  Dynamical systems that evolve continuously over time are ubiquitous
throughout science and engineering. Machine learning (ML) provides data-driven
approaches to model and predict the dynamics of such systems. A core issue with
this approach is that ML models are typically trained on discrete data, using
ML methodologies that are not aware of underlying continuity properties. This
results in models that often do not capture any underlying continuous dynamics
-- either of the system of interest, or indeed of any related system. To
address this challenge, we develop a convergence test based on numerical
analysis theory. Our test verifies whether a model has learned a function that
accurately approximates an underlying continuous dynamics. Models that fail
this test fail to capture relevant dynamics, rendering them of limited utility
for many scientific prediction tasks; while models that pass this test enable
both better interpolation and better extrapolation in multiple ways. Our
results illustrate how principled numerical analysis methods can be coupled
with existing ML training/testing methodologies to validate models for science
and engineering applications.
