We extend the methodology in Yang et al. [SIAM J. Appl. Dyn. Syst. 22, 269–310 (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.

We extend the methodology in [65] 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.

With complex throwing patterns of multiple objects, jugglers seemingly defy human limits of reaction time and throwing accuracy.