Many physical systems characterized by nonlinear multiscale interactions can be modelled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behaviour are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behaviour to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using forward and adjoint Fokker–Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by coloured noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply Langevin regression to experimental measurements of a turbulent bluff body wake and show that the statistical behaviour of the centre of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.

许多以非线性多尺度相互作用为特征的物理系统可以通过将未解决的自由度视为随机波动来建模。然而，即使知道微观控制方程和定性宏观行为，通常也很难推导出与观察结果一致的随机模型。对于诸如湍流之类的系统尤其如此，其中扰动的行为不像高斯白噪声，从而将非马尔可夫行为引入动力学。我们通过一个框架来解决这些挑战，该框架用于从实验数据中识别可解释的随机非线性动力学，使用正向和伴随 Fokker-Planck 方程来加强统计一致性。如果 Langevin 方程的形式未知，一个简单的稀疏过程可以提供适当的函数形式。我们证明该方法可以在两个人工示例中学习随机模型：恢复由有色噪声强制的非线性朗之万方程，并用相应的一阶分岔范式逼近双阱势中粒子的二阶动力学。最后，我们将朗之万回归应用于湍流钝体尾流的实验测量，并表明压力中心的统计行为可以通过由非线性状态相关噪声驱动的相应层流的动力学来描述。恢复由有色噪声强迫的非线性朗之万方程，并用相应的一阶分岔范式近似双阱势中粒子的二阶动力学。最后，我们将朗之万回归应用于湍流钝体尾流的实验测量，并表明压力中心的统计行为可以通过由非线性状态相关噪声驱动的相应层流的动力学来描述。恢复由有色噪声强迫的非线性朗之万方程，并用相应的一阶分岔范式近似双阱势中粒子的二阶动力学。最后，我们将朗之万回归应用于湍流钝体尾流的实验测量，并表明压力中心的统计行为可以通过由非线性状态相关噪声驱动的相应层流的动力学来描述。