Learning mean-field equations from particle data using WSINDy.

Area of Science:

• Computational physics
• Applied mathematics
• Complex systems

Background:

• Interacting particle systems (IPS) often involve high computational complexity.
• Existing system identification methods struggle with large particle numbers and limited experimental data.
• Noise robustness is crucial for realistic simulations of physical systems.

Purpose of the Study:

• To develop a computationally efficient and noise-robust system identification method for IPS.
• To enable the recovery of governing stochastic differential equations for large-scale IPS.
• To contrast the proposed method with existing strong-form approaches.

Main Methods:

• Utilizing mean-field theory concepts for IPS.
• Applying the weak-form sparse identification of nonlinear dynamics (WSINDy) algorithm.
• Developing a scheme for systems with thousands of particles and fewer than 100 experiments.

Main Results:

• Proving convergence rate under standard regularity assumptions in an ordinary least squares setting.
• Demonstrating numerical convergence rates in one and two spatial dimensions.
• Successfully applying the method to homogenization theory, swarm dynamics, and the Keller-Segel model.

Conclusions:

• The developed weak-form sparse identification method offers a fast and reliable approach for IPS.
• The method effectively reduces computational complexity for large particle numbers.
• This technique provides robustness to intrinsic and extrinsic noise in system identification.