The Adaptive Solution of High-Frequency Helmholtz Equations via Multi-Grade Deep Learning

TL;DR

FD-MGDL, an adaptive framework integrating finite difference schemes with Multi-Grade Deep Learning to efficiently resolve high-frequency solutions in complex domains, is established as a robust, scalable solver for high-frequency wave equations in complex domains.

Abstract

The Helmholtz equation is fundamental to wave modeling in acoustics, electromagnetics, and seismic imaging, yet high-frequency regimes remain challenging due to the ``pollution effect''. We propose FD-MGDL, an adaptive framework integrating finite difference schemes with Multi-Grade Deep Learning to efficiently resolve high-frequency solutions. While traditional PINNs struggle with spectral bias and automatic differentiation overhead, FD-MGDL employs a progressive training strategy, incrementally adding hidden layers to refine the solution and maintain stability. Crucially, when using ReLU activation, our algorithm recasts the highly non-convex training problem into a sequence of convex subproblems. Numerical experiments in 2D and 3D with wavenumbers up to $κ=200$ show that FD-MGDL significantly outperforms single-grade and conventional neural solvers in accuracy and speed. Applied to an inhomogeneous concave velocity model, the framework accurately resolves wave focusing and caustics, surpassing the 5-point finite difference method in capturing sharp phase transitions and amplitude spikes. These results establish FD-MGDL as a robust, scalable solver for high-frequency wave equations in complex domains.

Figures (12)

• Figure 1: Performance comparison of training loss curves of FD-MGDL with different structures for solving 2D Helmholtz equation \ref{['2D-Helmholtz-equation']}.
• Figure 2: Performance comparison of FD-MGDL and six baseline methods (FD-SGDL, Mscale, FBPINN, SIREN, PINN and Pre-PINN) for the 2D Helmholtz problem \ref{['2d-Helmholtz-Dirichlet']} with the exact solution \ref{['2d-sin-solution']} at $\kappa=50$: $(a)$ training loss curves; $(b)$ exact solution; $(c)-(i)$ error visualizations of the numerical solutions.
• Figure 3: Performance comparison of FD-MGDL and six baseline methods (FD-SGDL, Mscale, FBPINN, SIREN, PINN and Pre-PINN) for the 2D Helmholtz problem \ref{['2d-Helmholtz-Dirichlet']} with the exact solution \ref{['2d-sin-solution']} at $\kappa=100$: $(a)$ training loss curves; $(b)$ exact solution; $(c)-(i)$ error visualizations of the numerical solutions.
• Figure 4: Performance comparison of FD-MGDL and six baseline methods (FD-SGDL, Mscale, FBPINN, SIREN, PINN and Pre-PINN) for the 2D Helmholtz problem \ref{['2d-Helmholtz-Dirichlet']} with the exact solution \ref{['2d-sin-solution']} at $\kappa=150$: $(a)$ training loss curves; $(b)$ exact solution; $(c)-(i)$ error visualizations of the numerical solutions.
• Figure 5: Performance comparison of FD-MGDL and six baseline methods (FD-SGDL, Mscale, FBPINN, SIREN, PINN and Pre-PINN) for the 2D Helmholtz problem \ref{['2d-Helmholtz-Dirichlet']} with the exact solution \ref{['2d-sin-solution']} at $\kappa=200$: $(a)$ training loss curves; $(b)$ exact solution; $(c)-(i)$ error visualizations of the numerical solutions.

Theorems & Definitions (4)

• Theorem 1: Monotonicity of Loss
• Theorem 2: Convex–Nonconvex Equivalence
• proof
• proof