Stochastic differential equation (SDE in short) solvers find numerous applications across various fields. However, in practical simulations, we usually resort to using Ito-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 computational 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 conducted at varying time scales. Additionally, we evaluate its generalization performance to verify its good versatility in different scenarios.

翻译：随机微分方程求解器在多个领域有着广泛应用。然而在实际仿真中，我们通常采用基于伊藤-泰勒级数的方法（如欧拉-丸山法）。这类方法受限于固定时间步长及针对不同布朗运动需要重新计算，导致计算效率低下——尤其在生成模型与采样模型中。为解决上述问题，我们提出一种新方法：学习随机微分方程解与对应布朗运动之间的映射关系。该映射具有跨尺度的通用性，且训练所需路径数量极少。具体而言，我们采用DeepONet方法学习非线性映射，并通过不同时间尺度的仿真评估该方法效率。此外，我们测试了其泛化性能，验证了该方法在不同场景下的良好通用性。