Learning Dynamical Systems From Invariant Measures

We extend the methodology in [Yang et al., 2021] to learn autonomous continuous-time dynamical systems from invariant measures. We assume that our data accurately describes the dynamics' asymptotic statistics but that the available time history of observations is insufficient for approximating the Lagrangian velocity. Therefore, invariant measures are treated as the inference data and velocity learning is reformulated as a data-fitting, PDE-constrained optimization problem in which the stationary distributional solution to the Fokker--Planck equation is used as a differentiable surrogate forward model. We consider velocity parameterizations based upon global polynomials, piecewise polynomials, and fully connected neural networks, as well as various objective functions to compare synthetic and reference invariant measures. We utilize the adjoint-state method together with the backpropagation technique to efficiently perform gradient-based parameter identification. Numerical results for the Van der Pol oscillator and Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, are presented to demonstrate the effectiveness of the proposed approach.