Fourier Neural Operators (FNO) have emerged as promising solutions for efficiently solving partial differential equations (PDEs) by learning infinite-dimensional function mappings through frequency domain transformations. However, the sparsity of high-frequency signals limits computational efficiency for high-dimensional inputs, and fixed-pattern truncation often causes high-frequency signal loss, reducing performance in scenarios such as high-resolution inputs or long-term predictions. To address these challenges, we propose FreqMoE, an efficient and progressive training framework that exploits the dependency of high-frequency signals on low-frequency components. The model first learns low-frequency weights and then applies a sparse upward-cycling strategy to construct a mixture of experts (MoE) in the frequency domain, effectively extending the learned weights to high-frequency regions. Experiments on both regular and irregular grid PDEs demonstrate that FreqMoE achieves up to 16.6% accuracy improvement while using merely 2.1% parameters (47.32x reduction) compared to dense FNO. Furthermore, the approach demonstrates remarkable stability in long-term predictions and generalizes seamlessly to various FNO variants and grid structures, establishing a new ``Low frequency Pretraining, High frequency Fine-tuning'' paradigm for solving PDEs.
View on arXiv@article{chen2025_2505.06858,
title={ FreqMoE: Dynamic Frequency Enhancement for Neural PDE Solvers },
author={ Tianyu Chen and Haoyi Zhou and Ying Li and Hao Wang and Zhenzhe Zhang and Tianchen Zhu and Shanghang Zhang and Jianxin Li },
journal={arXiv preprint arXiv:2505.06858},
year={ 2025 }
}