A probabilistic white‐box model for PDE constrained inverse problems

Instead of employing deterministic solvers in a black‐box fashion, we seek to address the inherent challenges of uncertainty quantification by restating the solution of a PDE as a problem of probabilistic inference. In doing so, state variables are treated as random fields, constrained or mutually entangled by underlying physical laws. The resulting fully probabilistic model expresses the PDE operator as prior knowledge and allows the solution of the PDE‐constrained inverse problem simply by introducing the observed data as an additional source of information. We demonstrate the proposed framework for the solution of PDE‐constrained inverse problems in a solid mechanics setting and employ adjoint‐free, second‐order methods to infer the joint posterior over the Hilbert space of displacement, stress and unknown material parameters. The method generalizes to nonlinear problems and higher‐order function spaces, while the Maximum‐A‐Posteriori estimate recovers the results obtained by the mixed Finite Element method derived from the Hellinger‐Reissner variational principle.