标题： 基于神经网络增强的积分器用于模拟常微分方程 显示英文标题

标题： Neural network-enhanced integrators for simulating ordinary differential equations

摘要： 许多应用需要计算微分方程在广泛初值条件和系统参数下的数值解，这推动了对高效且准确的数值积分方法的需求。本研究提出了一种神经网络（NN）增强的经典数值积分器。 神经网络被训练以学习积分误差，这些误差随后作为加法校正项在数值方案中使用。 通过数值研究比较了这些增强积分器的性能与成熟的方法，特别强调计算效率。 从局部误差和后向误差分析的角度检查了分析性质。 然后采用嵌入式龙格-库塔格式开发增强积分器，以减轻泛化风险，确保神经网络在状态空间中未见过的区域内的评估不会使积分器不稳定。 保证增强积分器的表现至少与期望的经典龙格-库塔格式一样好。 通过使用基于已建立的模拟框架OpenFast得出参数的真实风力涡轮机模型进行了广泛的数值研究，证明了所提出的策略的有效性。 显示英文摘要

摘要： Numerous applications necessitate the computation of numerical solutions to differential equations across a wide range of initial conditions and system parameters, which feeds the demand for efficient yet accurate numerical integration methods.This study proposes a neural network (NN) enhancement of classical numerical integrators. NNs are trained to learn integration errors, which are then used as additive correction terms in numerical schemes. The performance of these enhanced integrators is compared with well-established methods through numerical studies, with a particular emphasis on computational efficiency. Analytical properties are examined in terms of local errors and backward error analysis. Embedded Runge-Kutta schemes are then employed to develop enhanced integrators that mitigate generalization risk, ensuring that the neural network's evaluation in previously unseen regions of the state space does not destabilize the integrator. It is guaranteed that the enhanced integrators perform at least as well as the desired classical Runge-Kutta schemes. The effectiveness of the proposed approaches is demonstrated through extensive numerical studies using a realistic model of a wind turbine, with parameters derived from the established simulation framework OpenFast.