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Accurately learning the temporal behavior of dynamical systems requires models with well-chosen learning biases. Recent innovations embed the Hamiltonian and Lagrangian formalisms into neural networks and demonstrate a significant improvement over other approaches in predicting trajectories of physical systems. These methods generally tackle autonomous systems that depend implicitly on time or systems for which a control signal is known apriori. Despite this success, many real world dynamical systems are non-autonomous, driven by time-dependent forces and experience energy dissipation. In this study, we address the challenge of learning from such non-autonomous systems by embedding the port-Hamiltonian formalism into neural networks, a versatile framework that can capture energy dissipation and time-dependent control forces. We show that the proposed emph{port-Hamiltonian neural network} can efficiently learn the dynamics of nonlinear physical systems of practical interest and accurately recover the underlying stationary Hamiltonian, time-dependent force, and dissipative coefficient. A promising outcome of our network is its ability to learn and predict chaotic systems such as the Duffing equation, for which the trajectories are typically hard to learn.
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The rapid growth of research in exploiting machine learning to predict chaotic systems has revived a recent interest in Hamiltonian Neural Networks (HNNs) with physical constraints defined by the Hamiltons equations of motion, which represent a major class of physics-enhanced neural networks. We introduce a class of HNNs capable of adaptable prediction of nonlinear physical systems: by training the neural network based on time series from a small number of bifurcation-parameter values of the target Hamiltonian system, the HNN can predict the dynamical states at other parameter values, where the network has not been exposed to any information about the system at these parameter values. The architecture of the HNN differs from the previous ones in that we incorporate an input parameter channel, rendering the HNN parameter--cognizant. We demonstrate, using paradigmatic Hamiltonian systems, that training the HNN using time series from as few as four parameter values bestows the neural machine with the ability to predict the state of the target system in an entire parameter interval. Utilizing the ensemble maximum Lyapunov exponent and the alignment index as indicators, we show that our parameter-cognizant HNN can successfully predict the route of transition to chaos. Physics-enhanced machine learning is a forefront area of research, and our adaptable HNNs provide an approach to understanding machine learning with broad applications.
Passivity-based control (PBC) for port-Hamiltonian systems provides an intuitive way of achieving stabilization by rendering a system passive with respect to a desired storage function. However, in most instances the control law is obtained without any performance considerations and it has to be calculated by solving a complex partial differential equation (PDE). In order to address these issues we introduce a reinforcement learning approach into the energy-balancing passivity-based control (EB-PBC) method, which is a form of PBC in which the closed-loop energy is equal to the difference between the stored and supplied energies. We propose a technique to parameterize EB-PBC that preserves the systemss PDE matching conditions, does not require the specification of a global desired Hamiltonian, includes performance criteria, and is robust to extra non-linearities such as control input saturation. The parameters of the control law are found using actor-critic reinforcement learning, enabling learning near-optimal control policies satisfying a desired closed-loop energy landscape. The advantages are that near-optimal controllers can be generated using standard energy shaping techniques and that the solutions learned can be interpreted in terms of energy shaping and damping injection, which makes it possible to numerically assess stability using passivity theory. From the reinforcement learning perspective, our proposal allows for the class of port-Hamiltonian systems to be incorporated in the actor-critic framework, speeding up the learning thanks to the resulting parameterization of the policy. The method has been successfully applied to the pendulum swing-up problem in simulations and real-life experiments.
We propose an effective and lightweight learning algorithm, Symplectic Taylor Neural Networks (Taylor-nets), to conduct continuous, long-term predictions of a complex Hamiltonian dynamic system based on sparse, short-term observations. At the heart of our algorithm is a novel neural network architecture consisting of two sub-networks. Both are embedded with terms in the form of Taylor series expansion designed with symmetric structure. The key mechanism underpinning our infrastructure is the strong expressiveness and special symmetric property of the Taylor series expansion, which naturally accommodate the numerical fitting process of the gradients of the Hamiltonian with respect to the generalized coordinates as well as preserve its symplectic structure. We further incorporate a fourth-order symplectic integrator in conjunction with neural ODEs framework into our Taylor-net architecture to learn the continuous-time evolution of the target systems while simultaneously preserving their symplectic structures. We demonstrated the efficacy of our Taylor-net in predicting a broad spectrum of Hamiltonian dynamic systems, including the pendulum, the Lotka--Volterra, the Kepler, and the Henon--Heiles systems. Our model exhibits unique computational merits by outperforming previous methods to a great extent regarding the prediction accuracy, the convergence rate, and the robustness despite using extremely small training data with a short training period (6000 times shorter than the predicting period), small sample sizes, and no intermediate data to train the networks.
A dynamical neural network consists of a set of interconnected neurons that interact over time continuously. It can exhibit computational properties in the sense that the dynamical systems evolution and/or limit points in the associated state space can correspond to numerical solutions to certain mathematical optimization or learning problems. Such a computational system is particularly attractive in that it can be mapped to a massively parallel computer architecture for power and throughput efficiency, especially if each neuron can rely solely on local information (i.e., local memory). Deriving gradients from the dynamical networks various states while conforming to this last constraint, however, is challenging. We show that by combining ideas of top-down feedback and contrastive learning, a dynamical network for solving the l1-minimizing dictionary learning problem can be constructed, and the true gradients for learning are provably computable by individual neurons. Using spiking neurons to construct our dynamical network, we present a learning process, its rigorous mathematical analysis, and numerical results on several dictionary learning problems.
Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but are solutions of an ordinary differential equation (ODE); however, these networks are still optimized via discrete methods (e.g. gradient descent). In this paper, we explore a different direction: namely, we propose a novel framework for learning in which the parameters themselves are solutions of ODEs. By viewing the optimization process as the evolution of a port-Hamiltonian system, we can ensure convergence to a minimum of the objective function. Numerical experiments have been performed to show the validity and effectiveness of the proposed methods.
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