误。
- 在热传导和伯格斯方程上，半群误差与长时滚动退化相关性达0.635。
- 适合评估物理模拟器长期稳定性，尤其对自动驾驶等任务重要。
学习型物理模拟器常通过单步或短时预测误差评估，但此类指标难以发现时间组合与长时滚动中的失效。对于自主、状态完备系统，精确解映射满足半群律：直接演化 s+t 时段应与先演化 s 再演化 t 一致。本文提出归一化半群误差作为事后、模型无关的诊断方法，比较直接与分步演化结果。在一维热传导与伯格斯动力学上，采用时间条件卷积网络与FNO基线模型，半群误差与滚动预测退化呈正相关，轨迹级斯皮尔曼相关系数ρ=0.635，95%置信区间[0.621, 0.649]。半群正则化效果参差不齐，表明半群一致性更适合作为评估诊断工具而非通用训练目标。
原文摘要 · Abstract (English)
Learned physics simulators are often evaluated by one-step or short-horizon prediction error, but these metrics can miss failures in temporal composition and long-horizon rollout. For autonomous, state-complete systems, exact solution maps satisfy a semigroup law: direct evolution over $s+t$ should agree with evolution over $s$ followed by $t$. We propose normalized semigroup error as a post hoc, model-agnostic diagnostic comparing these direct and composed learned predictions. On one-dimensional heat and Burgers dynamics with time-conditioned ConvNet and FNO baselines, semigroup error is positively associated with rollout degradation, with trajectory-level Spearman correlation $ρ= 0.635$ and $95%$ CI $[0.621, 0.649]$. Semigroup regularization has mixed effects, supporting semigroup consistency primarily as an evaluation diagnostic rather than a universally beneficial training objective.
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