一种具有随机物理信息正则化的绿色积分约束神经求解器
A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization
- University of the Basque Country (UPV/EHU)（巴斯克国家大学）
- Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）
- Ikerbasque, Basque Foundation for Science（Ikerbasque巴斯克科学基金会）
- King Abdullah University of Science and Technology (KAUST)（卡迪斯国王科学与技术大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种绿色积分约束神经求解器，通过积分表示强制波物理，解决高振荡亥姆霍兹解在异质介质中的模拟问题，采用FFT加速减少计算成本，提升局部精度。
AI中文摘要：
标准物理信息神经网络（PINNs）在模拟高振荡亥姆霍兹解时面临计算成本高、偏向光滑解和需要人工吸收边界层的挑战。为此，我们引入了绿色积分（GI）神经求解器，通过非局部约束强制波物理。通过积分核直接编码振荡行为和辐射，消除二阶空间导数，无需额外边界层即可获得物理解。理论上，通过神经网络优化GI损失相当于频谱调制的预条件迭代，使在经典Born级数发散的异质介质中实现收敛。通过FFT基于卷积加速GI损失评估，显著减少GPU内存使用和训练时间。然而，这种效率依赖于固定正则网格，可能限制局部分辨率。为提高强散射区域的局部精度，我们还提出GI+PDE混合损失，通过少量非均匀采样点施加轻量级亥姆霍兹残差。我们在具有结构对比和次波长异质性的地震基准模型上评估了该方法，在频率高达20Hz时，GI训练方法比PDE基PINNs减少计算成本超过十倍。在具有局部散射的模型中，混合损失提供最准确的重建，提供了一种稳定、高效且物理基础的替代方案。
英文摘要：
Standard physics-informed neural networks (PINNs) struggle to simulate highly oscillatory Helmholtz solutions in heterogeneous media because pointwise minimization of second-order PDE residuals is computationally expensive, biased toward smooth solutions, and requires artificial absorbing boundary layers to restrict the solution. To overcome these challenges, we introduce a Green-Integral (GI) neural solver for the acoustic Helmholtz equation. It departs from the PDE-residual-based formulation by enforcing wave physics through an integral representation that imposes a nonlocal constraint. Oscillatory behavior and outgoing radiation are encoded directly through the integral kernel, eliminating second-order spatial derivatives and enforcing physical solutions without additional boundary layers. Theoretically, optimizing this GI loss via a neural network acts as a spectrally tuned preconditioned iteration, enabling convergence in heterogeneous media where the classical Born series diverges. By exploiting FFT-based convolution to accelerate the GI loss evaluation, our approach substantially reduces GPU memory usage and training time. However, this efficiency relies on a fixed regular grid, which can limit local resolution. To improve local accuracy in strong scattering regions, we also propose a hybrid GI+PDE loss, enforcing a lightweight Helmholtz residual at a small number of nonuniformly sampled collocation points. We evaluate our method on seismic benchmark models characterized by structural contrasts and subwavelength heterogeneity at frequencies up to 20Hz. GI-based training consistently outperforms PDE-based PINNs, reducing computational cost by over a factor of ten. In models with localized scattering, the hybrid loss yields the most accurate reconstructions, providing a stable, efficient, and physically grounded alternative.
