DynGMA: a robust approach for learning stochastic differential equations from data
This work addresses a challenging problem in scientific computing and data-driven modeling for researchers dealing with stochastic systems, though it appears to be an incremental improvement over existing neural network-based methods.
The authors tackled the problem of learning unknown stochastic differential equations from observed data, particularly when data has low time resolution or variable time steps, by introducing DynGMA, a dynamical Gaussian mixture approximation for transition density. Their method demonstrated superior accuracy in learning drift and diffusion functions and computing invariant distributions compared to baseline approaches.
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.