Nonparametric estimation problem of uncertain differential equations based on artificial neural networks

Uncertain differential equations (UDEs) provide an effective framework for modeling uncertain dynamic systems. In recent years, the study of parameter estimation for UDEs has gained significant attention. However, in practical applications, parameterized models may not fully capture system behavior due to unknown external disturbances, noise, parameter uncertainty, or variations in initial conditions. To address these challenges, it is crucial to consider nonparametric estimation methods based on observational data. Artificial neural networks (ANNs) are mathematical models inspired by the structure and function of biological neural systems, possessing the universal approximation property, which allows them to approximate any continuous function on a compact domain to arbitrary accuracy. In this article, we propose two ANN-based approaches for the nonparametric estimation of UDEs. After that, some numerical examples are given, and the feasibility of nonparametric estimation method is illustrated by the use of residuals and uncertain hypothesis test. To further demonstrate effectiveness, we conduct a comparative analysis between the two proposed neural network-based approaches. Finally, the method is applied to the classic Snowshoe Hare and Canadian Lynx dataset from 1845 to 1935. The results show that the proposed method can effectively learn the nonlinear predator-prey cycles and accurately predict future population dynamics with reliable uncertainty bounds.