Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems
This method addresses the computational inefficiency of traditional Bayesian inverse problem solutions for parametric PDEs, though it appears incremental as an extension of DeepONets with RealNVP.
The authors tackled high-dimensional Bayesian inverse problems by developing a semi-supervised invertible neural operator that approximates full posteriors without additional forward solves or iterative sampling, achieving accurate results across three benchmarks including reaction-diffusion and porous media flow.
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.