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首页> 外文期刊>Journal of Computational Physics >Symplectic neural networks in Taylor series form for Hamiltonian systems
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Symplectic neural networks in Taylor series form for Hamiltonian systems

机译：哈密顿系统泰勒系列形式的杂交神经网络

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摘要

We propose an effective and light-weight 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. (C) 2021 Elsevier Inc. All rights reserved.
机译：我们提出了一种有效且轻量级的学习算法，即辛泰勒神经网络（Taylor nets），用于基于稀疏的短期观测对复杂哈密顿动力学系统进行连续、长期的预测。我们算法的核心是一种由两个子网络组成的新型神经网络结构。两者都嵌入了泰勒级数展开形式的项，采用对称结构设计。支撑我们基础设施的关键机制是泰勒级数展开式的强表达性和特殊对称性，它自然地适应了哈密顿量相对于广义坐标梯度的数值拟合过程，并保持了其辛结构。我们进一步将四阶辛积分器与神经ODEs框架结合到我们的泰勒网络结构中，以学习目标系统的连续时间演化，同时保持其辛结构。我们证明了我们的泰勒网在预测广泛的哈密顿动力学系统方面的有效性，包括摆、洛特卡-沃尔特拉、开普勒和海勒系统。我们的模型在预测精度、收敛速度和鲁棒性方面表现出独特的计算优势，在很大程度上优于以前的方法，尽管使用了非常小的训练数据，训练周期很短（比预测周期短6000倍），样本量小，并且没有中间数据来训练网络。（c）2021爱思唯尔公司保留所有权利。

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