  Numerical solvers of partial differential equations (PDEs) have been widely
employed for simulating physical systems. However, the computational cost
remains a major bottleneck in various scientific and engineering applications,
which has motivated the development of reduced-order models (ROMs). Recently,
machine-learning-based ROMs have gained significant popularity and are
promising for addressing some limitations of traditional ROM methods,
especially for advection dominated systems. In this chapter, we focus on a
particular framework known as Latent Space Dynamics Identification (LaSDI),
which transforms the high-fidelity data, governed by a PDE, to simpler and
low-dimensional latent-space data, governed by ordinary differential equations
(ODEs). These ODEs can be learned and subsequently interpolated to make ROM
predictions. Each building block of LaSDI can be easily modulated depending on
the application, which makes the LaSDI framework highly flexible. In
particular, we present strategies to enforce the laws of thermodynamics into
LaSDI models (tLaSDI), enhance robustness in the presence of noise through the
weak form (WLaSDI), select high-fidelity training data efficiently through
active learning (gLaSDI, GPLaSDI), and quantify the ROM prediction uncertainty
through Gaussian processes (GPLaSDI). We demonstrate the performance of
different LaSDI approaches on Burgers equation, a non-linear heat conduction
problem, and a plasma physics problem, showing that LaSDI algorithms can
achieve relative errors of less than a few percent and up to thousands of times
speed-ups.
