Sparse identification of nonlocal interaction kernels in nonlinear gradient flow equations via partial inversion
Year: 2025
Type: article
Abstract: We address the inverse problem of identifying nonlocal interaction potentials in nonlinear aggregation–diffusion equations from noisy discrete trajectory data. Our approach involves formulating and solving a regularized variational problem, which requires minimizing a quadratic error functional across a set of hypothesis functions, further augmented by a sparsity-enhancing regularizer. We employ a partial inversion algorithm, akin to the CoSaMP and subspace pursuit algorithms, to solve the basis pursuit problem. A key theoretical contribution is our novel stability estimate for the PDEs, validating the error functional ability in controlling the 2-Wasserstein distance between solutions generated using the true and estimated interaction potentials. Our work also includes an error analysis of estimators caused by discretization and observational errors in practical implementations. We demonstrate the effectiveness of the methods through various 1D and 2D examples showcasing collective behaviors.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Awards: DMS-EPSRC: Stability Analysis for Nonlinear Partial Differential Equations across Multiscale Applications, Spectral element methods for fractional differential equations, with applications in applied analysis and medical imaging, G1600321425, Data-Driven Discovery of Dynamics in Interacting Agent Systems and Linear Diffusion Processes, G2691952350, G434337789, G6700243956, EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation, G6860964818, G7742257476, CAREER: Solving Estimation Problems of Networked Interacting Dynamical Systems Via Exploiting Low Dimensional Structures: Mathematical Foundations, Algorithms and Applications, G8920660035
OpenAlex ID: W4407240046