CKNet: A Convolutional Neural Network Based on Koopman Operator for Modeling Latent Dynamics from Pixels

With the development of end-to-end control based on deep learning, it is important to study new system modeling techniques to realize dynamics modeling with high-dimensional inputs. In this paper, a novel Koopman-based deep convolutional network, called CKNet, is proposed to identify latent dynamics from raw pixels. CKNet learns an encoder and decoder to play the role of the Koopman eigenfunctions and modes, respectively. The Koopman eigenvalues can be approximated by eigenvalues of the learned state transition matrix. The deterministic convolutional Koopman network (DCKNet) and the variational convolutional Koopman network (VCKNet) are proposed to span some subspace for approximating the Koopman operator respectively. Because CKNet is trained under the constraints of the Koopman theory, the identified latent dynamics is in a linear form and has good interpretability. Besides, the state transition and control matrices are trained as trainable tensors so that the identified dynamics is also time-invariant. We also design an auxiliary weight term for reducing multi-step linearity and prediction losses. Experiments were conducted on two offline trained and four online trained nonlinear forced dynamical systems with continuous action spaces in Gym and Mujoco environment respectively, and the results show that identified dynamics are adequate for approximating the latent dynamics and generating clear images. Especially for offline trained cases, this work confirms CKNet from a novel perspective that we visualize the evolutionary processes of the latent states and the Koopman eigenfunctions with DCKNet and VCKNet separately to each task based on the same episode and results demonstrate that different approaches learn similar features in shapes.