A Convex Relaxation Approach for Learning the Robust Koopman Operator

Abstract

Although data-driven models, especially deep learning, have achieved astonishing results on many prediction tasks for nonlinear sequences, challenges still remain in finding an appropriate way to embed prior knowledge of physical dynamics in these models. In this work, we introduce a convex relaxation approach for learning robust Koopman operators of nonlinear dynamical systems, which are intended to construct approximate space spanned by eigenfunctions of the Koopman operator. Different from the classical dynamic mode decomposition, we use the layer weights of neural networks as eigenfunctions of the Koopman operator, providing intrinsic coordinates that globally linearize the dynamics. We find that the approximation of space can be regarded as an orthogonal Procrustes problem on the Stiefel manifold, which is highly sensitive to noise. The key contribution of this paper is to demonstrate that strict orthogonal constraint can be replaced by its convex relaxation, and the performance of the model can be improved without increasing the complexity when dealing with both clean and noisy data. After that, the overall model can be optimized via backpropagation in an end-to-end manner. The comparisons of the proposed method against several state-of-the-art competitors are shown on nonlinear oscillators and the lid-driven cavity flow.