Data-Adaptive Learning of Dynamical Systems by Matching Transfer Operators and Invariant Measures

Trajectory-based learning of dynamical systems is often fragile in the presence of noise, chaos, or sparse observations, as small pointwise errors can rapidly amplify. We introduce a transition-statistics approach to system identification that learns dynamics by matching the induced motion of probability mass across a data-adaptive mesh. Given trajectory data, we build an unstructured partition of state space and approximate the Perron--Frobenius operator with a regularized Ulam transition matrix. We replace hard cell indicators with continuous, piecewise-smooth partition-of-unity weights, yielding a Markov matrix supporting gradient-based optimization with respect to the parameters of a learned vector field. This enables two related training objectives: matching invariant measures through the stationary eigenvectors of the transition matrices, and matching the full transition matrices to capture transport between regions of state space. Numerical experiments on Lorenz-63, Lorenz-96, and a reduced-order NOAA sea surface temperature forecasting problem show that transition-statistics matching gives more reliable long-time dynamics than pointwise trajectory matching, particularly under measurement noise and sparse sampling. The approach provides a robust operator-theoretic alternative to trajectory-level losses for learning chaotic and partially observed dynamical systems.