  Enabling fast and accurate physical simulations with data has become an
important area of computational physics to aid in inverse problems,
design-optimization, uncertainty quantification, and other various
decision-making applications. This paper presents a data-driven framework for
parametric latent space dynamics identification procedure that enables fast and
accurate simulations. The parametric model is achieved by building a set of
local latent space model and designing an interaction among them. An individual
local latent space dynamics model achieves accurate solution in a trust region.
By letting the set of trust region to cover the whole parameter space, our
model shows an increase in accuracy with an increase in training data. We
introduce two different types of interaction mechanisms, i.e., point-wise and
region-based approach. Both linear and nonlinear data compression techniques
are used. We illustrate the framework of Latent Space Dynamics Identification
(LaSDI) enable a fast and accurate solution process on various partial
differential equations, i.e., Burgers' equations, radial advection problem, and
nonlinear heat conduction problem, achieving $O(100)$x speed-up and $O(1)\%$
relative error with respect to the corresponding full order models.
