POD-CNF: reduced order modeling of parametrized stochastic PDEs via normalizing flows and proper orthogonal decomposition

Reduced-order models (ROMs) learned from trusted numerical simulations enable the real-time simulation of complex physical systems, making computationally prohibitive tasks-such as uncertainty quantification and optimal control-tractable. In particular, data-driven ROMs are opening the opportunity to learn from both numerical simulations and experimental data. However, integrating these sources of information is far from straightforward. Real-world measurements often involve multiple sources of uncertainty, including non-Gaussian noise, latent heterogeneities, unknown or hidden system parameters, and multiscale phenomena occurring below the resolution of the data. Consequently, real-world data are better described by parametrized stochastic partial differential equations, for which the development of effective ROMs remains relatively unexplored. In this work, we introduce POD-CNF, a data-driven generative ROM for efficient uncertainty quantification of parametrized high-dimensional systems. The approach combines projection-based model reduction and deep generative modeling. First, Proper Orthogonal Decomposition (POD) is employed to obtain a low-dimensional representation of the solution manifold. A Conditional Normalizing Flow (CNF) is then trained to approximate the density of the reduced coefficients conditionally on the input parameters. This formulation allows the surrogate to capture complex, multimodal output distributions, going beyond standard regression-based ROMs and least-squares predictions. We assess the effectiveness of POD-CNF through numerical experiments, demonstrating its ability to accurately reproduce output distributions while substantially reducing computational costs. Finally, a proof-of-concept application illustrates how the generative paradigm facilitates integration into Bayesian inverse problems, highlighting promising directions for future research.