DEFT: Data-Efficient Frequency-domain Top-k Sampling via Inverse Discrete Fourier Transform for Spatiotemporal Dynamical Systems Modeling

2026-08-11

AI Summary

This work addresses the challenges of high computational cost in physics-based simulation, strong data dependency, and poor generalization of purely data-driven approaches for modeling partial differential equation (PDE)-governed spatiotemporal systems. To this end, the authors propose a frequency-domain dominant mode sampling method that identifies critical Fourier modes and manipulates their amplitudes and phases to generate physically consistent training data via inverse discrete Fourier transform. This approach uniquely integrates frequency-domain sampling with operator learning, providing theoretical generalization bounds and deriving a Top-k mode selection criterion. Experiments demonstrate that on PDEBench, the method achieves less than 2% accuracy loss using only 60% of the original data; on a battery degradation PDE task, it attains an R² exceeding 0.99, and when transferring features to a new system, requires only 20% of fine-tuning data.

Abstract

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.

Problem

• spatiotemporal dynamical systems
• partial differential equations
• data efficiency
• generalization
• operator learning

Innovation

• frequency-domain sampling
• inverse discrete Fourier transform
• data-efficient learning
• operator learning
• spatiotemporal dynamical systems