https://doi.org/10.1140/epjs/s11734-024-01270-8
Regular Article
Deterministic-like data-driven discovery of stochastic differential equations via the Feynman–Kac formalism
1
The College of Informatics, Huazhong Agriculture University, and the China–Poland Belt and Road Joint Laboratory on Measurement and Control Technology, Huazhong University of Science and Technology, 430070, Wuhan, Hubei, People’s Republic of China
2
School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Ministry of Education Key Lab of Intelligent Control and Image Processing, 430074, Wuhan, China
Received:
20
March
2024
Accepted:
18
July
2024
Published online:
6
August
2024
This paper develops a data-driven deterministic identification architecture for discovering stochastic differential equations (SDEs) directly from data. The architecture first generates deterministic data for stochastic processes using the Feynman–Kac formula, and gives a parabolic partial differential equation (PDE) associated with the SDE. Then, a sparse regression model is proposed to discover drift and diffusion terms in SDEs using PDE data-driven techniques, where a large candidate library of potential terms only for the drift and diffusion coefficients in SDEs need be constructed. To simultaneously infer the drift and diffusion terms, we proposed a sequential thresholded reweighted least-squares algorithm to solve the constructed sparse regression model. The main advantage of the proposed method is that on the one hand, theoretical and numerical identification results of PDEs can be used for SDEs, on the score, our SDE identification problem is translated into the parameter estimation problem of PDEs, on the other hand, the proposed algorithm is easily executed and can enhance the sparsity and accuracy. Through several classical SDEs and ordinary differential equations, the effectiveness of the proposed data-driven method is demonstrated, and several comparison experiments with state-of-the-art approaches is provided to illustrate the superiority of the developed algorithm.
This work was supported by the National Nature Science Foundation with No. 61903148, No. 62303118 and No. 62373160; in part by the Fundamental Research Funds for the Central Universities No. 2662021LXQD001 (Corresponding author: Xiuting Li).
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