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

2025-10-17

AI Summary

We address inverse problems governed by partial differential equations (PDEs), where unknown parameters and latent states must be jointly estimated—and their uncertainties quantified—from noisy, sparse, and indirect observations. To this end, we propose Bayesian-regularized Physics-Informed Deep Learning (B-ODIL), the first method to embed the discrete PDE residual loss of ODIL into a Bayesian prior framework. B-ODIL unifies physics-based modeling, variational inference, and Markov Chain Monte Carlo (MCMC) sampling to rigorously estimate full posterior distributions over both parameters and latent states. Unlike conventional approaches, it eliminates reliance on surrogate models or explicit Jacobian computations, thereby substantially improving both uncertainty quantification accuracy and computational efficiency. We validate B-ODIL on high-dimensional synthetic benchmarks and a clinically informed 3D glioblastoma growth model fitted to MRI data. Results demonstrate high-fidelity reconstruction of tumor concentration fields and robust uncertainty characterization, establishing a new paradigm for physics-driven, interpretable AI modeling.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Probabilistic Inference
• Intelligent Robots: State Estimation

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 with quantified uncertainties. We demonstrate the capabilities of B-ODIL in a series of synthetic benchmarks involving PDEs in one, two, and three dimensions. We showcase the application of B-ODIL in estimating tumor concentration and its uncertainty in a patient's brain from MRI scans using a three-dimensional tumor growth model.

Problem

• Bayesian inference for PDE-based inverse problems with uncertainties
• Optimizing discrete loss to infer parameters from noisy data
• Integrating PDE models as priors with data likelihood

Innovation

• Bayesian extension of ODIL for PDE-based inverse problems
• Integrates PDE loss as prior with data likelihood
• Infers solutions with quantified uncertainties in benchmarks