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Register a new userMastering high-dimensional dynamics with Hamiltonian neural networks
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We detail how incorporating physics into neural network design can significantly improve the learning and forecasting of dynamical systems, even nonlinear systems of many dimensions. A map building perspective elucidates the superiority of Hamiltonian neural networks over conventional neural networks. The results clarify the critical relation between data, dimension, and neural network learning performance.
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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.
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.
While gradient descent has proven highly successful in learning connection weights for neural networks, the actual structure of these networks is usually determined by hand, or by other optimization algorithms. Here we describe a simple method to make network structure differentiable, and therefore accessible to gradient descent. We test this method on recurrent neural networks applied to simple sequence prediction problems. Starting with initial networks containing only one node, the method automatically builds networks that successfully solve the tasks. The number of nodes in the final network correlates with task difficulty. The method can dynamically increase network size in response to an abrupt complexification in the task; however, reduction in network size in response to task simplification is not evident for reasonable meta-parameters. The method does not penalize network performance for these test tasks: variable-size networks actually reach better performance than fixed-size networks of higher, lower or identical size. We conclude by discussing how this method could be applied to more complex networks, such as feedforward layered networks, or multiple-area networks of arbitrary shape.
We introduce a method to train Quantized Neural Networks (QNNs) --- neural networks with extremely low precision (e.g., 1-bit) weights and activations, at run-time. At train-time the quantized weights and activations are used for computing the parameter gradients. During the forward pass, QNNs drastically reduce memory size and accesses, and replace most arithmetic operations with bit-wise operations. As a result, power consumption is expected to be drastically reduced. We trained QNNs over the MNIST, CIFAR-10, SVHN and ImageNet datasets. The resulting QNNs achieve prediction accuracy comparable to their 32-bit counterparts. For example, our quantized version of AlexNet with 1-bit weights and 2-bit activations achieves $51%$ top-1 accuracy. Moreover, we quantize the parameter gradients to 6-bits as well which enables gradients computation using only bit-wise operation. Quantized recurrent neural networks were tested over the Penn Treebank dataset, and achieved comparable accuracy as their 32-bit counterparts using only 4-bits. Last but not least, we programmed a binary matrix multiplication GPU kernel with which it is possible to run our MNIST QNN 7 times faster than with an unoptimized GPU kernel, without suffering any loss in classification accuracy. The QNN code is available online.
Although spiking neural networks (SNNs) take benefits from the bio-plausible neural modeling, the low accuracy under the common local synaptic plasticity learning rules limits their application in many practical tasks. Recently, an emerging SNN supervised learning algorithm inspired by backpropagation through time (BPTT) from the domain of artificial neural networks (ANNs) has successfully boosted the accuracy of SNNs and helped improve the practicability of SNNs. However, current general-purpose processors suffer from low efficiency when performing BPTT for SNNs due to the ANN-tailored optimization. On the other hand, current neuromorphic chips cannot support BPTT because they mainly adopt local synaptic plasticity rules for simplified implementation. In this work, we propose H2Learn, a novel architecture that can achieve high efficiency for BPTT-based SNN learning which ensures high accuracy of SNNs. At the beginning, we characterized the behaviors of BPTT-based SNN learning. Benefited from the binary spike-based computation in the forward pass and the weight update, we first design lookup table (LUT) based processing elements in Forward Engine and Weight Update Engine to make accumulations implicit and to fuse the computations of multiple input points. Second, benefited from the rich sparsity in the backward pass, we design a dual-sparsity-aware Backward Engine which exploits both input and output sparsity. Finally, we apply a pipeline optimization between different engines to build an end-to-end solution for the BPTT-based SNN learning. Compared with the modern NVIDIA V100 GPU, H2Learn achieves 7.38x area saving, 5.74-10.20x speedup, and 5.25-7.12x energy saving on several benchmark datasets.
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