A spectral-subspace-augmented POD-Galerkin method for parametrized PDEs with limited snapshot data

Parametrized partial differential equations (PDEs) arise in many-query simulation, optimization, control, and uncertainty quantification, where repeated full-order solves restrict the number of high-fidelity snapshots that can be generated. This limitation is particularly pronounced in computational energy science, where multiscale models of porous-media flow, transport, and energy materials often make large snapshot datasets impractical. Proper orthogonal decomposition (POD) constructs compact reduced bases from solution snapshots, but it may exhibit limited out-of-sample predictive capability when the snapshots insufficiently sample the solution manifold. To address this limitation, we propose a spectral-subspace-augmented POD-Galerkin method (SS-POD) tailored to limited-data regimes. SS-POD starts from a problem-adapted spectral approximation space, partitions it into orthogonal subspaces, and performs POD locally on the projected snapshot matrices. An energy-balancing rule determines the spectral partition so that the resulting local POD problems are assigned comparable amounts of snapshot energy. For nonlinear parametrized PDEs, SS-POD is coupled with the discrete empirical interpolation method (DEIM). Numerical experiments show that SS-POD improves out-of-sample accuracy over standard POD-Galerkin while retaining compact reduced bases in limited-snapshot regimes. In particular, for a Laplace-Beltrami problem on the unit sphere with only 5 snapshots, SS-POD achieves a relative error of $3.9*10^{-8}$ using 91 basis functions, whereas the standard POD error saturates at $7.8*10^{-4}$ and the spectral-Galerkin method requires 256 basis functions for comparable accuracy. These results indicate that SS-POD provides an effective strategy for high-fidelity reduced-order modeling from limited snapshot data.