Learning a class of stochastic differential equations via numerics-informed Bayesian denoising

Learning stochastic differential equations (SDEs) from observational data via neural networks is an important means of quantifying uncertainty in dynamical systems. The learning networks are typically built upon denoising the stochastic systems by harnessing their inherent deterministic nature, such as the Fokker-Planck equations related to SDEs. In this paper we propose the numerics-informed denoising by taking expectations on the Euler-Maruyama numerical scheme of SDEs, and then using the Bayesian neural networks (BNNs) to approximate the expectations through variational inference on the weights' posterior distribution. The approximation accuracy of the BNNs is analyzed. Meanwhiles we give a data acquisition method for learning non-autonomous differential equations (NADEs) which respects the time-variant nature of NADEs' flows. Numerical experiments on three models show effectiveness of the proposed methods.