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This paper introduces a Green-Integral (GI) neural solver for the acoustic Helmholtz equation to address the limitations of standard physics-informed neural networks (PINNs) in simulating highly oscillatory solutions. The GI solver enforces wave physics through an integral representation, encoding oscillatory behavior and outgoing radiation directly through the integral kernel, thus avoiding second-order spatial derivatives and artificial boundary layers. Experiments on seismic benchmark models demonstrate that the GI-based training outperforms PDE-based PINNs, reducing computational cost by over a factor of ten, with a hybrid GI+PDE loss further improving accuracy in models with localized scattering.
PINNs can now efficiently solve highly oscillatory wave equations in heterogeneous media, thanks to a Green's function-based integral formulation that cuts computation by 10x and avoids absorbing boundary layers.
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.
