Machine learning on manifolds for inverse scattering: Lipschitz stability analysis

TL;DR

This work establishes Lipschitz stability for the inverse of a nonlinear map defined on a learnable manifold, applied to inverse scattering problems for cracks in Helmholtz media. By restricting to finite-dimensional subspaces spanned by leading singular vectors of a parameter-dependent operator, the authors prove a Lipschitz lower bound that underpins stable recovery of crack geometry and forcing via neural networks. They implement a concrete ML pipeline using discretized forward operators $A_{m,app}$, train networks to recover crack parameters from measured fields, and demonstrate robustness to noise and model-related differences (avoiding inverse crime). The results provide a rigorous foundation for ML-based parameter estimation on manifolds in passive inverse problems with unbounded domains, with practical implications for nondestructive testing.

Abstract

Establishing Lipschitz stability estimates is crucial for ensuring the mathematical robustness of neural network (NN) approximations in machine learning (ML)-based parameter estimation, particularly in physics-informed settings. In this work, we derive such estimates for the inverse of a nonlinear map defined on a manifold that captures both unknown parameters and the nonlinear physical processes they influence. Our analysis is based on finite-dimensional, learnable representations of the manifold and provides Lipschitz stability estimates on the manifold-based subspaces, for a class of inverse maps associated with parameter dependent linear compact operators. Such operators model scattered and far-field data that can be used to detect structures such as cracks. We apply our theoretical ML manifold framework to inverse Helmholtz problems in unbounded regions exterior to cracks, addressing the scattered-field data-driven inverse problem while ensuring injectivity conditions on the manifold, a requirement for the Lipschitz stability. Our method accurately recovers crack-defining parameters without requiring prior knowledge of inputs such as incident wave types or external forces on the crack. Numerical experiments using NN approximations confirm the accuracy, efficiency, and robustness of the proposed approach.

Figures (6)

• Figure 1: Error versus epoch number. Left: with the $l^2$ (or MSE) loss function. Right: further trained with the $l^{\frac{2}{3}}$ loss function. The computed network for the $l^2$ loss function was used as a starting point for computing a new network with the $l^{\frac{2}{3}}$ loss function.
• Figure 2: Examples of configurations for the fault $\Gamma$ relative to the circle $S_R$ in Figure \ref{['fields']}. The real part of the total field is visualized. In each case, the crack $\Gamma$ is the black line segment. The circle $S_R$ is the dotted circle. The left plot corresponds to a Type 1 forcing term with incidence angle $\eta = (1, 0)$. The right plot corresponds to Type 3 forcing term with source $s=(6,0)$. The scale is different in order to facilitate visualization as the total field quickly decays from the source.
• Figure 3: Example of data in $\mathbb{C}^{N_S}$. Real and imaginary parts of $u$ at $(R \cos (j \frac{2 \pi }{N_S}), R\sin (j \frac{2 \pi }{N_S}))$, $j=1, ..., N_S$ are plotted with $j \frac{2 \pi }{N_S}$ on the horizontal axis. The smooth curves correspond to noise-free data. The jagged curves correspond to noisy data.
• Figure 4: Sorted absolute value errors in evaluating $\sin \theta$ (left) and $a$ (right) for 1000 random trials of $\theta, a$, support of the forcing term $g$, and random choice of type 1, 2, 3, or 4 forcing. The horizontal solid and dashed lines indicate average error for the 1000 trials using the two-step and standard (MSE) training, respectively. Blue: noise-free data. Red: noisy data.
• Figure 5: Example of a computed scattered field $u$ using a finite element code (left) versus using integral equation \ref{['Dir int']}. The scatterer is the thin rectangle (left) or the equivalent one-dimensional crack (right). Dirichlet conditions are applied in each case. The incoming wave in each case is the plane wave $e^{i k x_1}$, where $k=1.5$ as in the previous cases. Note the finer mesh in the vicinity of the thin rectangle.

Theorems & Definitions (7)

• Proposition 1
• Theorem 2
• Proposition 3
• Theorem 4
• Proposition 5
• Theorem 6
• Lemma 7