Smooth and Sparse Latent Dynamics in Operator Learning with Jerk
  Regularization
Xiaoyu Xie Saviz Mowlavi Mouhacine Benosman
Spatiotemporal modeling is critical for understanding complex systems across
various scientific and engineering disciplines, but governing equations are
often not fully known or computationally intractable due to inherent system
complexity. Data-driven reduced-order models (ROMs) offer a promising approach
for fast and accurate spatiotemporal forecasting by computing solutions in a
compressed latent space. However, these models often neglect temporal
correlations between consecutive snapshots when constructing the latent space,
leading to suboptimal compression, jagged latent trajectories, and limited
extrapolation ability over time. To address these issues, this paper introduces
a continuous operator learning framework that incorporates jerk regularization
into the learning of the compressed latent space. This jerk regularization
promotes smoothness and sparsity of latent space dynamics, which not only
yields enhanced accuracy and convergence speed but also helps identify
intrinsic latent space coordinates. Consisting of an implicit neural
representation (INR)-based autoencoder and a neural ODE latent dynamics model,
the framework allows for inference at any desired spatial or temporal
resolution. The effectiveness of this framework is demonstrated through a
two-dimensional unsteady flow problem governed by the Navier-Stokes equations,
highlighting its potential to expedite high-fidelity simulations in various
scientific and engineering applications.