复杂系统的随机物理约束算子推断

中文摘要 AI 辅助

由加性噪声驱动的二次非线性系统在科学和工程中广泛出现，它们既作为随机动力系统本身，也作为湍流的降阶模型（ROMs）。我们提出了随机物理约束算子推断（SPOpInf），这是一种非侵入式方法，从数据中学习随机微分方程，即二次-线性漂移和扩散幅度。与通过最小二乘拟合确定性漂移的算子推断（OpInf）不同，SPOpInf最大化欧拉-丸山增量似然，其中残差协方差本身是未知的，因此确定性和随机分量被联合校准；它通过优化中施加的线性等式约束精确地强制执行能量守恒二次结构；并且它采用上三角参数化，消除了完整二次张量的结构不可辨识性。由此产生的优化通过基于期望最大化（EM）的迭代学习算法求解，该算法轮流更新漂移系数和噪声水平：每次迭代仅需要廉价的加权最小二乘求解漂移和残差方差更新噪声。通过五个显示间歇性和混沌的随机基准系统，SPOpInf比OpInf更准确地恢复确定性算子，满足施加的约束达到机器精度，准确估计噪声幅度，并再现长时间统计。应用于随机Burgers方程和准地转双环流，相同的估计器产生非侵入式随机ROMs，在长时间内保持稳定，而无约束的替代方案则发散。

英文摘要

Quadratically nonlinear systems driven by additive noise arise throughout science and engineering, both as stochastic dynamical systems in their own right and as reduced-order models (ROMs) of turbulent flows. We propose stochastic physics-constrained operator inference (SPOpInf), a non-intrusive method that learns a stochastic differential equation, i.e., a quadratic--linear drift and a diffusion amplitude, from data. Unlike operator inference (OpInf), which fits a deterministic drift by least squares, SPOpInf maximizes an Euler--Maruyama increment likelihood in which the residual covariance is itself an unknown, so the deterministic and stochastic components are calibrated jointly; it enforces the energy-preserving quadratic structure exactly, through linear equality constraints imposed inside the optimization; and it works in an upper-triangular parametrization that removes the structural non-identifiability of the full quadratic tensor. The resulting optimization is solved by an expectation--maximization (EM)-based iterative learning algorithm that updates the drift coefficients and the noise levels in turn: each iteration requires only an inexpensive weighted least-squares solve for the drift and a residual-variance update for the noise. With five stochastic benchmark systems that show intermittency and chaos, SPOpInf recovers the deterministic operators more accurately than OpInf, satisfies the imposed constraints to machine precision, accurately estimates the noise amplitudes, and reproduces long-time statistics. Applied to a stochastic Burgers equation and a quasi-geostrophic double-gyre flow, the same estimator yields non-intrusive stochastic ROMs that remain stable over long times where unconstrained alternatives blow up.

发表机构

• Utah State University（犹他州立大学）

机构由 AI 辅助整理，请以论文原文为准。