用傅里叶特征迭代训练，让神经网络更准解高频物理方程。
Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Features
- 通过随机傅里叶特征增强隐空间，分两阶段训练。
- 在经典问题上逼近精度显著优于现有方法，全频段表现更好。
- 适合需高精度求解复杂波动、高频物理问题的研究者。
谱偏差指神经网络倾向于先学习低频特征，这是物理信息神经网络（PINNs）训练中的常见问题。为此，我们提出 IFeF-PINN 算法，通过随机傅里叶特征增强特征空间，实现 PINNs 的迭代训练。核心思想是在隐空间中引入高频成分，形成两阶段问题：(i) 估计特征空间的基函数，(ii) 回归确定增强基函数的系数。对于线性模型，该回归问题为凸优化，我们证明了迭代训练的收敛性。实验表明，随机傅里叶特征显著提升网络表达能力，可精准逼近高频偏微分方程（PDE）。在多个经典基准测试中，本方法性能全面超越现有先进算法，并在全频率范围内展现出更优逼近效果。
原文摘要 · Abstract (English)
Spectral bias, the tendency of neural networks to learn low-frequency features first, is a well-known issue with many training algorithms for physics-informed neural networks (PINNs). To overcome this issue, we propose IFeF-PINN, an algorithm for iterative training of PINNs with Fourier-enhanced features. The key idea is to enrich the latent space using high-frequency components through Random Fourier Features. This creates a two-stage training problem: (i) estimate a basis in the feature space, and (ii) perform regression to determine the coefficients of the enhanced basis functions. For an underlying linear model, it is shown that the latter problem is convex, and we prove that the iterative training scheme converges. Furthermore, we empirically establish that Random Fourier Features enhance the expressive capacity of the network, enabling accurate approximation of high-frequency PDEs. Through extensive numerical evaluation on classical benchmark problems, the superior performance of our method over state-of-the-art algorithms is shown, and the improved approximation across the frequency domain is illustrated.
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