Unveiling parsimonious differential equation models from corrupted and incomplete data

• Grant number: 24/02658-8
• Support Opportunities: Regular Research Grants
• Start date: June 01, 2024
• End date: May 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Mobility Program: SPRINT - Projetos de pesquisa - Mobilidade
• Principal Investigator: Cassio Machiaveli Oishi
• Grantee: Cassio Machiaveli Oishi
• Principal researcher abroad: GEOVANI NUNES GRAPIGLIA
• Institution abroad: Universitè Catolique de Louvain (UCL), Belgium
• Host Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil
• Associated research grant: 13/07375-0 - CeMEAI - Center for Mathematical Sciences Applied to Industry, AP.CEPID

Abstract

Mathematical models allow scientists to translate the behavior of complex systems into a set of equations describing the relation between the key variables. Among different mathematical models, differential equations constitute a powerful tool for modelling the evolution of systems over time, making them suitable to model dynamic phenomena such as population growth, spread of diseases and fluid flow. Usually, the derivation of the differential equations that describe a specific phenomenon is done from first principles. However, this can be a challenging task for nonlinear and multi-scale phenomena. Recent advances in Machine Learning have motivated several attempts to automate the discovery of differential equations directly from data available about the system A very promising data-driven approach is the Sparse Identification of Nonlinear Dynamics (SINDy), developed by Brunton et al (2016). Assuming that a given phenomenon can be described by a system of differential equations defined by an unknown mapping, SINDy tries to find this mapping as a linear combination of a finite number of candidate nonlinear functions. The coefficients of this linear combination are computed by minimizing a loss function formed by a data fidelity term and a regularization term that promotes sparsity. Despite its success in several cases, the basic formulation of SINDy described has two main drawbacks. Firstly, it uses the Euclidian norm in the data fidelity term, which is problematic for real-world datasets corrupted by noise and outliers. Secondly, SINDy requires complete data about all the state variables of the system under study, which are not always available. In the present project we will attack these two issues by proposing alternatives to the optimization problem by which the differential equation is identified. The goal is to make SINDy more robust to noise and outliers in the data, and also expand its applicability to systems with incomplete data about the state variables. (AU)