Invertible Koopman Network and its application in data-driven modeling for dynamic systems

Koopman operator, acting on an infinite-dimensional Hilbert space of the observables, provides a global systematic linear representation of nonlinear systems, which is a leading candidate for data-driven modeling. In practical applications, the observable function is required to have an invertible property to predict or control the original system. However, two approaches for constructing the invertible observable functions in current research, introducing the full-state observables and adopting the auto-encoder, have drawbacks such as overly strong assumptions and lossy reconstruction. To address these issues, we propose the Invertible Koopman Network (IKN) to approximate the Koopman operator. With special invertible blocks, IKN achieves structure-level reversibility while retaining sufficient nonlinear mapping capability. This network results in a transformation from the complex dynamic system's states to the observables with simpler dynamics and affords lossless reconstruction via its invertible process. Then, we use the IKN as a dynamical embedding model and adopt the Transformer based on an autoregressive task to learn the evolution pattern of the observables. Through the greedy decoding strategy, the trained IKN and Transformer model enable data-driven prediction where only initial value is provided. The method is tested on three typical chaotic dynamic systems (Lorenz system, Chen system, and Kuramoto-Sivashinsky equation) and is shown to capture the systems’ intrinsic evolution patterns, such as the structure of the strange attractor. Comparative experiments with alternative data-driven modeling approaches illustrate that our proposed method can model a longer evolutionary process with less error, further demonstrating its effectiveness and superiority.