用傅里叶与神经网络结合，突破物理信息神经网络精度瓶颈。
Breaking the Precision Ceiling in Physics-Informed Neural Networks: A Hybrid Fourier-Neural Architecture for Ultra-High Accuracy
- 将截断傅里叶级数与深度网络结合，分别捕捉模态特征和自适应残差修正。
- 在欧拉-伯努利梁方程上实现1.94×10⁻⁷的L2误差，比传统PINN提升17倍。
- 适合高精度科学计算场景，尤其需超精准模拟的工程与仿真领域。
物理信息神经网络（PINNs）在四阶偏微分方程求解中误差稳定在10⁻³–10⁻⁴量级，形成公认的精度天花板，制约其在工程中的应用。本文提出一种混合傅里叶-神经架构，针对欧拉-伯努利梁方程，实现了前所未有的L2误差1.94×10⁻⁷，较标准PINNs提升17倍，比传统数值方法高出15–500倍。该方法协同使用截断傅里叶级数捕捉主导模态行为，以及深度神经网络进行自适应残差修正。系统性谐波优化研究发现：恰好10个谐波时性能最优，超过此阈值精度骤降至10⁻¹。采用两阶段优化策略（Adam后接L-BFGS）与自适应权重平衡，确保超精度收敛稳定。GPU加速实现亚30分钟训练，克服四阶导数复杂性。通过解决现有方法中的12项关键缺陷——从架构僵化到优化景观问题——本工作证明，通过合理设计可实现超精度，为机器学习在科学计算中媲美甚至超越传统数值方法开辟新范式。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) have plateaued at errors of $10^{-3}$-$10^{-4}$ for fourth-order partial differential equations, creating a perceived precision ceiling that limits their adoption in engineering applications. We break through this barrier with a hybrid Fourier-neural architecture for the Euler-Bernoulli beam equation, achieving unprecedented L2 error of $1.94 \times 10^{-7}$-a 17-fold improvement over standard PINNs and \(15-500\times\) better than traditional numerical methods. Our approach synergistically combines a truncated Fourier series capturing dominant modal behavior with a deep neural network providing adaptive residual corrections. A systematic harmonic optimization study revealed a counter-intuitive discovery: exactly 10 harmonics yield optimal performance, with accuracy catastrophically degrading from $10^{-7}$ to $10^{-1}$ beyond this threshold. The two-phase optimization strategy (Adam followed by L-BFGS) and adaptive weight balancing enable stable ultra-precision convergence. GPU-accelerated implementation achieves sub-30-minute training despite fourth-order derivative complexity. By addressing 12 critical gaps in existing approaches-from architectural rigidity to optimization landscapes-this work demonstrates that ultra-precision is achievable through proper design, opening new paradigms for scientific computing where machine learning can match or exceed traditional numerical methods.
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