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

TL;DR

This work presents B-ODIL, a Bayesian extension of ODIL that incorporates PDE residuals as a prior and couples them with data likelihood to perform inverse problems with quantified uncertainty. By employing a Laplace approximation (and mode-based MAP approaches), it achieves scalable uncertainty estimation in high-dimensional PDE settings, validated across harmonic oscillator, diffusion, and reaction-diffusion benchmarks, and demonstrated on a 3D brain tumor growth scenario using MRI data. The method preserves consistency with ODIL at the MAP, while enabling posterior uncertainty characterization via a Gaussian approximation or low-dimensional marginalizations. The results show accurate uncertainty quantification and practical applicability to large-scale PDE problems, including clinically relevant brain tumor imaging where uncertainty informs treatment planning and margin decisions.

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.

Figures (9)

• Figure 1: Left: Prediction of the position (top row) and velocity (bottom row) of the oscillator given the data (crosses) using the UQ-ODIL framework with Laplace approximation (left column) and HMC (right column). The shaded area denotes the 5 to 95% quantiles of the posterior, and the solid line denotes the posterior mean. The dashed line represents the underlying process that was used to generate the data. Right: Covariance matrices of the full solution $(x_1,\dots, x_N, v_1, \dots, v_N)$ obtained from the Laplace approximation (left column) and estimated from HMC samples (right column).
• Figure 2: Left: Covariance matrices of the posterior distribution of the full solution $(\omega^2, x_1,\dots, x_N, v_1, \dots, v_N)$ obtained with the Laplace and HMC methods. Right: Marginal posterior probability of $\omega^2$ obtained from HMC (histogram), Laplace (solid line), mode approximation given by \ref{['eq:posterior:params:approx:MAP']} (dashed line), and exact solution (dots).
• Figure 3: Laplace approximation applied to the diffusion equation. From left to right: Data used for inferring the field; Exact solution, used to generate the data; MAP solution of the diffusion problem given the data; spread of uncertainty given by the Laplace approximation (5-95% quantiles).
• Figure 4: Reaction diffusion data. (A,D) Diffusion field $D(x, y)$ for cases 1 and 2, respectively black and grey regions have values $D=0.005$ and $D=0.1$, respectively. (B,C) Data with $\sigma=0.01$ and $\sigma=0.05$, respectively, for case 1. (E,F) Data with $\sigma=0.01$ and $\sigma=0.05$, respectively, for case 2.
• Figure 5: Samples from the posterior distribution of the initial conditions $x_0$, $y_0$ obtained with TMCMC. Colors indicate the rank of the log-posterior value of each sample. (A) case 1, $\sigma=0.05$, $\lambda_\mathrm{PDE} = 10$, $\lambda_\mathrm{IC}=100$. (B) case 1, $\sigma=0.01$, $\lambda_\mathrm{PDE} = 100$, $\lambda_\mathrm{IC}=1000$. (C) case 1, $\sigma=0.01$, $\lambda_\mathrm{PDE} = 10$, $\lambda_\mathrm{IC}=100$. (D) case 2, $\sigma=0.05$, $\lambda_\mathrm{PDE} = 10$, $\lambda_\mathrm{IC}=100$. (E) case 2, $\sigma=0.01$, $\lambda_\mathrm{PDE} = 100$, $\lambda_\mathrm{IC}=1000$. (F) case 2, $\sigma=0.01$, $\lambda_\mathrm{PDE} = 10$, $\lambda_\mathrm{IC}=100$.