Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces
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| Format: | Preprint |
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2026
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| _version_ | 1866910275137961984 |
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| author | Cheng, Zilan Wang, Li-Lian Wang, Zhongjian |
| author_facet | Cheng, Zilan Wang, Li-Lian Wang, Zhongjian |
| contents | We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime. Based on one-step generative transport, the method learns an amortized neural operator whose pushforward of a Gaussian source approximates the posterior distribution conditioned on each new observation. We show that white-noise sources are incompatible with the function-space limit, and therefore adopt a prior-aligned GRF as the source. We justify this choice through the Lipschitz regularity of the resulting one-step conditional posterior transport and numerical experiments on linear inverse and PDE-based inverse problems. The method is not distilled from MCMC: it is trained only with prior samples and simulated partial noisy observations. Once trained, it generates a $64\times64$ posterior sample in $\sim 10^{-3}$s, avoiding repeated forward-model evaluations in MCMC and repeated network evaluations in multistep generative samplers while matching key posterior summaries. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14798 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces Cheng, Zilan Wang, Li-Lian Wang, Zhongjian Machine Learning Numerical Analysis We propose a machine-learning algorithm for Bayesian inverse problems in the function-space regime. Based on one-step generative transport, the method learns an amortized neural operator whose pushforward of a Gaussian source approximates the posterior distribution conditioned on each new observation. We show that white-noise sources are incompatible with the function-space limit, and therefore adopt a prior-aligned GRF as the source. We justify this choice through the Lipschitz regularity of the resulting one-step conditional posterior transport and numerical experiments on linear inverse and PDE-based inverse problems. The method is not distilled from MCMC: it is trained only with prior samples and simulated partial noisy observations. Once trained, it generates a $64\times64$ posterior sample in $\sim 10^{-3}$s, avoiding repeated forward-model evaluations in MCMC and repeated network evaluations in multistep generative samplers while matching key posterior summaries. |
| title | Preconditioned One-Step Generative Modeling for Bayesian Inverse Problems in Function Spaces |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2603.14798 |