Extracting the governing stochastic differential equation model from elusive data is crucial to understand and forecast dynamics for various systems. We devise a method to extract the drift term and estimate the diffusion coefficient of a governing stochastic dynamical system, from its time-series data for the most probable transition trajectory. By the Onsager–Machlup theory, the most probable transition trajectory satisfies the corresponding Euler–Lagrange equation, which is a second-order deterministic ordinary differential equation (ODE) involving the drift term and diffusion coefficient. We first estimate the coefficients of the Euler–Lagrange equation based on the data of the most probable trajectory, and then calculate the drift and diffusion coefficient of the governing system. These two steps involve sparse regression and optimization for a loss function with parameters. We select the estimators from all the results with different parameters by comparing the errors caused by the difference between the two sides of the second-order ODE for the most probable transition trajectory. We finally illustrate our method, especially for parameter selection, with examples to verify the effectiveness of our proposed method.

从难以捉摸的数据中提取控制随机微分方程模型对于理解和预测各种系统的动态至关重要。我们设计了一种方法，从最可能的转变轨迹的时间序列数据中提取漂移项并估计控制随机动力系统的扩散系数。根据Onsager-Machlup理论，最可能的转变轨迹满足相应的Euler-Lagrange方程，该方程是一个涉及漂移项和扩散系数的二阶确定性常微分方程（ODE）。我们首先根据最可能轨迹的数据估计欧拉-拉格朗日方程的系数，然后计算控制系统的漂移和扩散系数。这两个步骤涉及稀疏回归和带参数的损失函数的优化。我们通过比较最可能转移轨迹的二阶 ODE 两侧差异所引起的误差，从具有不同参数的所有结果中选择估计量。最后，我们通过示例说明了我们的方法，特别是参数选择，以验证我们提出的方法的有效性。