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Life sciences · Preprint
arXiv · September 3, 2026
Raises a question worth testing. It does not answer one.
This is a preprint proposing an infinite-dimensional continuous normalizing flow method for Bayesian prior learning in inverse problems governed by PDEs. The authors establish a theoretical framework and demonstrate feasibility on three mathematical examples (smooth, scattering, and heat conduction problems), but the work remains at the proof-of-concept stage and has not been peer reviewed.
Theoretical framework development with numerical experiments. Intervention: Continuous normalizing flow based infinite-dimensional Bayesian prior model using neural ordinary differential equations.
A neural ODE framework is introduced to transform a simple reference measure into a complex prior in infinite-dimensional Hilbert space Theoretical well-posedness of the proposed Bayesian prior is established Training methods and sampling algorithms are developed for two data settings
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This is a methodological paper proposing a novel computational framework for Bayesian inverse problems; it presents theoretical development and numerical validation but addresses mathematical technique rather than a clinical or applied outcome.
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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.
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