用低精度数据训练的神经微分方程模型，竟比高精度数值求解器更准。
Neural Emulator Superiority: When Machine Learning for PDEs Surpasses its Training Data
- 基于低精度求解器数据训练神经算子，利用其隐式正则化优势
- 多步推演中误差累积更优，精度反超高精度参考解
- 适用于需要高物理保真度的仿真场景，如流体、传热建模
传统观点认为，基于数值求解器数据训练的微分方程神经算子受限于训练数据的精度。我们提出“模拟器超越”现象：仅在低精度求解器数据上训练的神经网络，在多步滚动预测中，反而可达到高于该求解器的精度，甚至优于高精度参考解。理论分析揭示，模拟器归纳偏置、训练目标与数值误差特性的协同作用，使模型在长期演化中展现出更优的误差传播特性。我们在多种典型偏微分方程上验证该现象，使用标准神经架构证明，模拟器能隐式学习到更具规律性或更利于误差抑制的动力学行为，突破训练数据限制，缓解数值伪影。这项工作挑战了现有评估范式，提示神经模拟器在特定条件下可能实现比训练源更高的物理保真度。项目页面：https://tum-pbs.github.io/emulator-superiority
原文摘要 · Abstract (English)
Neural operators or emulators for PDEs trained on data from numerical solvers are conventionally assumed to be limited by their training data's fidelity. We challenge this assumption by identifying "emulator superiority," where neural networks trained purely on low-fidelity solver data can achieve higher accuracy than those solvers when evaluated against a higher-fidelity reference. Our theoretical analysis reveals how the interplay between emulator inductive biases, training objectives, and numerical error characteristics enables superior performance during multi-step rollouts. We empirically validate this finding across different PDEs using standard neural architectures, demonstrating that emulators can implicitly learn dynamics that are more regularized or exhibit more favorable error accumulation properties than their training data, potentially surpassing training data limitations and mitigating numerical artifacts. This work prompts a re-evaluation of emulator benchmarking, suggesting neural emulators might achieve greater physical fidelity than their training source within specific operational regimes. Project Page: https://tum-pbs.github.io/emulator-superiority
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。