Modeling data obtained from dynamical systems has gained attention in recent years as a challenging task for machine learning models. Previous approaches assume the measurements to be distributed on a grid. However, for real-world applications like weather prediction, the observations are taken from arbitrary locations within the spatial domain. In this paper, we propose TaylorPDENet - a novel machine learning method that is designed to overcome this challenge. Our algorithm uses the multidimensional Taylor expansion of a dynamical system at each observation point to estimate the spatial derivatives to perform predictions. TaylorPDENet is able to accomplish two objectives simultaneously: accurately forecast the evolution of a complex dynamical system and explicitly reconstruct the underlying differential equation describing the system. We evaluate our model on a variety of advection-diffusion equations with different parameters and show that it performs similarly to equivalent approaches on grid-structured data while being able to process unstructured data as well.

翻译：近年来，从动力系统获取的数据建模已成为机器学习领域的一项具有挑战性的任务。以往的方法假设测量数据分布在网格上。然而，在诸如天气预报等实际应用中，观测数据往往取自空间域内的任意位置。本文提出了一种新颖的机器学习方法TaylorPDENet，旨在解决这一挑战。该算法利用动力系统在每个观测点处的多维泰勒展开来估计空间导数，从而实现预测。TaylorPDENet能够同时达成两个目标：准确预测复杂动力系统的演化过程，以及显式重建描述该系统的潜在微分方程。我们在具有不同参数的对流-扩散方程上评估了该模型，结果表明其在网格结构化数据上的性能与现有方法相当，同时还能处理非结构化数据。