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

Inverse problems are ubiquitous in science, engineering, and medicine, in particular for problems where observations provide only indirect or incomplete information about a system [1]. Inverse problems are central in a wide range of applications such as flow field reconstruction [2, 3, 4], data assimilation [5], medical imaging [6, 7], and parameters estimation of material properties [8, 9, 10]. A particularly challenging class of inverse problems arises when the forward model is governed by ordinary differential equations (ODEs) or partial differential equations (PDEs) [11]. Incorporating physical knowledge through this approach reduces the space of possible solutions, avoiding the need for arbitrary regularization as is often the case in inverse problems [12, 13, 14]. However, this approach can suffer from the high dimensionality of the problem, stiffness, noisy measurements, and sensitivity to parameters. In particular, quantifying the uncertainties of solutions is challenging with standard techniques for inverse PDE problems such as Bayesian inference [15, 14], variational methods [16], ensemble Kalman methods [17], and adjoint-based optimization [18], which can be limited with issues of scalability, robustness, and computational cost. In parallel, operator learning approaches based on DeepONets [19], Fourier neural operators [20], and graph neural networks [21, 22] have been extended to inverse problems and uncertainty quantification [23, 24, 25]. Similar Bayesian techniques rely on training data to build prior knowledge [26]. However, the application of these operator learning techniques to large-scale problems is limited by the cost of their training and the difficulty of generating sufficient high-fidelity data.