GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics
  Identification through Deep Autoencoder
Christophe Bonneville Youngsoo Choi Debojyoti Ghosh Jonathan L. Belof
Numerically solving partial differential equations (PDEs) can be challenging
and computationally expensive. This has led to the development of reduced-order
models (ROMs) that are accurate but faster than full order models (FOMs).
Recently, machine learning advances have enabled the creation of non-linear
projection methods, such as Latent Space Dynamics Identification (LaSDI). LaSDI
maps full-order PDE solutions to a latent space using autoencoders and learns
the system of ODEs governing the latent space dynamics. By interpolating and
solving the ODE system in the reduced latent space, fast and accurate ROM
predictions can be made by feeding the predicted latent space dynamics into the
decoder. In this paper, we introduce GPLaSDI, a novel LaSDI-based framework
that relies on Gaussian process (GP) for latent space ODE interpolations. Using
GPs offers two significant advantages. First, it enables the quantification of
uncertainty over the ROM predictions. Second, leveraging this prediction
uncertainty allows for efficient adaptive training through a greedy selection
of additional training data points. This approach does not require prior
knowledge of the underlying PDEs. Consequently, GPLaSDI is inherently
non-intrusive and can be applied to problems without a known PDE or its
residual. We demonstrate the effectiveness of our approach on the Burgers
equation, Vlasov equation for plasma physics, and a rising thermal bubble
problem. Our proposed method achieves between 200 and 100,000 times speed-up,
with up to 7% relative error.