用简单架构+傅里叶特征，让PINN更快更准地解偏微分方程
Enhancing Physics-Informed Neural Networks Through Feature Engineering
- 采用单层网络与傅里叶特征工程提升模型表达力
- 参数少65%、训练时间不到30%且误差低一个数量级
- 适合追求高效求解科学计算问题的研究者
物理信息神经网络（PINNs）通过深度学习求解偏微分方程（PDEs）。主流全连接深层网络需长时间训练才能达到中等精度，而近期特征工程方法可实现更高精度和更快收敛。本文提出SAFE-NET，一种单层自适应特征工程网络，在参数量远少于基线方法的前提下，实现数量级更低的误差。SAFE-NET回归机器学习基础：使用傅里叶特征、简化单隐层结构及高效优化器，改善了PINN优化问题的条件性。数值结果表明，SAFE-NET收敛更快，通常优于更深网络和更复杂架构。其平均参数量比对比方法减少65%，在不足30%训练轮次内达成相当精度，且每轮训练速度提升95%。这些发现挑战了现代PINN能有效学习科学任务特征的普遍认知，凸显特征工程带来的效率优势。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) seek to solve partial differential equations (PDEs) with deep learning. Mainstream approaches that deploy fully-connected multi-layer deep learning architectures require prolonged training to achieve even moderate accuracy, while recent work on feature engineering allows higher accuracy and faster convergence. This paper introduces SAFE-NET, a Single-layered Adaptive Feature Engineering NETwork that achieves orders-of-magnitude lower errors with far fewer parameters than baseline feature engineering methods. SAFE-NET returns to basic ideas in machine learning, using Fourier features, a simplified single hidden layer network architecture, and an effective optimizer that improves the conditioning of the PINN optimization problem. Numerical results show that SAFE-NET converges faster and typically outperforms deeper networks and more complex architectures. It consistently uses fewer parameters -- on average, 65% fewer than the competing feature engineering methods -- while achieving comparable accuracy in less than 30% of the training epochs. Moreover, each SAFE-NET epoch is 95% faster than those of competing feature engineering approaches. These findings challenge the prevailing belief that modern PINNs effectively learn features in these scientific applications and highlight the efficiency gains possible through feature engineering.
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