  We extend the methodology in [Yang et al., 2023] to learn autonomous
continuous-time dynamical systems from invariant measures. The highlight of our
approach is to reformulate the inverse problem of learning ODEs or SDEs from
data as a PDE-constrained optimization problem. This shift in perspective
allows us to learn from slowly sampled inference trajectories and perform
uncertainty quantification for the forecasted dynamics. Our approach also
yields a forward model with better stability than direct trajectory simulation
in certain situations. We present numerical results for the Van der Pol
oscillator and the Lorenz-63 system, together with real-world applications to
Hall-effect thruster dynamics and temperature prediction, to demonstrate the
effectiveness of the proposed approach.
