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.

翻译：神经常微分方程（Neural ODEs）是从非均匀采样时间序列数据学习动力系统的有效框架。这些模型为潜在动力系统提供连续时间隐表示，其中任意时间点的新观测值可用于更新动力系统的隐表示。现有Neural ODEs动力函数的参数化方法限制了模型保留时间序列全局信息的能力；具体而言，观测值之间隐过程的逐段积分可能导致先前观测数据点动态模式的记忆丧失。我们提出PolyODE，一种将潜在连续时间过程建模为向正交多项式基投影的Neural ODE。该公式强制实现长程记忆并保留潜在动力系统的全局表示。我们的构造具有可靠的理论保证，通过一系列实验证明，该方法在重构过去与未来数据以及下游预测任务中均优于先前工作。