Networks are landmarks of many complex phenomena where interweaving interactions between different agents transform simple local rule-sets into nonlinear emergent behaviors. While some recent studies unveil associations between the network structure and the underlying dynamical process, identifying stochastic nonlinear dynamical processes continues to be an outstanding problem. Here we develop a simple data-driven framework based on operator-theoretic techniques to identify and control stochastic nonlinear dynamics taking place over large-scale networks. The proposed approach requires no prior knowledge of the network structure and identifies the underlying dynamics solely using a collection of two-step snapshots of the states. This data-driven system identification is achieved by using the Koopman operator to find a low dimensional representation of the dynamical patterns that evolve linearly. Further, we use the global linear Koopman model to solve critical control problems by applying to model predictive control (MPC)--typically, a challenging proposition when applied to large networks. We show that our proposed approach tackles this by converting the original nonlinear programming into a more tractable optimization problem that is both convex and with far fewer variables.

A low-dimensional dynamical system is observed in an experiment as a high-dimensional signal; For example, a video of a chaotic pendulums system. Assuming that we know the dynamical model up to some unknown parameters, can we estimate the underlying system's parameters by measuring its time-evolution only once? The key information for performing this estimation lies in the temporal inter-dependencies between the signal and the model. We propose a kernel-based score to compare these dependencies. Our score generalizes a maximum likelihood estimator for a linear model to a general nonlinear setting in an unknown feature space. We estimate the system's underlying parameters by maximizing the proposed score. We demonstrate the accuracy and efficiency of the method using two chaotic dynamical systems - the double pendulum and the Lorenz '63 model.

We study the ability of neural networks to steer or control trajectories of dynamical systems on graphs. In particular, we introduce a neural-network control (NNC) framework, which represents dynamical systems by neural ordinary different equations (neural ODEs), and find that NNC can learn control signals that drive networked dynamical systems into desired target states. To identify the influence of different target states on the NNC performance, we study two types of control: (i) microscopic control and (ii) macroscopic control. Microscopic control minimizes the L2 norm between the current and target state and macroscopic control minimizes the corresponding Wasserstein distance. We find that the proposed NNC framework produces low-energy control signals that are highly correlated with those of optimal control. Our results are robust for a wide range of graph structures and (non-)linear dynamical systems.

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 dynamic 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 re-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.

One of the core challenges in the field of control theory and reinforcement learning is adaptive control. It is the problem of controlling dynamical systems when the dynamics of the systems are unknown to the decision-making agents. In adaptive control, agents interact with given systems in order to explore and control them while the long-term objective is to minimize the overall average associated costs. The agent has to balance between exploration and exploitation, learn the dynamics, strategize for further exploration, and exploit the estimation to minimize the overall costs. The sequential nature of agent-system interaction results in challenges in the system identifying, estimation, and control under uncertainty, and these challenges are magnified when the systems are partially observable, i.e. contain hidden underlying dynamics. In the linear systems, when the underlying dynamics are fully observable, the asymptotic optimality of estimation methods has been the topic of study in the last decades [Lai et al., 1982, Lai and Wei, 1987]. Recently, novel techniques and learning algorithms have been developed to study the finite-time behavior of adaptive control algorithms and shed light on the design of optimal methods [Peña et al., 2009, Fiechter, 1997, Abbasi-Yadkori and Szepesvári, 2011]. In particular, Abbasi-Yadkori and Szepesvári [2011] proposes to use the principle of optimism in the face of uncertainty (OFU) to balance exploration and exploitation in LQR, where the state of the system is observable.

This work presents a framework of constructing the neural networks preserving the symplectic structure, so-called symplectic networks (SympNets). With the symplectic networks, we show some numerical results about (\romannumeral1) solving the Hamiltonian systems by learning abundant data points over the phase space, and (\romannumeral2) predicting the phase flows by learning a series of points depending on time. All the experiments point out that the symplectic networks perform much more better than the fully-connected networks that without any prior information, especially in the task of predicting which is unable to do within the conventional numerical methods.

It is organized in a tensor of the shape (M,J,N), where M 10, 000 is the total ensemble size, J 32 is the spatial discretization size, and N 50 is the sampled time instants with the time step Δt 0.01. Thus the initial data for training are given as the tKdV solutions in the time window 0,0.5.

Despite the availability of ever more data enabled through modern sensor and computer technology, it still remains an open problem to learn dynamical systems in a sample-efficient way. We propose active learning strategies that leverage information-theoretical properties arising naturally during Gaussian process regression, while respecting constraints on the sampling process imposed by the system dynamics. Sample points are selected in regions with high uncertainty, leading to exploratory behavior and data-efficient training of the model. All results are finally verified in an extensive numerical benchmark.

We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.

Attempts from different disciplines to provide a fundamental understanding of deep learning have advanced rapidly in recent years, yet a unified framework remains relatively limited. In this article, we provide one possible way to align existing branches of deep learning theory through the lens of dynamical system and optimal control. By viewing deep neural networks as discrete-time nonlinear dynamical systems, we can analyze how information propagates through layers using mean field theory. When optimization algorithms are further recast as controllers, the ultimate goal of training processes can be formulated as an optimal control problem. In addition, we can reveal convergence and generalization properties by studying the stochastic dynamics of optimization algorithms. This viewpoint features a wide range of theoretical study from information bottleneck to statistical physics. It also provides a principled way for hyper-parameter tuning when optimal control theory is introduced. Our framework fits nicely with supervised learning and can be extended to other learning problems, such as Bayesian learning, adversarial training, and specific forms of meta learning, without efforts. The review aims to shed lights on the importance of dynamics and optimal control when developing deep learning theory.