Abstract: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.
| Subjects: | Machine Learning (cs.LG); Geophysics (physics.geo-ph) |
| Cite as: | arXiv:2604.21411 [cs.LG] |
| (or arXiv:2604.21411v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2604.21411 arXiv-issued DOI via DataCite (pending registration) |
From: Mohammad Mahdi Abedi [view email]
[v1]
Thu, 23 Apr 2026 08:24:10 UTC (8,478 KB)
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