A Morphology-Adaptive Random Feature Method for Inverse Source Problem of the Helmholtz Equation

TL;DR

This work tackles the inverse source problem for the Helmholtz equation with complex, potentially discontinuous sources, where traditional methods are either costly or struggle with sharp morphologies. It introduces the Morphology-Adaptive Random Feature Method (MA-RFM), a two-stage, physics-informed framework that first localizes sources via Integral Adaptive RFM (IA-RFM) and then enriches the solution basis with morphology-aware functions to capture intricate geometries, all within a convex, Tikhonov-regularized integral-equation formulation using the fundamental solution. Theoretical results establish uniqueness for multi-frequency data with partial boundary information and provide a stability bound for the regularized solution, ensuring robust, convergent behavior. Numerical experiments across 2D and 3D problems demonstrate substantial computational savings (often 2–3 orders of magnitude) and improved accuracy, even under noise and limited aperture data, confirming the method’s effectiveness and potential for broader PDE applications.

Abstract

The inverse source problem for the Helmholtz equation poses significant challenges, particularly when sources exhibit complex or discontinuous geometries. Traditional numerical methods suffer from prohibitive computational costs, while machine learning-based approaches such as Physics-Informed Neural Networks (PINNs) and the Random Feature Method (RFM) -- though computationally efficient for inverse problems -- lack the intrinsic machinery to handle the sharp morphological features in such singular problems, leading to inaccurate solutions. To address this issue, we propose the Morphology-Adaptive Random Feature Method (MA-RFM), a novel two-phase framework that adaptively locates critical regions and adds morphology activation functions for tackling the multi-frequency inverse source problem with complex geometry. Our framework recasts the ill-posed inverse problem into a well-posed, strictly convex optimization problem by reformulating the governing Helmholtz equation as a Tikhonov-regularized integral equation via its fundamental solution. In the first stage, the Integral Adaptive RFM (IA-RFM), employs an adaptive algorithm to rapidly localize the source support, thereby reducing computational overhead and accelerating convergence. In the second stage, posterior geometric information is progressively integrated into the solver via hybrid basis functions, enabling a precise reconstruction of complex morphologies. The MA-RFM extends the capabilities of RFM to handle PDEs with singular solutions while preserving its mesh-free efficiency. We demonstrate the superior performance of our approach through ample challenging 2D and 3D benchmark problems, even under limited and noisy measurement conditions, highlighting its robustness and accuracy in reconstructing complex and disjoint sources.

Figures (16)

• Figure 1: Integral Random Feature method (IRFM), Integral Adaptive Random Feature Method (IA-RFM), Morphology-Adaptive Random Feature Method (MA-RFM).
• Figure 2: Example \ref{['ex:Prior parameters']}: Left) Prior parameters: $\eta_\text{M}$ versus $\delta_{\text{all}}$; Middle) $\lambda_{\text{reg}}^2$, $\|s^{\delta}-s^*\|_2$ and the theoretical error bounds; Right) $\|S^{\delta}-S^*_M\|_2$ for the training set versus the test set.
• Figure 3: Example \ref{['ex:Prior parameters']}: Prior parameters: IRFM results with $\eta_\text{M}=1$, $\nu=1$, $\delta=5\%$, $\delta_{\text{all}}=0.031$, $M_0=3200$, $N_s=15$, yield $3.01\%$$E_{l^2}(S)$.
• Figure 4: Example \ref{['ex:Mountain shape source function']}: Mountain shape source function: (a). PINN solution with $u_{NN}$: [2, 50, 50, 2] and $S_{NN}$: [2, 50, 50, 1] yields 99.68% $E_{l^2}(S)$. (b). The true solution () and PINN solution () at($x_1$,0).
• Figure 5: Example \ref{['ex:Mountain shape source function']}: Mountain shape source function: IA-RFM results with $\delta=10\%$, $M_0=3200$, $N_s=50$, $N_{\text{generate}}$=100, $\lambda_{\text{reg}}^2=7.84e\text{-}2$ yield 1.39% $E_{l^2}(S)$.

Theorems & Definitions (6)

• Theorem 2.1
• Remark 1
• Theorem 3.1: Uniqueness and stability of Tikhonov regularization solution
• Remark 2
• proof
• proof