The ability to accurately approximate trajectories of dynamical systems enables their analysis, prediction, and control. Neural network (NN)-based approximations have attracted significant interest due to fast evaluation with good accuracy over long integration time steps. In contrast to established numerical approximation schemes such as Runge-Kutta methods, the estimation of the error of the NN-based approximations proves to be difficult. In this work, we propose to use the NN's predictions in a high-order implicit Runge-Kutta (IRK) method. The residuals in the implicit system of equations can be related to the NN's prediction error, hence, we can provide an error estimate at several points along a trajectory. We find that this error estimate highly correlates with the NN's prediction error and that increasing the order of the IRK method improves this estimate. We demonstrate this estimation methodology for Physics-Informed Neural Network (PINNs) on the logistic equation as an illustrative example and then apply it to a four-state electric generator model that is regularly used in power system modelling.

翻译：准确逼近动力系统轨迹的能力使其分析、预测和控制成为可能。基于神经网络（NN）的近似方法因在长时间积分步长内兼具快速评估与良好精度而备受关注。与龙格-库塔法等经典数值逼近方案不同，神经网络近似的误差估计被证明颇具挑战性。本研究提出将神经网络的预测结果应用于高阶隐式龙格-库塔（IRK）方法。隐式方程组中的残差可与神经网络的预测误差相关联，从而在轨迹上的多个点提供误差估计。我们发现该误差估计与神经网络的预测误差高度相关，且提高IRK方法的阶数可改善估计效果。我们以逻辑斯蒂方程为例展示了针对物理信息神经网络（PINNs）的该估计方法，随后将其应用于电力系统建模中常用的四状态发电机模型。