Exploiting Nonlinearity in Transfer Operator Approaches to Achieve more Robust Data-Driven Predictors
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The conversion of a finite-dimensional nonlinear problem into an infinite-dimensional linear problem is a key benefit of Koopman/transfer operator approaches to dynamics. The operator-theoretic approach has been highly fruitful, particularly in revealing macroscopic structures in phase space which govern system behaviour. However, when striving for robust trajectory reconstruction, the (nonlinear) characteristics of the original problem can sometimes be of much use. We show how to leverage nonlinearity in the transfer operator approach, demonstrating connections to highly successful schemes such as ’sparse identification of nonlinear dynamics’ (SINDy). In particular we offer explainability for their success and rigorous convergence proofs.
