An approximate operator-based learning method for the numerical solution of stochastic differential equations

Stochastic differential equation (SDE in short) solvers find numerous ap- plications across various fields. However, in practical simulations, we usu- ally resort to using Itˆo-Taylor series-based methods like the Euler-Maruyama method. These methods often suffer from the limitation of fixed time scales and recalculations for different Brownian motions, which lead to computa- tional inefficiency, especially in generative and sampling models. To address these issues, we propose a novel approach: learning a mapping between the solution of SDE and corresponding Brownian motion. This mapping exhibits versatility across different scales and requires minimal paths for training. Specifically, we employ the DeepONet method to learn a nonlinear mapping. And we also assess the efficiency of this method through simulations con- ducted at varying time scales. Additionally, we evaluate its generalization performance to verify its good versatility in different scenarios