标题： 神经常微分方程在刚性系统模型降阶中的应用 显示英文标题

标题： Neural Ordinary Differential Equations for Model Order Reduction of Stiff Systems

摘要： 神经常微分方程（ODEs）代表了机器学习与动力系统交叉领域的一项重大进展，为离散神经网络提供了连续时间的类比。 尽管它们具有潜力，但在实际应用中部署神经ODEs时常会遇到刚性问题的挑战，即解的某些分量的快速变化需要显式求解器采用过小的时间步长。 本文通过在时间上引入适当的重参数化来解决在模型降阶中使用神经ODEs时的刚性问题。 所考虑的映射是数据驱动的，并由隐式求解器在参考解上的自适应时间步长诱导产生。 我们证明该映射生成了一个非刚性的系统，可以用廉价的显式时间积分方案求解。 通过一个连接状态空间到时间重参数化的神经网络，可以恢复原始的、刚性的时态动态。 我们通过广泛的实验验证了我们的方法，在保证鲁棒性和准确性的同时，提高了神经ODE推理的效率。 此外，神经网络模型对于超出训练数据范围的时间也展示了良好的泛化性能。 显示英文摘要

摘要： Neural Ordinary Differential Equations (ODEs) represent a significant advancement at the intersection of machine learning and dynamical systems, offering a continuous-time analog to discrete neural networks. Despite their promise, deploying neural ODEs in practical applications often encounters the challenge of stiffness, a condition where rapid variations in some components of the solution demand prohibitively small time steps for explicit solvers. This work addresses the stiffness issue when employing neural ODEs for model order reduction by introducing a suitable reparametrization in time. The considered map is data-driven and it is induced by the adaptive time-stepping of an implicit solver on a reference solution. We show the map produces a nonstiff system that can be cheaply solved with an explicit time integration scheme. The original, stiff, time dynamic is recovered by means of a map learnt by a neural network that connects the state space to the time reparametrization. We validate our method through extensive experiments, demonstrating improvements in efficiency for the neural ODE inference while maintaining robustness and accuracy. The neural network model also showcases good generalization properties for times beyond the training data.