Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces

2026-03-15

AI Summary

This work addresses the computational burden and poor scalability of posterior sampling in Bayesian inverse problems over function spaces, particularly in PDE-driven settings. The authors propose an amortized sampling approach based on one-step generative transport, which leverages a prior-aligned anisotropic Gaussian reference distribution and constructs a fully conditional transport map parameterized by a neural operator. This map directly transforms reference samples into approximate posterior samples, circumventing repeated PDE solves. The method requires no MCMC distillation and is trained solely on prior samples paired with simulated observations, while guaranteeing stability and Lipschitz regularity in the function space limit. Experiments demonstrate that the model generates high-quality posterior samples at 64×64 resolution in approximately 10⁻³ seconds, achieving significantly higher efficiency than conventional MCMC methods while accurately recovering key posterior statistics.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime based on one-step generative transport. Building on the Mean Flows, we learn a fully conditional amortized sampler with a neural-operator backbone that maps a reference Gaussian noise to approximate posterior samples. We show that while white-noise references may be admissible at fixed discretization, they become incompatible with the function-space limit, leading to instability in inference for Bayesian problems arising from PDEs. To address this issue, we adopt a prior-aligned anisotropic Gaussian reference distribution and establish the Lipschitz regularity of the resulting transport. Our method is not distilled from MCMC: training relies only on prior samples and simulated partial and noisy observations. Once trained, it generates a $64\times64$ posterior sample in $\sim 10^{-3}$s, avoiding the repeated PDE solves of MCMC while matching key posterior summaries.

Problem

• Bayesian inverse problems
• function spaces
• generative modeling
• instability
• PDEs

Innovation

• one-step generative modeling
• function-space Bayesian inference
• neural operator
• anisotropic Gaussian reference
• amortized posterior sampling