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

Jose A. Carrillo, Gissell Estrada-Rodriguez, Laszlo Mikolas, Sui Tang

TL;DR

The paper addresses the problem of identifying nonlocal interaction kernels $W$ in nonlinear gradient-flow PDEs from noisy trajectory data. It introduces a regularized variational approach that couples a quadratic data-fidelity term with an $\ell^1$ sparsity penalty, solved via a tailored Basis Pursuit method called PartInv to cope with highly coherent regression matrices. The authors establish stability results bounding the $d_2$ (2-Wasserstein) distance between the true and learned solutions through the error functional $\tilde{\mathcal{E}}_{\infty}$, and they provide a Gamma-convergence viewpoint in the no-diffusion setting along with discretization and observation-noise error analyses. Extensive 1D and 2D numerical experiments demonstrate that PartInv can robustly recover sparse interaction kernels under nonlinear and linear diffusion, including noisy and downsampled data, and it often outperforms standard BP methods, highlighting its practical potential for data-driven kernel discovery in aggregation-diffusion PDEs.

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 [57] and subspace pursuit algorithms [31], 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.

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

TL;DR

The paper addresses the problem of identifying nonlocal interaction kernels in nonlinear gradient-flow PDEs from noisy trajectory data. It introduces a regularized variational approach that couples a quadratic data-fidelity term with an sparsity penalty, solved via a tailored Basis Pursuit method called PartInv to cope with highly coherent regression matrices. The authors establish stability results bounding the (2-Wasserstein) distance between the true and learned solutions through the error functional , and they provide a Gamma-convergence viewpoint in the no-diffusion setting along with discretization and observation-noise error analyses. Extensive 1D and 2D numerical experiments demonstrate that PartInv can robustly recover sparse interaction kernels under nonlinear and linear diffusion, including noisy and downsampled data, and it often outperforms standard BP methods, highlighting its practical potential for data-driven kernel discovery in aggregation-diffusion PDEs.

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 [57] and subspace pursuit algorithms [31], 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.
Paper Structure (35 sections, 9 theorems, 151 equations, 11 figures, 9 tables, 1 algorithm)

This paper contains 35 sections, 9 theorems, 151 equations, 11 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A regularised variational approach via basis pursuit
  3. Notation
  4. The error functional
  5. The vanilla least squares solutions
  6. $\ell^1$ regularization via Basis Pursuit
  7. Stability estimates and $\Gamma$-convergence
  8. No diffusion, the aggregation equation case
  9. Mean-field dynamics and $\Gamma$-convergence
  10. Linear diffusion
  11. Nonlinear diffusion
  12. Numerical Schemes
  13. Discrete error functional
  14. Error bounds
  15. Implications for Optimal Estimation Accuracy:
  16. ...and 20 more sections

Key Result

Proposition 2.1

Let $\mu, \widehat{\mu} \in C([0,T],\mathcal{P}^2(\mathbb{R}^d))$ be solutions of nonlocal with the interaction potential and external potential $(W,V), (\widehat{W},\widehat{V})$ respectively. Then, under suitable regularity conditions on the velocity fields of $\mu$ and $\widehat{\mu}$, we have th where $C>0$ is a suitable constant depending only on the final time $T$ and the Lipschitz constant

Figures (11)

  • Figure 1: Profile of a subset of trajectory data used in our training where we choose $\Delta x = 6 \delta x$ and $\Delta t =50 \delta t$. A spectrum of colors transitioning from red to blue to symbolize the passage of time. (a) data generated from the numerical solver (b) data with 3% noise added.
  • Figure 2: Results with piecewise constant basis where we choose $\Delta x = 6 \delta x$ and $\Delta t =50 \delta t$. From (b), we clearly see that sparsity effectively regularizes the inverse problem and finds a solution that closely aligns with the ground truth coefficient vector $[5,5]$ with respect to the basis $[\psi_1,\psi_2]$.
  • Figure 3: Results of PartInv with sparsity $K=2$ using piecewise constant basis. (a) Accuracy for different levels of noise where we display the mean and standard deviation of relative errors over 100 trials. (b) Relation between the relative reconstruction error, given by the different color intensities, and the mesh size $(\Delta x,{\Delta t})$.
  • Figure 4: Results for PartInv with piecewise linear basis where we use the same training data as in Figure \ref{['fig:NCWleastsquare']}. (a)-(b) are the cases without support pruning. The case with support pruning with $K=3$ is presented in (c). We see it produced the most accurate estimation of the true coefficient $[5,5]$ with respect to the basis $(\psi_1,\psi_3)$.
  • Figure 5: Profile of the solution for $\Delta x = 5 \delta x, \Delta t = 2500 \delta t$. (a) a subset of solution data generated from the numerical solver (b) the solution data with $1\%$ noise added.
  • ...and 6 more figures

Theorems & Definitions (31)

  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Proposition 3.6
  • proof
  • ...and 21 more