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This paper introduces a data-driven algorithm that utilizes a neural power iteration scheme to approximate the dominant eigenfunctions of the Koopman operator for nonlinear dynamical systems. By directly learning these modes without constructing explicit projections, the method effectively mitigates the curse of dimensionality and provides theoretical guarantees for convergence as sample size and network width increase. Numerical experiments validate the approach, showing it achieves accurate and smooth approximations, outperforming traditional methods like extended dynamic mode decomposition.
A novel neural power iteration algorithm can accurately approximate dominant Koopman modes without the curse of dimensionality, revolutionizing the analysis of nonlinear dynamical systems.
This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require ``anti-collapse mechanisms''to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.
