Identifiability of interaction kernels in mean-field equations of interacting particles

Quanjun Lang; Fei Lu

doi:10.3934/fods.2023007

Article Contents

2023, Volume 5, Issue 4: 480-502. Doi: 10.3934/fods.2023007

This issue Previous Article Fast computation of persistent homology representatives with involuted persistent homology Next Article A log-Gaussian Cox process with sequential Monte Carlo for line narrowing in spectroscopy

Identifiability of interaction kernels in mean-field equations of interacting particles

Quanjun Lang and
Fei Lu^,

Department of Mathematics, Johns Hopkins University, USA

^*Corresponding author: Fei Lu
^*Corresponding author: Fei Lu

Received: June 2022

Revised: April 2023

Early access: June 2023

Published: December 2023

FL is supported by NSF DMS-1913243, FA9550-20-1-0288 and DE-SC0021361.

Abstract / Introduction Full Text(HTML) Figure(3) / Table(2) Related Papers Cited by

Abstract

This study examines the identifiability of interaction kernels in mean-field equations of interacting particles or agents, an area of growing interest across various scientific and engineering fields. The main focus is identifying data-dependent function spaces where a quadratic loss functional possesses a unique minimizer. We consider two data-adaptive $ L^2 $ spaces: one weighted by a data-adaptive measure and the other using the Lebesgue measure. In each $ L^2 $ space, we show that the function space of identifiability is the closure of the RKHS associated with the integral operator of inversion. Alongside prior research, our study completes a full characterization of identifiability in interacting particle systems with either finite or infinite particles, highlighting critical differences between these two settings. Moreover, the identifiability analysis has important implications for computational practice. It shows that the inverse problem is ill-posed, necessitating regularization. Our numerical demonstrations show that the weighted $ L^2 $ space is preferable over the unweighted $ L^2 $ space, as it yields more accurate regularized estimators.

Keywords:

Mathematics Subject Classification: Primary: 35R30; Secondary: 68Q32.

Citation:

Full Text(HTML)

Figure 1. Regularized estimators via truncated SVD for the three examples, superimposed with the exploration measure $ \bar \rho_T $. The weighted SVD ("$ H_R $ Estimator") has smaller errors than the unweighted SVD ("$ H_G $ Estimator"), while both are significantly more accurate than the un-regularized estimator ("all spline")

Download: Full-size image PowerPoint slide

Figure 2. Eigenfunctions in the estimation via weighted and unweighted SVD in Figure 1. The weighted operator $ \mathcal{L}_R $ has smoother eigenfunctions than the unweighted operator $ \mathcal{L}_G $

Download: Full-size image PowerPoint slide

Figure 3. SVD analysis of the regression in three examples. Here $ R $ represents the weighted SVD and $ G $ represents the unweighed SVD. In all three examples, the weighted SVD has larger eigenvalues than those of the unweighted SVD; and it has slightly smaller ratios $ \frac{ \boldsymbol{{u}}_i^\top b}{\sigma_i} $

Download: Full-size image PowerPoint slide

Table 1. Notations

	Radial kernel	Non-radial kernel
Interaction potential	$ \Psi(\|x\|) $ and $ \psi = \Psi' $;	$ \Psi(x) $
Interaction kernel	$ K_\psi(x) = \psi(\|x\|)\frac{x}{\|x\|} $	$ K_\psi(x) = \nabla \Psi(x) $
Loss functional	$ \mathcal{E}(\psi) $	$ \mathcal{E}(K_\psi) $
Density of exploration measure	$ \bar \rho_T $, $ \mathcal{X} =\mathrm{support}( \bar \rho_T) $, in (9)
Function space of learning	$ L^2( \mathcal{X}) $ and $ L^2_{ \bar \rho_T} $
Mercer kernel & RKHS in $ L^2( \mathcal{X}) $	$ {\overline G_T} $ in (11) and $ \mathcal{H}_G $	$ {\overline F_T} $ in (24) and $ \mathcal{H}_F $
Mercer kernel & RKHS in $ L^2_{ \bar \rho_T} $	$ {\overline R_T} $ in (19) and $ \mathcal{H}_R $	$ {\overline Q_T} $ in (24) and $ \mathcal{H}_Q $

