发表机构
School of Computer Science and Engineering, UESTC; Electronics and Information Convergence Engineering, KHU; Department of Mathematical Sciences, UOL; School of Computer Science and Technology, XDU; School of Mechanical and Electrical Engineering, UESTC; College of Computer and Information Engineering, XAUAT（电子科技大学计算机科学与工程学院; 明知大学电子信息融合工程系; 利物浦大学数学科学系; 西安电子科技大学计算机科学与技术学院; 电子科技大学机械与电气工程学院; 西安建筑科技大学计算机与信息工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出HERO方法，通过结合模型优化历史的相对监督改进回滚训练，在9个PDE基准上提升了长跨度预测精度、稳定回滚长度与分布外鲁棒性，且无推理时间成本。
AI 中文摘要
神经算子通过将学习到的演化算子递归应用于自身预测，为随时间变化的偏微分方程（PDE）提供快速替代模型，但这种自回归回滚会将每一个预测误差作为输入反馈，导致局部误差累积。现有的回滚训练策略减少了训练输入与自生成状态之间的不匹配，然而其监督仍仅衡量与真实轨迹的绝对偏差，因此无法判断算子是否克服了优化过程中早期表现出的长跨度失效行为。我们提出历史增强回滚训练（HERO），该方法用模型优化历史衍生的相对监督增强传统的绝对轨迹监督。HERO通过回滚误差、谱偏差、能量漂移和误差增长，对定期更新的滞后算子、当前模型及扰动输入的分离候选回滚进行排序，并选择最强失效轨迹作为参考。该参考作为固定比较基准进入基于间隔的目标函数，对真实回滚梯度进行有界、样本依赖的重加权，而非独立梯度方向，我们对此进行了理论分析。在9个带有谱和注意力骨干的PDE基准上的实验表明，HERO在无推理时间成本的情况下，持续提升了长跨度预测精度、稳定回滚长度和分布外鲁棒性。
英文摘要
Neural operators provide fast surrogates for time-dependent partial differential equations (PDEs) by applying a learned evolution operator recursively to its own predictions, but this autoregressive rollout feeds every prediction error back as input, so local errors accumulate. Existing rollout-training strategies reduce the mismatch between training inputs and self-generated states, yet their supervision still measures only the absolute discrepancy from the ground-truth trajectory. Such supervision is therefore uninformative about whether the operator has overcome the long-horizon failure behaviors it exhibited earlier during optimization. We propose history-enriched rollout training (HERO), which augments conventional absolute trajectory supervision with relative supervision derived from the model's optimization history. HERO ranks detached candidate rollouts from a periodically refreshed lagged operator, the current model, and a perturbed input by rollout error, spectral discrepancy, energy drift, and error growth, and selects the strongest failure trajectory as reference. This reference enters a margin-based objective as a fixed comparison baseline, inducing a bounded, sample-dependent reweighting of the ground-truth rollout gradient rather than an independent gradient direction, which we further analyze theoretically. Experiments on nine PDE benchmarks with spectral and attention-based backbones show that HERO consistently improves long-horizon accuracy, stable rollout length, and out-of-distribution robustness at no inference-time cost. These results indicate that history-enriched relative supervision is effective for stabilizing long-horizon autoregressive prediction.
