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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Learning particle swarming models from data with Gaussian processes
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by Jinchao Feng, Charles Kulick, Yunxiang Ren and Sui Tang;
Math. Comp. 93 (2024), 2391-2437
DOI: https://doi.org/10.1090/mcom/3915
Published electronically: November 15, 2023

Abstract:

Interacting particle or agent systems that exhibit diverse swarming behaviors are prevalent in science and engineering. Developing effective differential equation models to understand the connection between individual interaction rules and swarming is a fundamental and challenging goal. In this paper, we study the data-driven discovery of a second-order particle swarming model that describes the evolution of $N$ particles in $\mathbb {R}^d$ under radial interactions. We propose a learning approach that models the latent radial interaction function as Gaussian processes, which can simultaneously fulfill two inference goals: one is the nonparametric inference of the interaction function with pointwise uncertainty quantification, and the other is the inference of unknown scalar parameters in the noncollective friction forces of the system. We formulate the learning problem as a statistical inverse learning problem and introduce an operator-theoretic framework that provides a detailed analysis of recoverability conditions, establishing that a coercivity condition is sufficient for recoverability. Given data collected from $M$ i.i.d trajectories with independent Gaussian observational noise, we provide a finite-sample analysis, showing that our posterior mean estimator converges in a Reproducing Kernel Hilbert Space norm, at an optimal rate in $M$ equal to the one in the classical 1-dimensional Kernel Ridge regression. As a byproduct, we show we can obtain a parametric learning rate in $M$ for the posterior marginal variance using $L^{\infty }$ norm and that the rate could also involve $N$ and $L$ (the number of observation time instances for each trajectory) depending on the condition number of the inverse problem. We provide numerical results on systems exhibiting different swarming behaviors, highlighting the effectiveness of our approach in the scarce, noisy trajectory data regime.
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Bibliographic Information
  • Jinchao Feng
  • Affiliation: Department of Mathematics, School of Sciences, Great Bay University, Dongguan, Guangdong 523000, People’s Republic of China
  • MR Author ID: 1531758
  • ORCID: 0000-0001-9330-4983
  • Email: jcfeng@gbu.edu.cn
  • Charles Kulick
  • Affiliation: Departments of Mathematics, University of California, Santa Barbara, Isla Vista, CA 93107, USA
  • MR Author ID: 1556839
  • ORCID: 0000-0002-2623-9349
  • Email: charles@math.ucsb.edu
  • Yunxiang Ren
  • Affiliation: Department of Physics, Harvard University, Cambridge, MA 02138, USA
  • MR Author ID: 1250599
  • Email: yren@g.harvard.edu
  • Sui Tang
  • Affiliation: Departments of Mathematics, University of California, Santa Barbara, Isla Vista, CA 93107, USA
  • MR Author ID: 1107641
  • ORCID: 0000-0003-3284-5123
  • Email: suitang@ucsb.edu
  • Received by editor(s): June 8, 2022
  • Received by editor(s) in revised form: June 12, 2022, March 6, 2023, and August 16, 2023
  • Published electronically: November 15, 2023
  • Additional Notes: The second author and fourth author were partially supported by Faculty Early Career Development Award sponsored by University of California Santa Barbara, Hellman Family Faculty Fellowship, and the NSF DMS-2111303. The second author was partly supported by NSF grant NSF DMS-2111303.
    The fourth author is the corresponding author. The authors are listed in alphabetical order.
  • © Copyright 2023 American Mathematical Society
  • Journal: Math. Comp. 93 (2024), 2391-2437
  • MSC (2020): Primary 62G05, 70F17, 65F22, 68Q32, 65Cxx; Secondary 70F40, 65Gxx
  • DOI: https://doi.org/10.1090/mcom/3915
  • MathSciNet review: 4759379