  Machine learning models have emerged as powerful tools in physics and
engineering. Although flexible, a fundamental challenge remains on how to
connect new machine learning models with known physics. In this work, we
present an autoencoder with latent space penalization, which discovers finite
dimensional manifolds underlying the partial differential equations of physics.
We test this method on the Kuramoto-Sivashinsky (K-S), Korteweg-de Vries (KdV),
and damped KdV equations. We show that the resulting optimal latent space of
the K-S equation is consistent with the dimension of the inertial manifold. The
results for the KdV equation imply that there is no reduced latent space, which
is consistent with the truly infinite dimensional dynamics of the KdV equation.
In the case of the damped KdV equation, we find that the number of active
dimensions decreases with increasing damping coefficient. We then uncover a
nonlinear basis representing the manifold of the latent space for the K-S
equation.
