Physics-Informed Deep Orthogonal Decomposition with Calibrated Uncertainty Quantification for Reduced-Order Modelling of Parametrized Partial Differential Equations

Abstract

Solving parametrized partial differential equations thousands of times is the bottleneck of design, control, inversion, and digital-twin workflows. Reduced-order models answer it by replacing the expensive discretization with a fast surrogate. Adaptive-basis surrogates, such as the deep orthogonal decomposition, break the slow decay of the Kolmogorov width for problems whose parameters interact with space. This work develops physics-informed deep orthogonal decomposition, which supervises a continuously adaptive basis and the reduced coefficients with the discrete equation residual at unlabelled collocation parameters. The surrogate learns from points the high-fidelity solver never visits. A calibrated uncertainty layer couples a heteroscedastic aleatoric estimate with a deep-ensemble epistemic estimate. The linear decoder is proven to transport this uncertainty to the physical field exactly and without inflation. A solver-free trust indicator fuses the predictive variance, the equation residual, and a Grassmann distance to the training manifold, backed by an exact error decomposition, two-sided residual bounds, and a distribution-free coverage guarantee.