  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 H\'enon--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.
