Spatio-temporal Latent Denoising Diffusion Probabilistic Models for Reduced-order Modeling of Parametrized Dynamical Systems
Many scientific problems require accurate modeling of complex physical phenomena, such as fluid dynamics or climate modeling. These phenomena often result in high-dimensional and thus computationally expensive computational models that can limit their application to real-time and multi-query problems. Model-order reduction (MOR) is an approach that seeks to approximate full-order models (FOMs) using reduced-order models (ROMs), trading a minor reduction in accuracy for a major reduction in computational cost. In this work, we propose a non-intrusive MOR method using generative machine learning by means of denoising diffusion probabilistic models (DDPMs) for generating solutions of the dynamical systems under different instances of their parameters. Unlike conventional DDPMs, which often operate purely in the spatial domain, we aim to generate spatio-temporal solutions to improve quality and temporal coherence. In addition, we embed the DDPM in a latent space obtained by sequentially applying proper orthogonal decomposition and an autoencoder to reduce the data dimensionality and the computational cost of the DDPM. We test our approach on a parametrized 2D fluid flow around an obstacle. The numerical experiments demonstrate that our latent DDPM can (i) produce accurate and temporally coherent solutions, (ii) achieve strong generalization capabilities to scenarios involving unseen parameter values, and (iii) extrapolate in time beyond the training horizon.