Hamilton系统的泰勒级数形式的辛神经网络
Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systems
摘要：
我们提出了一个有效的和轻量级的学习算法，辛泰勒神经网络（泰勒网），进行连续的，长期的预测一个复杂的哈密顿动力系统的基础上稀疏，短期的观察。在我们的算法的核心是一个新的神经网络架构，由两个子网络。两者都嵌入了泰勒级数展开形式的对称结构设计的条款。支撑我们的基础设施的关键机制是强大的表达能力和特殊的对称性质的泰勒级数展开，这自然适应的数值拟合过程中的梯度的Hamilton广义坐标，以及保持其辛结构。我们进一步将四阶辛积分器与神经ODE的框架结合到我们的泰勒网络架构中，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络在预测广泛的哈密顿动力系统，包括钟摆，洛特卡-Volterra，开普勒，和Hénon-Heiles系统的功效。我们的模型表现出独特的计算性能优于以前的方法在很大程度上的预测精度，收敛速度，和鲁棒性，尽管使用非常小的训练数据，训练周期短（比预测周期短6000倍），小样本量，没有中间数据训练网络。
英文摘要：
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 Hénon--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.
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