Paired Autoencoders for Likelihood-free Estimation in Inverse Problems

TL;DR

This work tackles nonlinear PDE-based inverse problems by introducing paired autoencoders that perform likelihood-free estimation while enabling data-fit validation and solution refinement. The core idea is to learn two coupled autoencoders for the model and data, linked by a latent-space mapping, so that forward and inverse surrogates can be computed without repeated forward PDE solves, and to use latent-space regularization to improve inversion via latent-space inversion (LSI). The authors provide theoretical bounds on residuals and model errors, develop practical OOD-detection metrics (RRE and RMA), and demonstrate improved inversion accuracy on seismic full waveform inversion and inverse electromagnetic imaging. The approach reduces dependence on expensive PDE solves, offers reliability checks, and supports refinement through latent-space optimization, with broad applicability to large-scale ill-posed PDE-based inversions.

Abstract

We consider the solution of nonlinear inverse problems where the forward problem is a discretization of a partial differential equation. Such problems are notoriously difficult to solve in practice and require minimizing a combination of a data-fit term and a regularization term. The main computational bottleneck of typical algorithms is the direct estimation of the data misfit. Therefore, likelihood-free approaches have become appealing alternatives. Nonetheless, difficulties in generalization and limitations in accuracy have hindered their broader utility and applicability. In this work, we use a paired autoencoder framework as a likelihood-free estimator for inverse problems. We show that the use of such an architecture allows us to construct a solution efficiently and to overcome some known open problems when using likelihood-free estimators. In particular, our framework can assess the quality of the solution and improve on it if needed. We demonstrate the viability of our approach using examples from full waveform inversion and inverse electromagnetic imaging.

Figures (6)

• Figure 1: Architecture of our proposed paired autoencoder framework. Two autoencoders and a (linear) mapping between the latent spaces are simultaneously learned, thereby creating a paired system.
• Figure 2: Seismic Inversion Example: An acoustic velocity model and the corresponding data for a single source. There are $30$ sources in total.
• Figure 3: Seismic reconstructions for BI and LSI, with the basic initial guess and the warm start, are provided in the second row of figures. Both BI approaches suffer from 'wavefront' like artifacts.
• Figure 4: Relative model errors per iteration for the four approaches for seismic inversion.
• Figure 5: The density in terms of RRE and RMA for the validation set is plotted in gray-scale (with large densities in black and low densities in white). Scattered points correspond to metrics obtained from OOD data and are color-coded based on reconstruction error $\|\widehat{\mathbf{q}} - \mathbf{q} \|$. Two model predictions are provided from the paired autoencoder.

Theorems & Definitions (6)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 3
• proof