A Wiener path integral variational formulation with free boundaries is developed for determining the stochastic response of high-dimensional nonlinear dynamical systems in a computationally efficient manner. Specifically, a Wiener path integral representation of a marginal or lower-dimensional joint response probability density function is derived. Due to this a priori marginalization, the associated computational cost of the technique becomes independent of the degrees of freedom (d.f.) or stochastic dimensions of the system, and thus, the ‘curse of dimensionality’ in stochastic dynamics is circumvented. Two indicative numerical examples are considered for highlighting the capabilities of the technique. The first relates to marine engineering and pertains to a structure exposed to nonlinear flow-induced forces and subjected to non-white stochastic excitation. The second relates to nano-engineering and pertains to a 100-d.f. stochastically excited nonlinear dynamical system modelling the behaviour of large arrays of coupled nano-mechanical oscillators. Comparisons with pertinent Monte Carlo simulation data demonstrate the computational efficiency and accuracy of the developed technique.

中文翻译：

---

解决随机动力学中的维数灾难：具有自由边界的维纳路径积分变分公式

开发了具有自由边界的维纳路径积分变分公式，用于以计算效率高的方式确定高维非线性动力系统的随机响应。具体而言，导出边缘或低维联合响应概率密度函数的维纳路径积分表示。由于这种先验边缘化，该技术的相关计算成本变得独立于系统的自由度 (df) 或随机维度，因此，可以避免随机动力学中的“维度灾难”。考虑了两个指示性数值示例以突出该技术的能力。第一个与海洋工程有关，涉及暴露于非线性流动诱导力并受到非白色随机激励的结构。第二个与纳米工程有关，涉及模拟大型耦合纳米机械振荡器阵列的行为的 100-df 随机激励非线性动力系统。与相关蒙特卡罗模拟数据的比较证明了所开发技术的计算效率和准确性。