Deep Identification of Nonlinear Systems in Koopman Form

TL;DR

This paper introduces a deep learning method for identifying nonlinear dynamical systems using Koopman operators, eliminating the need for pre-selected lifting functions and enabling efficient, accurate long-term predictions.

Contribution

It presents a novel deep state-space encoder approach that automatically learns the lifting functions for Koopman analysis, improving flexibility and prediction accuracy.

Findings

Effective long-term prediction demonstrated on benchmark examples

Reduces computational load via subsequence training approach

Automatically learns lifting functions without pre-specification

Abstract

The present paper treats the identification of nonlinear dynamical systems using Koopman-based deep state-space encoders. Through this method, the usual drawback of needing to choose a dictionary of lifting functions a priori is circumvented. The encoder represents the lifting function to the space where the dynamics are linearly propagated using the Koopman operator. An input-affine formulation is considered for the lifted model structure and we address both full and partial state availability. The approach is implemented using the the deepSI toolbox in Python. To lower the computational need of the simulation error-based training, the data is split into subsections where multi-step prediction errors are calculated independently. This formulation allows for efficient batch optimization of the network parameters and, at the same time, excellent long term prediction capabilities of the obtained models. The performance of the approach is illustrated by nonlinear benchmark examples.