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This work considers the problem of learning the Markov parameters of a linear system from ob- served data. Recent non-asymptotic system identification results have characterized the sample complexity of this problem in the single and multi-rollout setting. In both instances, the number of samples required in order to obtain acceptable estimates can produce optimization problems with an intractably large number of decision variables for a second-order algorithm. We show that a randomized and distributed Newton algorithm based on Hessian-sketching can produce ε-optimal solutions and converges geometrically. Moreover, the algorithm is trivially parallelizable. Our re- sults hold for a variety of sketching matrices and we illustrate the theory with numerical examples. 

Effective inclusion of physics-based knowledge into deep neural network models of dynamical sys- tems can greatly improve data efficiency and generalization. Such a priori knowledge might arise from physical principles (e.g., conservation laws) or from the system’s design (e.g., the Jacobian matrix of a robot), even if large portions of the system dynamics remain unknown. We develop a framework to learn dynamics models from trajectory data while incorporating a priori system knowledge as inductive bias. More specifically, the proposed framework uses physics-based side information to inform the structure of the neural network itself, and to place constraints on the values of the outputs and the internal states of the model. It represents the system’s vector field as a composition of known and unknown functions, the latter of which are parametrized by neural networks. The physics-informed constraints are enforced via the augmented Lagrangian method during the model’s training. We experimentally demonstrate the benefits of the proposed approach on a variety of dynamical systems – including a benchmark suite of robotics environments featur- ing large state spaces, non-linear dynamics, external forces, contact forces, and control inputs. By exploiting a priori system knowledge during training, the proposed approach learns to predict the system dynamics two orders of magnitude more accurately than a baseline approach that does not include prior knowledge, given the same training dataset. 

We develop a learning-based control algorithm for unknown dynamical systems under very severe data limitations. Specifically, the algorithm has access to streaming and noisy data only from a sin- gle and ongoing trial. It accomplishes such performance by effectively leveraging various forms of side information on the dynamics to reduce the sample complexity. Such side information typically comes from elementary laws of physics and qualitative properties of the system. More precisely, the algorithm approximately solves an optimal control problem encoding the system’s desired be- havior. To this end, it constructs and iteratively refines a data-driven differential inclusion that contains the unknown vector field of the dynamics. The differential inclusion, used in an interval Taylor-based method, enables to over-approximate the set of states the system may reach. Theo- retically, we establish a bound on the suboptimality of the approximate solution with respect to the optimal control with known dynamics. We show that the longer the trial or the more side infor- mation is available, the tighter the bound. Empirically, experiments in a high-fidelity F-16 aircraft simulator and MuJoCo’s environments illustrate that, despite the scarcity of data, the algorithm can provide performance comparable to reinforcement learning algorithms trained over millions of environment interactions. Besides, we show that the algorithm outperforms existing techniques combining system identification and model predictive control.