Zecheng Gan

Tianhao Hu, Zecheng Gan

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.

Haoze Song, Zhihao Li, Xiaobo Zhang et al.

Neural Operators have emerged as powerful tools for learning mappings between function spaces. Among them, the kernel integral operator has been widely validated on universally approximating various operators. Although many advancements following this definition have developed effective modules to better approximate the kernel function defined on the original domain (with $d$ dimensions, $d=1, 2, 3\dots$), the unclarified evolving mechanism in the embedding spaces blocks researchers' view to design neural operators that can fully capture the target system evolution. Drawing on the Schrödingerisation method in quantum simulations of partial differential equations (PDEs), we elucidate the linear evolution mechanism in neural operators. Based on that, we redefine neural operators on a new $d+1$ dimensional domain. Within this framework, we implement a Schrödingerised Kernel Neural Operator (SKNO) aligning better with the $d+1$ dimensional evolution. In experiments, the $d+1$ dimensional evolving designs in our SKNO consistently outperform other baselines across ten benchmarks of increasing difficulty, ranging from the simple 1D heat equation to the highly nonlinear 3D Rayleigh-Taylor instability. We also validate the resolution-invariance of SKNO on mixing-resolution training and zero-shot super-resolution tasks. In addition, we show the impact of different lifting and recovering operators on the prediction within the redefined NO framework, reflecting the alignment between our model and the underlying $d+1$ dimensional evolution.