用最小二乘法加速声波模拟中PINN的收敛速度
Least-Squares-Embedded Optimization for Accelerated Convergence of PINNs in Acoustic Wavefield Simulations
- 将最小二乘求解器嵌入梯度下降损失函数，优化输出层更新
- 在高频率波场下收敛更快，精度提升，标准方法失败时仍有效
- 计算开销小，适合大规模声波模拟，支持有无完美匹配层场景
物理信息神经网络（PINNs）在求解偏微分方程（PDEs）方面展现出潜力，包括频域赫姆霍兹方程。然而，使用梯度下降（GD）训练标准PINNs在高频率波场下存在收敛慢和不稳定的缺陷。针对基于赫姆霍兹方程的散射声波场模拟，我们推导出一种混合优化框架，通过将最小二乘（LS）求解器直接嵌入GD损失函数中，加速训练收敛。该形式使线性输出层获得最优更新。本方法适用于有或无完美匹配层（PML）的情况，并提供了两种场景下的实用张量实现。基准速度模型的数值实验表明，相比传统PINN训练，该方法收敛更快、精度更高、稳定性更强。尤其在标准GD训练失败的情况下，LS增强方法仍能快速收敛。LS求解器作用于一个小的正规矩阵，确保计算开销极低，使方法可扩展至大规模波场模拟。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) have shown promise in solving partial differential equations (PDEs), including the frequency-domain Helmholtz equation. However, standard training of PINNs using gradient descent (GD) suffers from slow convergence and instability, particularly for high-frequency wavefields. For scattered acoustic wavefield simulation based on Helmholtz equation, we derive a hybrid optimization framework that accelerates training convergence by embedding a least-squares (LS) solver directly into the GD loss function. This formulation enables optimal updates for the linear output layer. Our method is applicable with or without perfectly matched layers (PML), and we provide practical tensor-based implementations for both scenarios. Numerical experiments on benchmark velocity models demonstrate that our approach achieves faster convergence, higher accuracy, and improved stability compared to conventional PINN training. In particular, our results show that the LS-enhanced method converges rapidly even in cases where standard GD-based training fails. The LS solver operates on a small normal matrix, ensuring minimal computational overhead and making the method scalable for large-scale wavefield simulations.
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