Speaker
Description
We introduce a new class of generative deep learning based reduced-order models (DL-ROMs) for uncertainty quantification and data-driven modeling of complex physical systems with hidden physics and/or partially observed parameters. Indeed, while DL-ROMs have been extensively shown capable of learning from numerical simulations, existing approaches are predominantly deterministic and assume full knowledge of the system, including physics and parameters. Although this is not problematic in traditional settings, we argue that some challenges naturally arise when trying to extend the idea to ROMs that can learn both from numerical simulations and real data. In fact, real-world phenomena typically deviate from simplified model-based simulations due to external uncertainties and an intrinsic lack of knowledge about the true nature of the system. To overcome these limitations, we develop probabilistic extensions of established DL-ROM architectures, including POD–NN and autoencoder-based models, designed to learn probability distributions over solution manifolds, thus capturing the additional variability induced by parametric uncertainty, model inadequacy, and noise in the observed data. In the same spirit of recent trends in the literature, our construction combines traditional DL-ROM architectures with ideas from generative AI,. We train our probabilistic ROMs using a variational loss, resulting in computationally efficient surrogates capable of producing multiple physically plausible solutions for any given value of the problem parameters. A rigorous theoretical analysis accompanies the methodological developments, addressing convergence of the learned distributions toward the laws of true solutions. The analysis relies on tools from empirical process theory, optimal transport, and approximation theory for deep neural networks. We demonstrate the proposed approach on stationary and time-dependent partial differential equations, including applications to flow in porous media and elastic deformation under uncertainty. The results show that generative probabilistic ROMs can offer a new interesting perspective, allowing us to reduce the gap between numerical simulations and real-world phenomena.