通过自适应频率增强，提升张量神经网络解高维多尺度问题能力
Frequency-adaptive tensor neural networks for high-dimensional multi-scale problems
- 基于一维分量傅里叶变换提取高维函数频率特征
- 在多个测试问题上显著提升求解精度与收敛速度
- 适合处理高维、多尺度的科学计算与物理建模任务
张量神经网络（TNNs）在求解高维问题上表现出优势。然而，与传统神经网络类似，其受频率原则影响，难以准确捕捉解的高频特征。本文通过傅里叶分析研究TNNs训练动态，引入随机傅里叶特征以增强表达能力。利用TNNs的固有张量结构，提出对一维分量函数执行离散傅里叶变换，有效提取高维函数频率特征，缓解维度灾难。基于此，提出频率自适应张量神经网络算法，显著提升TNNs在复杂多尺度问题上的求解能力。大量数值实验验证了该算法的有效性与鲁棒性。
原文摘要 · Abstract (English)
Tensor neural networks (TNNs) have demonstrated their superiority in solving high-dimensional problems. However, similar to conventional neural networks, TNNs are also influenced by the Frequency Principle, which limits their ability to accurately capture high-frequency features of the solution. In this work, we analyze the training dynamics of TNNs by Fourier analysis and enhance their expressivity for high-dimensional multi-scale problems by incorporating random Fourier features. Leveraging the inherent tensor structure of TNNs, we further propose a novel approach to extract frequency features of high-dimensional functions by performing the Discrete Fourier Transform to one-dimensional component functions. This strategy effectively mitigates the curse of dimensionality. Building on this idea, we propose a frequency-adaptive TNNs algorithm, which significantly improves the ability of TNNs in solving complex multi-scale problems. Extensive numerical experiments are performed to validate the effectiveness and robustness of the proposed frequency-adaptive TNNs algorithm.
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