[2510.15664] Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss

Statistics > Methodology arXiv:2510.15664 (stat) [Submitted on 17 Oct 2025 (v1), last revised 5 Mar 2026 (this version, v2)] Title:Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss Authors:Lucas Amoudruz, Sergey Litvinov, Costas Papadimitriou, Petros Koumoutsakos View a PDF of the paper titled Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss, by Lucas Amoudruz and Sergey Litvinov and Costas Papadimitriou and Petros Koumoutsakos View PDF HTML (experimental) Abstract:Inverse problems are crucial for many applications in science, engineering and medicine that involve data assimilation, design, and imaging. Their solution infers the parameters or latent states of a complex system from noisy data and partially observable processes. When measurements are an incomplete or indirect view of the system, additional knowledge is required to accurately solve the inverse problem. Adopting a physical model of the system in the form of partial differential equations (PDEs) is a potent method to close this gap. In particular, the method of optimizing a discrete loss (ODIL) has shown great potential in terms of robustness and computational cost. In this work, we introduce B-ODIL, a Bayesian extension of ODIL, that integrates the PDE loss of ODIL as prior knowledge and combines it with a likelihood describing the data. B-ODIL employs a Bayesian formulation of PDE-based inverse problems to infer solutions wit...