Extended Dynamic Mode Decomposition with Invertible Dictionary Learning.

The Koopman operator has received attention for providing a potentially global linearization representation of the nonlinear dynamical system. To estimate or control the original system, the invertibility problem is introduced into the data-driven modeling, i.e., the observables are required to be reconstructed the original system's states. Existing methods cannot solve this problem perfectly. Only linear or nonlinear but lossy reconstruction can be achieved. This paper proposed a novel data-driven modeling approach, denoted as the Extended Dynamic Mode Decomposition with Invertible Dictionary Learning (EDMD-IDL) to address this issue, which can be interpreted as a further extension of the classical Extended Dynamic Mode Decomposition (EDMD). The Invertible Neural Network (INN) is introduced in the proposed method, where its inverse process provides the explicit inverse on the dictionary functions, thus allowing the nonlinear and lossless reconstruction. An iterative algorithm is designed to solve the extended optimization problem defined by the Koopman operator and INN by combining the optimization algorithm based on the gradient descent and the classical EDMD method, making the method successfully obtain the finite-dimensional approximation of the Koopman operator. The method is tested on various canonical nonlinear dynamical systems and is shown that the predictions obtained in a linear fashion and the ground truth match well over the long-term, where only the initial status is provided. Comparison experiments highlight the superiority of the proposed method over the other EDMD-based methods. Notably, a typical example in fluid dynamics, cylinder wake, illustrates the potential of the method to be further extended to the high-dimensional system with tens of thousands of states. By combining the Proper Orthogonal Decomposition technique, nontrivial Kármán vortex sheet phenomenon is perfectly reconstructed. Our proposed method provides a new paradigm for solving the finite-dimensional approximation of the Koopman operator and applying it to data-driven modeling.