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This work introduces a non-intrusive model reduction approach for learningreduced models from partially observed state trajectories of high-dimensionaldynamical systems. The proposed approach compensates for the loss ofinformation due to the partially observed states by constructing non-Markovianreduced models that make future-state predictions based on a history of reducedstates, in contrast to traditional Markovian reduced models that rely on thecurrent reduced state alone to predict the next state. The core contributionsof this work are a data sampling scheme to sample partially observed statesfrom high-dimensional dynamical systems and a formulation of a regressionproblem to fit the non-Markovian reduced terms to the sampled states. Undercertain conditions, the proposed approach recovers from data the very samenon-Markovian terms that one obtains with intrusive methods that require thegoverning equations and discrete operators of the high-dimensional dynamicalsystem. Numerical results demonstrate that the proposed approach leads tonon-Markovian reduced models that are predictive far beyond the trainingregime. Additionally, in the numerical experiments, the proposed approachlearns non-Markovian reduced models from trajectories with only 20% observedstate components that are about as accurate as traditional Markovian reducedmodels fitted to trajectories with 99% observed components.

Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectories