  Reduced order modeling lowers the computational cost of solving PDEs by
learning a low-order spatial representation from data and dynamically evolving
these representations using manifold projections of the governing equations.
While commonly used, linear subspace reduced-order models (ROMs) are often
suboptimal for problems with a slow decay of Kolmogorov $n$-width, such as
advection-dominated fluid flows at high Reynolds numbers. There has been a
growing interest in nonlinear ROMs that use state-of-the-art representation
learning techniques to accurately capture such phenomena with fewer degrees of
freedom. We propose smooth neural field ROM (SNF-ROM), a nonlinear reduced
modeling framework that combines grid-free reduced representations with
Galerkin projection. The SNF-ROM architecture constrains the learned ROM
trajectories to a smoothly varying path, which proves beneficial in the
dynamics evaluation when the reduced manifold is traversed in accordance with
the governing PDEs. Furthermore, we devise robust regularization schemes to
ensure the learned neural fields are smooth and differentiable. This allows us
to compute physics-based dynamics of the reduced system nonintrusively with
automatic differentiation and evolve the reduced system with classical
time-integrators. SNF-ROM leads to fast offline training as well as enhanced
accuracy and stability during the online dynamics evaluation. Numerical
experiments reveal that SNF-ROM is able to accelerate the full-order
computation by up to $199\times$. We demonstrate the efficacy of SNF-ROM on a
range of advection-dominated linear and nonlinear PDE problems where we
consistently outperform state-of-the-art ROMs.
