Sparse identification of nonlocal interaction kernels in nonlinear gradient flow equations via partial inversion

Abstract

We address the inverse problem of identifying nonlocal interaction potentials in nonlinearaggregation–di¿usion equations from noisy discrete trajectory data. Our approachinvolves formulating and solving a regularized variational problem, which requires min-imizing a quadratic error functional across a set of hypothesis functions, further aug-mented 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, validatingthe error functional ability in controlling the 2-Wasserstein distance between solutions generated using the true and estimated interaction potentials. Our work also includes anerror analysis of estimators caused by discretization and observational errors in practicalimplementations. We demonstrate the e¿ectiveness of the methods through various 1Dand 2D examples showcasing collective behaviors.

J.A.C. and G.E.R. were supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 883363). J.A.C. was also partially supported by EPSRC grant numbers EP/T022132/1 and EP/V051121/1. G.E.R. acknowledges the support from the research group 2021 SGR 00087 and the project macroKNIGHTs (PID2022-143012NA-100) funded by the Spanish Ministry of Science and Innovation. L.M. was supported by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation (EP/S023925/1). S.T. received partial support from the Hellman Faculty Fellowship and the Faculty Early Career Development Awards, funded by the University of California Santa Barbara and the NSF DMS under grant numbers 2111303 and 2340631. S.T. extends gratitude to Ben Adcock for valuable discussions on LASSO. Additionally, a portion of this research was conducted during visits by J.A.C. and S.T. to the Simons Institute for the Theory of Computing. L.M. wants to thank Ben Hambly and Markus Schmidtchen for their helpful comments and suggestions.

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