| Show Table

DownLoad: CSV

Table 2. Notation of variables in the eigenvalue problems on $ \mathcal{H} $

	in $ L^2( \mathcal{X}) $	in $ L^2_{ \bar \rho_T} $
integral kernel and operator	$ \widehat{ {\overline G_T}}\approx {\overline G_T} $, $ \mathcal{L}_{\widehat{ {\overline G_T}}} \approx \mathcal{L}_{ {\overline G_T}} $	$ \widehat{ {\overline R_T}} \approx {\overline R_T} $, $ \mathcal{L}_{\widehat{ {\overline R_T}}}\approx \mathcal{L}_{ {\overline R_T}} $
eigenfunction and eigenvalue	$ \mathcal{L}_{\widehat{ {\overline G_T}}} \widehat{\varphi_k} =\widehat{\lambda}_k \widehat{\varphi_k} $	$ \mathcal{L}_{\widehat{ {\overline R_T}}} \widehat{\psi_k} =\widehat{\gamma}_k \widehat{\psi_k} $
eigenvector and eigenvalue	$ A_n \overrightarrow{\varphi_k} =\widehat{\lambda}_k B_n^G \overrightarrow{\varphi_k} $	$ A_n \overrightarrow{\psi_k}= \widehat{\gamma}_k B_n^R\overrightarrow{\psi_k} $

