PosteriorBench: From Point Estimates to Posterior Matching in Evaluating Generative Inverse Solvers
PosteriorBench evaluates distributional accuracy of generative inverse solvers, revealing neural operators improve resolution robustness.
Key Findings
Methodology
PosteriorBench constructs high-fidelity reference posteriors to evaluate generative inverse solvers across four physics-based inverse problems. It uses rejection sampling and Markov chain Monte Carlo methods to generate reference posteriors, assessed by five metrics for posterior matching.
Key Results
- Neural operators excel in resolution robustness; guidance weights and generation noise are crucial for posterior variance calibration.
- Experiments reveal significant distribution-matching gaps across current solvers; FunDPS performs exceptionally across tasks.
- DDIS stands out as a strong posterior sampler across several scientific inverse tasks.
Significance
PosteriorBench provides a comprehensive evaluation framework for generative inverse solvers, addressing the inadequacy of focusing solely on single reconstructions and highlighting the importance of posterior uncertainty in scientific decision-making.
Technical Contribution
PosteriorBench introduces a distribution-centered evaluation protocol, combining high-fidelity reference posteriors with diverse inverse tasks, offering new engineering possibilities and theoretical guarantees.
Novelty
PosteriorBench is the first to systematically evaluate the posterior matching capability of generative inverse solvers, distinguishing itself from traditional single-point estimate evaluations by emphasizing distributional consistency.
Limitations
- Current methods lack effective posterior calibration at high noise levels, potentially leading to inaccurate distribution predictions.
- Some solvers perform poorly under multimodal priors, requiring further optimization.
Future Work
Future research could explore more efficient posterior construction methods and calibration mechanisms under varying noise levels.
AI Executive Summary
Generative models are increasingly used to solve scientific inverse problems, but existing evaluations primarily focus on whether a method can produce a single plausible reconstruction, which is insufficient for ill-posed problems with multiple solutions. PosteriorBench addresses this evaluation gap by assessing the distributional accuracy of generative inverse solvers. The framework evaluates four physics-based inverse problems and constructs high-fidelity reference posteriors using rejection sampling and Markov chain Monte Carlo methods. Experimental results show that neural operators excel in improving resolution robustness, with guidance weights and generation noise being key to posterior variance calibration. PosteriorBench not only reveals significant distribution-matching gaps across current solvers but also provides new directions for future research. While some methods lack effective posterior calibration at high noise levels, the framework offers a comprehensive evaluation platform for generative inverse solvers, emphasizing the importance of posterior uncertainty in scientific decision-making. Future research could explore more efficient posterior construction methods and calibration mechanisms under varying noise levels.
Deep Analysis
Background
Generative models are widely applied in scientific inverse problems, but existing evaluations primarily focus on single reconstruction plausibility. Traditional methods like InverseBench evaluate single solutions, ignoring the possibility of multiple solutions in ill-posed problems.
Core Problem
Scientific inverse problems often have non-uniqueness, and existing evaluations fail to effectively capture the diversity and uncertainty of posterior distributions, leading to insufficient information for decision-making.
Innovation
PosteriorBench constructs high-fidelity reference posteriors, providing a distribution-centered evaluation framework that emphasizes the importance of posterior uncertainty. It uses rejection sampling and Markov chain Monte Carlo methods to generate reference posteriors.
Methodology
- �� Construct high-fidelity reference posteriors using rejection sampling and Markov chain Monte Carlo methods.
- �� Evaluate four physics-based inverse problems: Darcy flow inversion, Poisson source recovery, carbon capture and storage, light transport material inference.
- �� Use five metrics to assess posterior matching capability: posterior-mean error, posterior-standard-deviation error, maximum mean discrepancy, sliced Wasserstein distance, radially averaged power-spectrum error.
Experiments
Experimental design includes comparing current solvers' performance under different noise levels and multimodal priors, using high-fidelity reference posteriors for evaluation, revealing significant distribution-matching gaps.
Results
Neural operators excel in improving resolution robustness; guidance weights and generation noise are crucial for posterior variance calibration. FunDPS performs exceptionally across tasks.
Applications
PosteriorBench can be used to evaluate generative inverse solvers in scientific decision-making, particularly in fields where posterior uncertainty must be considered.
Limitations & Outlook
Some methods lack effective posterior calibration at high noise levels, potentially leading to inaccurate distribution predictions. Further optimization is needed for performance under multimodal priors.
Plain Language Accessible to non-experts
Imagine you're in a kitchen, and generative models are like a smart chef assistant. Traditional methods only focus on whether a single dish is tasty, while PosteriorBench evaluates whether the entire menu is reasonable. It assesses the chef assistant's performance under different ingredients and cooking conditions, ensuring each dish meets the expected taste and style. This way, it helps us make more informed decisions when facing complex cooking challenges.
ELI14 Explained like you're 14
Imagine you're playing a game with many levels, each with different tasks. PosteriorBench is like a super guide that not only tells you how to pass the level but also reveals hidden treasures and secrets. It evaluates different characters' performance in the game, helping you choose the best one to complete the tasks. This way, you can score higher and earn more rewards in the game!
Glossary
Posterior Distribution
The probability distribution of unknown parameters given observed data, used to evaluate uncertainty and multiple solution possibilities.
Used to assess the distributional accuracy of generative inverse solvers.
Rejection Sampling
A Monte Carlo method that generates target distributions by rejecting samples that do not meet conditions.
Used to construct high-fidelity reference posteriors.
Markov Chain Monte Carlo
A sampling method that simulates complex distributions by constructing a Markov chain.
Used to generate high-fidelity reference posteriors.
Neural Operator
A machine learning method that solves partial differential equations by learning function space mappings.
Used to improve resolution robustness.
Maximum Mean Discrepancy
A statistic used to evaluate the difference between two distributions.
Used to assess posterior matching capability.
Open Questions Unanswered questions from this research
- 1 How to effectively calibrate posterior distributions at high noise levels? Current methods perform poorly in this scenario, requiring further research.
- 2 Significant distribution matching gaps remain under multimodal priors, necessitating stronger solvers.
Applications
Immediate Applications
Scientific Decision Support
Helps scientists make more informed decisions when facing uncertainty, especially in fields requiring consideration of multiple solution possibilities.
Long-term Vision
Intelligent Inverse Problem Solutions
Develop stronger generative inverse solvers capable of providing reliable distribution predictions in complex physical scenarios.
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.