Weak-Form Latent Space Dynamics Identification
April Tran Xiaolong He Daniel A. Messenger Youngsoo Choi David M. Bortz
Recent work in data-driven modeling has demonstrated that a weak formulation
of model equations enhances the noise robustness of a wide range of
computational methods. In this paper, we demonstrate the power of the weak form
to enhance the LaSDI (Latent Space Dynamics Identification) algorithm, a
recently developed data-driven reduced order modeling technique.
  We introduce a weak form-based version WLaSDI (Weak-form Latent Space
Dynamics Identification). WLaSDI first compresses data, then projects onto the
test functions and learns the local latent space models. Notably, WLaSDI
demonstrates significantly enhanced robustness to noise. With WLaSDI, the local
latent space is obtained using weak-form equation learning techniques. Compared
to the standard sparse identification of nonlinear dynamics (SINDy) used in
LaSDI, the variance reduction of the weak form guarantees a robust and precise
latent space recovery, hence allowing for a fast, robust, and accurate
simulation. We demonstrate the efficacy of WLaSDI vs. LaSDI on several common
benchmark examples including viscid and inviscid Burgers', radial advection,
and heat conduction. For instance, in the case of 1D inviscid Burgers'
simulations with the addition of up to 100% Gaussian white noise, the relative
error remains consistently below 6% for WLaSDI, while it can exceed 10,000% for
LaSDI. Similarly, for radial advection simulations, the relative errors stay
below 15% for WLaSDI, in stark contrast to the potential errors of up to
10,000% with LaSDI. Moreover, speedups of several orders of magnitude can be
obtained with WLaSDI. For example applying WLaSDI to 1D Burgers' yields a 140X
speedup compared to the corresponding full order model.
  Python code to reproduce the results in this work is available at
(https://github.com/MathBioCU/PyWSINDy_ODE) and
(https://github.com/MathBioCU/PyWLaSDI).