Learning Dynamical Systems from Noisy Sensor Measurements using Multiple Shooting

Learning dynamical systems offers the possibility to forecast the future, which plays a fundamental role in our understanding of complex physical phenomena in a large variety of fields: physics, chemistry, biology, engineering. An accurate model of the system dynamics also plays a central role in optimal control or model-based reinforcement learning [1]. In real applications, model learning has to be done with measurements that are usually partial, indirect and corrupted by noise, and, consequently, a latent representation is needed. In this work, we introduce a generic approach to train neural networks forming a discrete nonlinear deterministic state space model (SSM). Such representations have been widely studied in the system identification and control communities with parametric physical models and offer many beneficial properties for analysis and control synthesis. However, due to the resulting complex maximum likelihood estimator, it remains challenging to learn such representations with deep neural networks. This paper shows that state space representations are compatible with neural networks and proposes a generic, yet computationally simple way to learn them. Our framework can consider complex measurements such as raw images and can capture complex dynamics such as chaotic dynamics.