Residual correction for POD-Galerkin reduced order models via conditional VAEs: Application to the bias-aware solution of forward and inverse problems

POD-Galerkin reduced order models (PGROMs) significantly accelerate the solution of parameterized partial differential equations (PDEs) but can suffer from accuracy limitations across the parameter space. We propose a hybrid framework that augments a PGROM with a data-driven residual corrector based on a conditional variational autoencoder (cVAE). In the offline stage, we (i) construct a standard PGROM from high-fidelity snapshots and (ii) train a cVAE on PGROM residuals defined as the differences between PGROM approximations and the corresponding highfidelity solutions. The cVAE learns a mapping from time-parameter pairs to residual fields. In the online stage, for a new parameter instance, the PGROM provides a baseline approximation while the cVAE decoder predicts the residual; adding the two yields a corrected, high-accuracy solution. The effectiveness of the proposed hybrid approach is demonstrated on three parameterized PDE problems: a heat dissipation problem with varying thermal conductivity, a pollutant transport problem with varying convection velocity, and a plane-strain elastodynamics problem with a circular inclusion whose stiffness and density are parameterized.