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This paper introduces a continuous normalizing flow-based approach for Bayesian inference in infinite-dimensional Hilbert spaces, specifically targeting inverse problems related to partial differential equations. By employing a neural ordinary differential equation, the authors transform a simple reference measure into a complex measure that effectively incorporates prior information. The framework is validated through numerical experiments on three inverse problems, demonstrating both theoretical soundness and practical efficiency of the proposed algorithms.
Transforming simple measures into complex Bayesian priors in infinite-dimensional spaces could revolutionize how we tackle inverse problems in PDEs.
This paper addresses infinite-dimensional Bayesian inference for inverse problem of partial differential equations with model parameters in infinite-dimensional Hilbert space. To effectively incorporate prior information, we propose a novel continuous normalizing flows based infinite-dimensional model. Specifically, by introducing a well-defined neural ordinary differential equation in infinite-dimensional space, a simple reference measure can be transformed into a more complex measure which encodes the prior information. A corresponding theoretical framework is established to ensure the well-posedness of our proposed Bayesian prior in infinite-dimensional space. We also provide training methods of the prior for two distinct data settings, along with two sampling algorithms for the resulting Bayesian posterior. The proposed framework is applied to three representative inverse problems: the simple smooth inverse problem, inverse scattering problem, and the inverse heat conduction problem. Numerical experiments support the theoretical analysis and demonstrate the efficiency of the proposed algorithms.
