PosteriorBench: A Benchmark for Evaluating Posterior Matching in Generative Inverse Problem Solvers

Forum topic · 小凯 · 2026-09-19

Summary

PosteriorBench (arXiv:2609.20794) is a new benchmark evaluating generative inverse problem solvers on distributional accuracy rather than single-reconstruction quality. Since scientific inverse problems are often ill-posed, multiple solutions may fit the same sparse or noisy observations, and methods with strong pointwise accuracy can still fail via mode collapse, overconfident uncertainty, or averaging incompatible solutions. The benchmark covers four physics-based inverse problems: Darcy flow inversion, Poisson source recovery, carbon capture and storage, and light transport material inference. High-fidelity reference posteriors are built with established but computationally intensive methods such as rejection sampling and MCMC. Evaluation uses five posterior metrics: posterior mean error, posterior standard deviation error, maximum mean discrepancy, sliced Wasserstein distance, and radially averaged power spectrum error, assessing point accuracy, marginal uncertainty, distribution alignment, and frequency fidelity. Experiments reveal significant distribution-matching gaps in current solvers; neural operators improve resolution robustness, while guidance weights and generative noise are key for posterior variance calibration.

Overview

Field: Machine Learning Authors: Jiachen Yao, Zi-Siang Hsu, Xi Deng, Aditi Gupta, Xin Ju, Sally M Benson, Gege Wen, Anima Anandkumar arXiv: 2609.20794

Abstract (translation)

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.

The authors introduce PosteriorBench, a benchmark for evaluating the distributional accuracy of generative inverse solvers. It evaluates four physics-based inverse problems:

Darcy flow inversion
Poisson source recovery
Carbon capture and storage
Light transport material inference

For each task, high-fidelity reference posteriors are constructed using established but computationally intensive procedures such as rejection sampling and Markov chain Monte Carlo, enabling direct assessment of whether a solver recovers the full solution set rather than a single best sample.

Evaluation suite

The reference posteriors are paired with a five-metric posterior evaluation suite:

Posterior mean error
Posterior standard deviation error
Maximum mean discrepancy (MMD)
Sliced Wasserstein distance
Radially averaged power spectrum error

These metrics assess pointwise accuracy, marginal uncertainty, distributional alignment, and global frequency fidelity. The benchmark covers sparse sensing, low-resolution observations, nonlinear forward models, varying noise levels, and multimodal priors, and provides a unified pipeline for distribution matching and uncertainty quantification.

Findings

Experiments reveal significant distribution-matching gaps in current solvers. Neural operators improve resolution robustness, while guidance weights and generative noise are key to calibrating posterior variance.

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