NEURAL DYNAMICAL SYSTEMS: BALANCING STRUC-TURE AND FLEXIBILITY IN PHYSICAL PREDICTION

Abstract

We introduce Neural Dynamical Systems (NDS), a method of learning dynamical models in various gray-box settings which incorporates prior knowledge in the form of systems of ordinary differential equations. NDS uses neural networks to estimate free parameters of the system, predicts residual terms, and numerically integrates over time to predict future states. A key insight is that many real dynamical systems of interest are hard to model because the dynamics may vary across rollouts. We mitigate this problem by taking a trajectory of prior states as the input to NDS and train it to dynamically estimate system parameters using the preceding trajectory. We find that NDS learns dynamics with higher accuracy and fewer samples than a variety of deep learning methods that do not incorporate the prior knowledge and methods from the system identification literature which do. We demonstrate these advantages first on synthetic dynamical systems and then on real data captured from deuterium shots from a nuclear fusion reactor. Finally, we demonstrate that these benefits can be utilized for control in small-scale experiments.

1. INTRODUCTION

The use of function approximators for dynamical system modeling has become increasingly common. This has proven quite effective when a substantial amount of real data is available relative to the complexity of the model being learned (Chua et al., 2018; Janner et al., 2019; Chen et al., 1990) . These learned models are used for downstream applications such as model-based reinforcement learning (Nagabandi et al., 2017; Ross & Bagnell, 2012) or model-predictive control (MPC) (Wang & Ba, 2019) . Model-based control techniques are exciting as we may be able to solve new classes of problems with improved controllers. Problems like dextrous robotic manipulation (Nagabandi et al., 2019 ), game-playing (Schrittwieser et al., 2019) , and nuclear fusion are increasingly being approached using model-based reinforcement learning techniques. However, learning a dynamics model using, for example, a deep neural network can require large amounts of data. This is especially problematic when trying to optimize real physical systems, where data collection can be expensive. As an alternative to data-hungry machine learning methods, there is also a long history of fitting models to a system using techniques from system identification, some of which include prior knowledge about the system drawn from human understanding (Nelles, 2013; Ljung et al., 2009; Sohlberg & Jacobsen, 2008) . These models, especially in the gray-box setting, are typically data-efficient and often contain interpretable model parameters. However, they are not well suited for the situation where the given prior knowledge is approximate or incomplete in nature. They also do not generally adapt to the situation where trajectories are drawn from a variety of parameter settings at test time. This is an especially crucial point as many systems of interest exhibit path-dependent dynamics, which we aim to recover on the fly. In total, system identification methods are sample efficient but inflexible given changing parameter settings and incomplete or approximate knowledge. Conversely, deep learning methods are more flexible at the cost of many more samples. In this paper, we aim to solve both of these problems by biasing the model class towards our physical model of dynamics. Physical models of dynamics are often given in the form of systems of ordinary differential equations (ODEs), which are ubiquitious and may have free parameters that specialize them to a given physical system. We develop a model that uses neural networks to predict the free parameters of an ODE system from the previous 1