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	A. S. Baumgarten and K. Kamrin, A general constitutive model for dense, fine-particle suspensions validated in many geometries, Proc. Natl. Acad. Sci. USA, 116 (2019), 20828-20836. doi: 10.1073/pnas.1908065116.
[2]	N. Bell, Y. Yu and P. J. Mucha, Particle-based simulation of granular materials, In Proceedings of the 2005 ACM SIGGRAPH/Eurographics Symposium on Computer Animation - SCA '05, Los Angeles, California, 2005. ACM Press, 77-86. doi: 10.1145/1073368.1073379.
[3]	C. Berg, J. P. R. Christensen and P. Ressel, Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions, volume 100. New York: Springer, 1984. doi: 10.1007/978-1-4612-1128-0.
[4]	M. Bongini, M. Fornasier, M. Hansen and M. Maggioni, Inferring interaction rules from observations of evolutive systems Ⅰ: The variational approach, Mathematical Models and Methods in Applied Sciences, 27 (2017), 909-951. doi: 10.1142/S0218202517500208.
[5]	J. A. Carrillo, K. Craig and Y. Yao, Aggregation-diffusion equations: Dynamics, asymptotics, and singular limits, In Active Particles, Volume 2, Springer, 2019, 65-108.
[6]	J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271. doi: 10.1215/00127094-2010-211.
[7]	X. Chen, Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data, Electron. Commun. Probab., 26 (2021), Paper No. 45, 13 pp. arXiv: 2007.11048, Math Stat, 2021. doi: 10.1214/21-ecp416.
[8]	F. Cucker and D. X. Zhou, Learning Theory: An Approximation Theory Viewpoint, volume 24, Cambridge University Press, 2007. doi: 10.1017/CBO9780511618796.
[9]	L. Della Maestra and M. Hoffmann, The LAN property for McKean-Vlasov models in a mean-field regime, Stochastic Process. Appl., 155 (2023), 109-146. doi: 10.1016/j.spa.2022.10.002.
[10]	L. Della Maestra and M. Hoffmann, Nonparametric estimation for interacting particle systems: McKean–Vlasov models, Probability Theory and Related Fields, 182 (2022), 551-613. doi: 10.1007/s00440-021-01044-6.
[11]	J. Fan and Q. Yao, Nonlinear Time Series: Nonparametric and Parametric Methods, Springer, New York, NY, 2003. doi: 10.1007/b97702.
[12]	P. C. Hansen, Regularization Tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer Algor, 6 (1994), 1-35. doi: 10.1007/BF02149761.
[13]	P. C. Hansen, The L-curve and its use in the numerical treatment of inverse problems, In Computational Inverse Problems in Electrocardiology, ed. P. Johnston, Advances in Computational Bioengineering, WIT Press, 2000, 119-142.
[14]	P.-E. Jabin and Z. Wang, Mean field limit for stochastic particle systems, In N. Bellomo, P. Degond, and E. Tadmor, editors, Active Particles, Volume 1, Springer International Publishing, Cham, 2017, 379-402.
[15]	P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with $w^{-1, \infty}$ kernels, Invent. Math., 214 (2018), 523-591. doi: 10.1007/s00222-018-0808-y.
[16]	R. A. Kasonga, Maximum likelihood theory for large interacting systems, SIAM J. Appl. Math., 50 (1990), 865-875. doi: 10.1137/0150050.
[17]	Q. Lang and F. Lu, Learning interaction kernels in mean-field equations of first-order systems of interacting particles, SIAM Journal on Scientific Computing, 44 (2022), A260-A285. doi: 10.1137/20M1377072.
[18]	Z. Li and F. Lu, On the coercivity condition in the learning of interacting particle systems, arXiv preprint, arXiv: 2011.10480, 2020.
[19]	Z. Li, F. Lu, M. Maggioni, S. Tang and C. Zhang, On the identifiability of interaction functions in systems of interacting particles, Stochastic Processes and their Applications, 132 (2021), 135-163. doi: 10.1016/j.spa.2020.10.005.
[20]	F. Lu, Q. Lang and Q. An, Data adaptive RKHS Tikhonov regularization for learning kernels in operators, Proceedings of Mathematical and Scientific Machine Learning, PMLR 190, 158-172, 2022.
[21]	F. Lu, M. Maggioni and S. Tang, Learning interaction kernels in heterogeneous systems of agents from multiple trajectories, Journal of Machine Learning Research, 22 (2021), 1-67.
[22]	F. Lu, M. Maggioni and S. Tang, Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories, Foundations of Computational Mathematics, 22 (2022), 1013-1067. doi: 10.1007/s10208-021-09521-z.
[23]	F. Lu, M. Zhong, S. Tang and M. Maggioni, Nonparametric inference of interaction laws in systems of agents from trajectory data, Proceedings of the National Academy of Sciences of the United States of America, 116 (2019), 14424-14433. doi: 10.1073/pnas.1822012116.
[24]	F. Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes, Ann. Appl. Probab., 13 (2003), 540-560. doi: 10.1214/aoap/1050689593.
[25]	S. Méléard, Asymptotic Behaviour of Some Interacting Particle Systems; McKean-Vlasov and Boltzmann Models, volume 1627, Springer Berlin Heidelberg, Berlin, Heidelberg, 1996, 42-95. doi: 10.1007/BFb0093177.
[26]	S. Mostch and E. Tadmor, Heterophilious dynamics enhances consensus, Siam Review, 56 (2014), 577-621. doi: 10.1137/120901866.
[27]	C. E. Rasmussen, Gaussian processes in machine learning, In Summer School on Machine Learning, Springer, 2003, 63-71.
[28]	L. Sharrock, N. Kantas, P. Parpas and G. A. Pavliotis, Parameter estimation for the McKean-Vlasov stochastic differential equation, arXiv preprint, arXiv: 2106.13751, 2021.
[29]	H. Sun, Mercer theorem for RKHS on noncompact sets, Journal of Complexity, 21 (2005), 337-349. doi: 10.1016/j.jco.2004.09.002.
[30]	A.-S. Sznitman, Topics in Propagation of Chaos, volume 1464, Springer Berlin Heidelberg, Berlin, Heidelberg, 1991, 165-251. doi: 10.1007/BFb0085169.
[31]	T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004.
[32]	R. Yao, X. Chen, and Y. Yang, Mean-field nonparametric estimation of interacting particle systems, arXiv preprint, arXiv: 2205.07937, 2022.