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Numerically solving partial differential equations (PDEs) can be challengingand computationally expensive. This has led to the development of reduced-ordermodels (ROMs) that are accurate but faster than full order models (FOMs).Recently, machine learning advances have enabled the creation of non-linearprojection methods, such as Latent Space Dynamics Identification (LaSDI). LaSDImaps full-order PDE solutions to a latent space using autoencoders and learnsthe system of ODEs governing the latent space dynamics. By interpolating andsolving the ODE system in the reduced latent space, fast and accurate ROMpredictions can be made by feeding the predicted latent space dynamics into thedecoder. In this paper, we introduce GPLaSDI, a novel LaSDI-based frameworkthat relies on Gaussian process (GP) for latent space ODE interpolations. UsingGPs offers two significant advantages. First, it enables the quantification ofuncertainty over the ROM predictions. Second, leveraging this predictionuncertainty allows for efficient adaptive training through a greedy selectionof additional training data points. This approach does not require priorknowledge of the underlying PDEs. Consequently, GPLaSDI is inherentlynon-intrusive and can be applied to problems without a known PDE or itsresidual. We demonstrate the effectiveness of our approach on the Burgersequation, Vlasov equation for plasma physics, and a rising thermal bubbleproblem. Our proposed method achieves between 200 and 100,000 times speed-up,with up to 7% relative error.

GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder