Learning unknown stochastic differential equations (SDEs) from observed data
is a significant and challenging task with applications in various fields.
Current approaches often use neural networks to represent drift and diffusion
functions, and construct likelihood-based loss by approximating the transition
density to train these networks. However, these methods often rely on one-step
stochastic numerical schemes, necessitating data with sufficiently high time
resolution. In this paper, we introduce novel approximations to the transition
density of the parameterized SDE: a Gaussian density approximation inspired by
the random perturbation theory of dynamical systems, and its extension, the
dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust
density approximation, our method exhibits superior accuracy compared to
baseline methods in learning the fully unknown drift and diffusion functions
and computing the invariant distribution from trajectory data. And it is
capable of handling trajectory data with low time resolution and variable, even
uncontrollable, time step sizes, such as data generated from Gillespie's
stochastic simulations. We then conduct several experiments across various
scenarios to verify the advantages and robustness of the proposed method.

本文介绍了一种从轨迹数据学习未知随机微分方程的方法，通过近似参数化 SDE 的过渡密度的高斯密度近似和动力学高斯混合近似方法，实现了对完全未知漂移和扩散函数的精确学习和计算不变分布，适用于低时间分辨率和可变的，甚至是不可控的时间步长的轨迹数据。通过在不同场景下进行多个实验，验证了所提方法的优势和鲁棒性。