  Learning dynamics from dissipative chaotic systems is notoriously difficult
due to their inherent instability, as formalized by their positive Lyapunov
exponents, which exponentially amplify errors in the learned dynamics. However,
many of these systems exhibit ergodicity and an attractor: a compact and highly
complex manifold, to which trajectories converge in finite-time, that supports
an invariant measure, i.e., a probability distribution that is invariant under
the action of the dynamics, which dictates the long-term statistical behavior
of the system. In this work, we leverage this structure to propose a new
framework that targets learning the invariant measure as well as the dynamics,
in contrast with typical methods that only target the misfit between
trajectories, which often leads to divergence as the trajectories' length
increases. We use our framework to propose a tractable and sample efficient
objective that can be used with any existing learning objectives. Our Dynamics
Stable Learning by Invariant Measure (DySLIM) objective enables model training
that achieves better point-wise tracking and long-term statistical accuracy
relative to other learning objectives. By targeting the distribution with a
scalable regularization term, we hope that this approach can be extended to
more complex systems exhibiting slowly-variant distributions, such as weather
and climate models.
