Stochastic collocation (SC) is a well-known non-intrusive method of constructing surrogate models for uncertainty quantification. In dynamical systems, SC is especially suited for full-field uncertainty propagation that characterizes the distributions of the high-dimensional primary solution fields of a model with stochastic input parameters. However, due to the highly nonlinear nature of the parameter-to-solution map in even the simplest dynamical systems, the constructed SC surrogates are often inaccurate. This work presents an alternative approach, where we apply the SC approximation over the dynamics of the model, rather than the solution. By combining the data-driven sparse identification of nonlinear dynamics (SINDy) framework with SC, we construct dynamics surrogates and integrate them through time to construct the surrogate solutions. We demonstrate that the SC-over-dynamics framework leads to smaller errors, both in terms of the approximated system trajectories as well as the model state distributions, when compared against full-field SC applied to the solutions directly. We present numerical evidence of this improvement using three test problems: a chaotic ordinary differential equation, and two partial differential equations from solid mechanics.

翻译：随机配置法（Stochastic Collocation, SC）是一种构建不确定性量化代理模型的经典非侵入式方法。在动力系统中，SC特别适用于全场不确定性传播，可表征具有随机输入参数的模型的高维主解场分布。然而，即使是最简单的动力系统，其参数-解映射也具有高度非线性特性，这导致构建的SC代理模型常存在精度不足的问题。本文提出一种替代方案：将SC近似应用于模型动力学过程而非最终解。通过结合数据驱动的非线性动力学稀疏辨识（SINDy）框架与SC，我们构建动力学代理模型，并对其进行时间积分以获得代理解。与直接应用于解的全场SC相比，我们证明SC-动力学框架在近似系统轨迹和模型状态分布两方面均能实现更小的误差。通过三个测试问题——混沌常微分方程及固体力学中的两个偏微分方程——我们提供了该改进效果的数值证据。