Summary
Inverse problems happen when you try to figure out hidden causes from limited or noisy observations, but multiple different answers might fit the data. The authors point out that simply checking if one best guess looks right isn’t enough because methods can miss the full range of possible solutions. They created PosteriorBench, a set of tests that compares how well different methods capture the full range of possible answers—not just one—using carefully computed reference solutions. These tests measure not only the average guess but also how well methods represent uncertainty and the overall distribution of solutions. Their experiments show current methods often fall short, but some neural network approaches help improve accuracy and better estimate uncertainty.
What this means in practice
- •For environmental engineers: Evaluate and improve models that estimate underground water flow and pollutant sources with more reliable uncertainty assessments.
- •For computer vision developers: Enhance algorithms that infer material properties from images by better capturing the range of possible solutions and uncertainties.
Authors
Jiachen Yao, Zi-Siang Hsu, Xi Deng, Aditi Gupta, Xin Ju, Sally M Benson, Gege Wen, Anima Anandkumar
Abstract
Generative models are increasingly used to solve scientific inverse problems, but existing evaluations still focus primarily on whether a method can produce a single plausible reconstruction. This is insufficient for ill-posed problems, where multiple solutions may be consistent with the same sparse or noisy observations. In these settings, a method can achieve strong pointwise accuracy while still failing to capture the true posterior through mode collapse, overconfident uncertainty, or averaging incompatible solutions. We introduce PosteriorBench, a benchmark for evaluating the distributional accuracy of generative inverse solvers. PosteriorBench evaluates four physics-based inverse problems: Darcy flow inversion, Poisson source recovery, carbon capture and storage, and light transport material inference. For each task, we construct high-fidelity reference posteriors using computationally heavy but established procedures such as rejection sampling and Markov chain Monte Carlo, enabling direct assessment of whether solvers recover the full set of solutions rather than the single best sample. We pair these references with a five-metric posterior evaluation suite: posterior-mean error, posterior-standard-deviation error, maximum mean discrepancy, sliced Wasserstein distance, and radially averaged power-spectrum error. These metrics assess pointwise accuracy, marginal uncertainty, distributional alignment, and global frequency fidelity. The benchmark spans sparse sensing, low-resolution observations, nonlinear forward models, varying noise levels, and multimodal priors, with a unified pipeline for distribution matching and uncertainty quantification. Our experiments reveal substantial distribution-matching gaps across current solvers, while showing that neural operators improve resolution robustness, and guidance weights and generation noise are key to posterior-variance calibration.