Semi-supervised invertible neural operators for Bayesian inverse problems

Abstract

Neural Operators offer a powerful, data-driven tool for solving parametric PDEs as they can represent maps between infinite-dimensional function spaces. In this work, we employ physics-informed Neural Operators in the context of high-dimensional, Bayesian inverse problems. Traditional solution strategies necessitate an enormous, and frequently infeasible, number of forward model solves, as well as the computation of parametric derivatives. In order to enable efficient solutions, we extend Deep Operator Networks (DeepONets) by employing a RealNVP architecture which yields an invertible and differentiable map between the parametric input and the branch-net output. This allows us to construct accurate approximations of the full posterior, irrespective of the number of observations and the magnitude of the observation noise, without any need for additional forward solves nor for cumbersome, iterative sampling procedures. We demonstrate the efficacy and accuracy of the proposed methodology in the context of inverse problems for three benchmarks: an anti-derivative equation, reaction-diffusion dynamics and flow through porous media.

Efficient Bayesian inference using physics-informed invertible neural networks for inverse problems

“…Kaltenbach et al [43] proposed the utilization of an INN to construct the branch net within DeepONet, thereby achieving simultaneous learning of deterministic forward and inverse operators between the solution function u and the parameter field κ. This model is referred to as invertible DeepONet.…”

“…As discussed in section 2.3 and at the beginning of section 3, our proposed PI-INN can be regarded as an extension of the invertible DeepONet [43]. The primary difference lies in how the INN is employed.…”

“…Furthermore, similar to the invertible DeepONet, in scenarios where the solution function u is not precisely known but only observed with noise at certain locations, we can calculate the unnormalized posterior distribution of (λ u , z) based on their joint prior given by PI-INN, along with the likelihood of observations. Subsequently, following the approach outlined in [43], we can utilize the variational Bayesian method to estimate and sample the posterior distributions of u and κ. Further discussion and investigation on this topic will be conducted in future research.…”

“…Furthermore, models based on invertible neural networks (INNs) [35][36][37] have exhibited potential in Bayesian inference. This is attributed to their efficiency and accuracy advantages in both sampling and density estimation of complex target distributions and dynamics through bijective transformations, as evidenced in the works of [38][39][40][41][42][43][44][45][46]. Nonetheless, certain limitations persist within the current research.…”

“…For instance, INN models in [38][39][40][41] demand substantial quantities of labeled data for pretraining, which is often unavailable in many practical inverse problems. The invertible DeepONet introduced in [43] alleviates the need for labeled data, allowing for the construction of accurate approximations of the full Bayesian posterior in scenarios with noisy observational data. However, it does not quantify the aleatoric uncertainty inherent in a physical system, which pertains to systematic uncertainties.…”