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The paper addresses Bayesian inferences in inverse problems with uncertaintyquantification involving a computationally expensive forward map associatedwith solving a partial differential equations. To mitigate the computationalcost, the paper proposes a new surrogate model informed by the physics of theproblem, specifically when the forward map involves solving a linear ellipticpartial differential equation. The study establishes the consistency of theposterior distribution for this surrogate model and demonstrates itseffectiveness through numerical examples with synthetic data. The resultsindicate a substantial improvement in computational speed, reducing theprocessing time from several months with the exact forward map to a fewminutes, while maintaining negligible loss of accuracy in the posteriordistribution.

A Physics Based Surrogate Model in Bayesian Uncertainty Quantification involving Elliptic PDEs