Operator learning meets inverse problems: A probabilistic perspective

TL;DR

The chapter surveys how operator learning can accelerate and stabilize solving inverse problems in function spaces, highlighting two main directions: end-to-end learning of inverse maps and learning priors/regularizers to augment traditional solvers. It develops a probabilistic and measure-centric framework, reviews neural-operator architectures (e.g., DeepONet, Fourier Neural Operators), and introduces data-to-posterior maps via measure transport and AMINO/ NIO approaches. Theoretical results are presented for EIT and linear/nonlinear inverse problems through finite-dimensional projections, together with convergence analyses and stability considerations for learned regularizers and plug-and-play denoisers. The work emphasizes discretization-invariant designs, measure-valued data, and uncertainty quantification, while identifying open problems in theory, stability, and measure-centric architectures for broad PDE-based inversions.

Abstract

Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys methodological and theoretical developments at the intersection of operator learning and inverse problems. It begins by summarizing the probabilistic and deterministic approaches to inverse problems, and pays special attention to emerging measure-centric formulations that treat observed data or unknown parameters as probability distributions. The discussion then turns to operator learning by covering essential components such as data generation, loss functions, and widely used architectures for representing function-to-function maps. The core of the chapter centers on the end-to-end inverse operator learning paradigm, which aims to directly map observed data to the solution of the inverse problem without requiring explicit knowledge of the forward map. It highlights the unique challenge that noise plays in this data-driven inversion setting, presents structure-aware architectures for both point predictions and posterior estimates, and surveys relevant theory for linear and nonlinear inverse problems. The chapter also discusses the estimation of priors and regularizers, where operator learning is used more selectively within classical inversion algorithms.

Figures (5)

• Figure 1: Overall structure of the chapter. The dotted lines indicate the connections associated with end-to-end learning, whereas the dashed lines correspond to regularizer learning.
• Figure 2: Electrical impedance tomography from \ref{['ex:eit']}. The domain $\Omega$ is the unit disk in $\mathbb{R}^2$. The boundary manifold $\partial\Omega$ is identified as the one-dimensional unit torus $\mathbb{T}$ by transforming into polar coordinates. \ref{['subfig:eit_finite']} displays two pairs of randomly sampled Neumann data ($g_m$, black) and corresponding Dirichlet data ($\Lambda_\gamma g_m$, red) on the torus. In \ref{['subfig:eit_in']}, the full NtD map $\Lambda_\gamma$ is represented numerically using its $L^2(\partial\Omega\times\partial\Omega)$ integral kernel function; this kernel function is further shifted, re-scaled, and transformed to $\mathbb{T}^2$ to aid visualization. The true realization of the conductivity $\gamma$ generating these measurements under the forward map $\gamma\mapsto\Lambda_\gamma$ is shown in \ref{['subfig:eit_out']}. Solving the inverse conductivity problem amounts to evaluating the inverse map $\Lambda_\gamma\mapsto\gamma$.
• Figure 3: Stability within the range of $\mathcal{G}\colon\gamma\mapsto\Lambda_\gamma$ from \ref{['ex:eit']}, \ref{['eqn:stability_eit_inside_range']}.
• Figure 4: Noisy data $y^\delta$ lives outside of the domain $\mathop{\mathrm{Dom}}\nolimits(\Psi^\star)$ of the true inverse map $\Psi^\star\colon \mathop{\mathrm{Dom}}\nolimits(\Psi^\star)\to \mathcal{U}$. In the illustration, the distance between $y^\delta$ and $\mathop{\mathrm{Dom}}\nolimits(\Psi^\star)$ is less than $\delta$. A continuous extension $\widetilde{\Psi}^\star\colon\mathcal{Y}\to\mathcal{U}$ of $\Psi^\star$ is well-defined on noisy data $y^\delta$. As $\delta\to 0$, it holds that $\widetilde{\Psi}^\star(y^\delta)\to \Psi^\star(\mathcal{G}(u))=u$ with a certain convergence rate. Learned inverse maps $\widehat{\Psi}\colon\mathcal{Y}\to\mathcal{U}$ conceptually aim to approximate the extension $\widetilde{\Psi}^\star$.
• Figure 5: The stability of prior learning concerns the relationship between the closeness of the posteriors, measured by $\mathsf{d}(\mu^y, \nu^y)$, and the closeness of the priors, measured by $\mathsf{d}_0(\mu, \nu)$.

Theorems & Definitions (4)

• Example 1: electrical impedance tomography
• Proposition 1: prior-to-posterior local Lipschitz stability in Dudley metric
• proof
• Remark 1