The Path-dependent Neural Jump ODE (PD-NJ-ODE) is a model for online prediction of generic (possibly non-Markovian) stochastic processes with irregular (in time) and potentially incomplete (with respect to coordinates) observations. It is a model for which convergence to the $L^2$-optimal predictor, which is given by the conditional expectation, is established theoretically. Thereby, the training of the model is solely based on a dataset of realizations of the underlying stochastic process, without the need of knowledge of the law of the process. In the case where the underlying process is deterministic, the conditional expectation coincides with the process itself. Therefore, this framework can equivalently be used to learn the dynamics of ODE or PDE systems solely from realizations of the dynamical system with different initial conditions. We showcase the potential of our method by applying it to the chaotic system of a double pendulum. When training the standard PD-NJ-ODE method, we see that the prediction starts to diverge from the true path after about half of the evaluation time. In this work we enhance the model with two novel ideas, which independently of each other improve the performance of our modelling setup. The resulting dynamics match the true dynamics of the chaotic system very closely. The same enhancements can be used to provably enable the PD-NJ-ODE to learn long-term predictions for general stochastic datasets, where the standard model fails. This is verified in several experiments.

翻译：路径依赖神经跳跃常微分方程（PD-NJ-ODE）是一种用于在线预测具有不规则（时间上）且可能不完整（坐标上）观测值的通用（可能非马尔可夫）随机过程的模型。该模型在理论上被证明收敛于由条件期望给出的$L^2$最优预测器。因此，模型的训练完全基于底层随机过程实现的数据集，无需了解过程规律。当底层过程为确定性时，条件期望与过程本身一致。因此，该框架可等效地用于仅通过具有不同初始条件的动力系统实现来学习常微分方程或偏微分方程系统的动力学。我们通过将方法应用于双摆混沌系统来展示其潜力。在训练标准PD-NJ-ODE方法时，我们观察到预测在约一半评估时间后开始偏离真实轨迹。本研究通过两个新颖的改进思路增强模型，这些思路各自独立地提升了建模框架的性能。改进后的动力学与混沌系统的真实动力学高度吻合。相同的增强方法可证明使PD-NJ-ODE能够学习通用随机数据集的长期预测，而标准模型在此方面存在不足。多个实验验证了该结论。