We propose an effective and light-weighted 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 that are designed with a symmetric structure. The key mechanism underpinning our infrastructure is the strong expressiveness and special symmetric property of the Taylor series expansion, which can inherently accommodate the numerical fitting process of the spatial derivatives of the Hamiltonian 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 preserving their symplectic structures simultaneously. We demonstrated the efficacy of our Tayler-net in predicting a broad spectrum of Hamiltonian dynamic systems, including the pendulum, the Lotka-Volterra, the Kepler, and the H\`enon-Heiles systems. Compared with previous methods, our model exhibits its unique computational merits by using extremely small training data with short training period (6000 times shorter than the predicting period), small sample sizes (5 times smaller compared with the state-of-the-art methods), and no intermediary data to train the networks, while outperforming others to a great extent regarding the prediction accuracy, the convergence rate, and the robustness.

提出一种有效且轻量级的学习算法——辛泰勒神经网络（Taylor-nets），用于基于稀疏、短期观察进行连续、长期预测复杂的哈密顿动态系统。该算法基于一种新颖的神经网络架构，它包含两个嵌入对称Taylor级数展开形式术语的子网络，并将四阶辛普勒积分器与神经ODE框架相结合，以学习目标系统的连续时间演化，同时保持其辛结构。在较小的训练数据、短训练周期（预测周期的6000倍）的情况下，该模型表现出了独特的计算优点，具有较高的预测精度、收敛速度和鲁棒性。

Taylor级数形式的辛神经网络用于哈密顿系统