Neural ordinary differential equations (Neural ODEs) are an effective
framework for learning dynamical systems from irregularly sampled time series
data. These models provide a continuous-time latent representation of the
underlying dynamical system where new observations at arbitrary time points can
be used to update the latent representation of the dynamical system. Existing
parameterizations for the dynamics functions of Neural ODEs limit the ability
of the model to retain global information about the time series; specifically,
a piece-wise integration of the latent process between observations can result
in a loss of memory on the dynamic patterns of previously observed data points.
We propose PolyODE, a Neural ODE that models the latent continuous-time process
as a projection onto a basis of orthogonal polynomials. This formulation
enforces long-range memory and preserves a global representation of the
underlying dynamical system. Our construction is backed by favourable
theoretical guarantees and in a series of experiments, we demonstrate that it
outperforms previous works in the reconstruction of past and future data, and
in downstream prediction tasks.

本研究提出了 PolyODE，一种基于正交多项式投影的神经常微分方程模型，用于学习动态系统，以实现长期记忆和整体表示，优于先前的模型在数据重建和下游预测任务中的性能。