DEFT：用于时空动力系统建模的、基于逆离散傅里叶变换的数据高效频域Top-k采样方法

中文摘要 AI 辅助

对由偏微分方程（PDE）描述的时空动力系统进行建模存在两大核心挑战：要么需要昂贵的基于物理的模拟器，这类模拟器需进行迭代数值求解，计算成本极高；要么依赖大量训练数据，但纯数据驱动模型往往对下游动态运行条件的泛化能力较差。我们提出了DEFT，一种频域数据采样方法，该方法可识别物理系统的主导傅里叶模式，并通过逆离散傅里叶变换系统地调整相应的振幅和相位，以生成物理一致性的训练数据。此外，我们推导了该方法的泛化界，还注意到它为选择K提供了理论上有原则性的准则。我们通过三组实验对所提方法进行评估，每组实验针对其效用的不同方面。首先，我们在经典PDE求解任务上验证该框架，结果表明当系统由少数显著频率分量主导时，其性能优于传统方法。其次，我们将DEFT作为数据价值过滤器应用于PDEBench的扩散-吸附方程和伯格斯方程，显示其可减少40%的数据需求，同时预测精度损失不足2%。第三，为评估DEFT在更具挑战性和实际意义的问题上的表现，我们在电池退化PDE系统中对其进行验证，在各种测试数据集上均实现了超过0.99的R²值，预测精度始终很高。此外，学习到的频域特征仅用20%的微调数据即可迁移至其他电池化学体系。这些结果表明，DEFT是一种用于高效算子学习的有效数据采样方法。

英文摘要

Modeling spatiotemporal dynamical systems governed by partial differential equations (PDEs) poses two major challenges: it either requires expensive physics-based simulators that entail iterative numerical solving at high computational cost, or it depends on abundant training data, yet purely data-driven models often generalize poorly to downstream dynamic operating conditions. We propose DEFT, a frequency-domain data sampling method that identifies the dominant Fourier modes of a physical system and systematically varies the corresponding amplitudes and phases to generate physically consistent training data via the inverse discrete Fourier transform. In addition, we derive a generalization bound of this method. We note that it also provides a theoretically principled criterion for selecting $K$. We evaluate the proposed method through three sets of experiments, each targeting a distinct aspect of its utility. First, we validate the framework on canonical PDEs solving demonstrating that it outperforms traditional methods when the system is dominated by a few prominent frequency components. Second, we employ DEFT as a data-value filter on the diffusion--sorption and Burgers equations of PDEBench, showing that it reduces data requirements by $40\%$ while sacrificing less than $2\%$ in predictive accuracy. Third, to evaluate DEFT for more challenging and practically relevant problems, we validate it in the battery degradation PDE system, achieving consistently high predictive accuracy across various test datasets with $R^2$ values exceeding $0.99$. Moreover, the learned frequency-domain features transfer to other battery chemistries with only $20\%$ of the fine-tuning data. These results demonstrate that DEFT is an effective data-sampling method for efficient operator learning.

发表机构

• Peking University（北京大学）

机构由 AI 辅助整理，请以论文原文为准。