标题： 泰勒模型物理信息神经网络（PINNs）用于常微分方程 显示英文标题

标题： Taylor-Model Physics-Informed Neural Networks (PINNs) for Ordinary Differential Equations

摘要： 我们研究了在参数不确定性下学习常微分方程（ODE）的神经网络模型的问题。这种神经网络模型能够捕捉给定一组参数、初始条件和时间范围内的ODE解。物理信息神经网络（PINNs）作为一种有前途的方法被提出，用于以系统的方式结合数据驱动的深度学习与符号物理模型来学习此类模型。然而，当PINNs用于求解由变化的参数和初始条件表征的整个初始值问题家族时，其准确性会下降。在本文中，我们结合符号微分和泰勒级数方法，提出了一类更高阶的模型来捕捉ODE的解。这些模型结合了神经网络和符号项：它们使用高阶李导数和通过符号获得的泰勒级数展开式，余项则建模为神经网络。关键见解是，余项本身可以建模为一个一阶ODE的解。我们展示了如何使用这些高阶PINNs通过有趣但具有挑战性的ODE基准来提高准确性。我们还表明，所得到的模型在控制作为ODE建模的不确定物理系统等情况下非常有用。 显示英文摘要

摘要： We study the problem of learning neural network models for Ordinary Differential Equations (ODEs) with parametric uncertainties. Such neural network models capture the solution to the ODE over a given set of parameters, initial conditions, and range of times. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for learning such models that combine data-driven deep learning with symbolic physics models in a principled manner. However, the accuracy of PINNs degrade when they are used to solve an entire family of initial value problems characterized by varying parameters and initial conditions. In this paper, we combine symbolic differentiation and Taylor series methods to propose a class of higher-order models for capturing the solutions to ODEs. These models combine neural networks and symbolic terms: they use higher order Lie derivatives and a Taylor series expansion obtained symbolically, with the remainder term modeled as a neural network. The key insight is that the remainder term can itself be modeled as a solution to a first-order ODE. We show how the use of these higher order PINNs can improve accuracy using interesting, but challenging ODE benchmarks. We also show that the resulting model can be quite useful for situations such as controlling uncertain physical systems modeled as ODEs.