Sparse Identification of Stochastic Differential Equations from Kramers Moyal approximations
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SINDy algorithm (Sparse Identification of Nonlinear Dynamics) uses sparse regression techniques and machine learning to identify the governing equations in dynamical systems described by autonomous ODEs ([3, 4]). A crucial step involves the construction of a dictionary Θ of candidate functions from data measurements, aiming to represent each component of the dynamics f as a linear combination of these functions. This method has been used to identify the governing equations in ODE and PDE systems, and it has also been extended to the analysis of SDE systems. In the latter case, the columns of the dictionary Θ are constructed from data measurements of both drift and diffusion functions using Kramers-Moyal approximations ([1, 5]). In this work, we show how to recover the right-hand side of an SDE by means of a stochastic version of the SINDy algorithm. The MATLAB implementation will be described and numerical simulations will highlight the relevance of the proposed approach ([2]). This work falls within the activities of PRIN-MUR 2022 project 20229P2HEA “Stochastic numerical modelling for sustainable innovation”, CUP: E53D23017940001, granted by the Italian Ministry of University and Research within the framework of the Call relating to the scrolling of the final rankings of the PRIN 2022 call. References [1] L. Boninsegna, F. Nüske, C. Clementi, Sparse learning of stochastic dynamical equations, J. Chem. Phys., 148 (2018). [2] D. Breda, D. Conte, R. D’Ambrosio, I. Santaniello, M. Tanveer, Sparse Identification of Nonlinear Dynamics for Delay and Stochastic Differential Equations, 9th European Congress on Computational Methods in Applied Sciences and Engineering, Lisboa, Portugal, 3–7 June 2024, Session: MS178 - Numerical Modeling and Data Analysis for Advancing Sustainable Innovation. [3] S. L. Brunton, J. N. Kutz, Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control, Cambridge University Press, Singapore, 2019. [4] S. L. Brunton, J. L. Proctor, J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, PNAS, 113 (2016), pp. 3932–3937. [5] M. Wanner, I. Mezić, On Numerical Methods for Stochastic SINDy, 2023.
