Parameter calibration with stochastic gradient descent for interacting particle systems driven by neural networks

We propose a neural network approach to model general interaction dynamics and an adjoint-based stochastic gradient descent algorithm to calibrate its parameters. The parameter calibration problem is considered as optimal control problem that is investigated from a theoretical and numerical point of view. We prove the existence of optimal controls, derive the corresponding first-order optimality system and formulate a stochastic gradient descent algorithm to identify parameters for given data sets. To validate the approach, we use real data sets from traffic and crowd dynamics to fit the parameters. The results are compared to forces corresponding to well-known interaction models such as the Lighthill–Whitham–Richards model for traffic and the social force model for crowd motion.

Space mapping-based optimization with the macroscopic limit of interacting particle systems

We propose a space mapping-based optimization algorithm for microscopic interacting particle dynamics which are infeasible for direct optimization. This is of relevance for example in applications with bounded domains for which the microscopic optimization is difficult. The space mapping algorithm exploits the relationship of the microscopic description of the interacting particle system and a corresponding macroscopic description as partial differential equation in the “many particle limit”. We validate the approach with the help of a toy problem that allows for direct optimization. Then we study the performance of the algorithm in two applications. A pedestrian flow is considered and the transportation of goods on a conveyor belt is optimized. The numerical results underline the feasibility of the proposed algorithm.

Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems

Population cross-diffusion systems of Shigesada–Kawasaki–Teramoto type are derived in a mean-field-type limit from stochastic, moderately interacting many-particle systems for multiple population species in the whole space. The diffusion term in the stochastic model depends nonlinearly on the interactions between the individuals, and the drift term is the gradient of the environmental potential. In the first step, the mean-field limit leads to an intermediate nonlocal model. The local cross-diffusion system is derived in the second step in a moderate scaling regime, when the interaction potentials approach the Dirac delta distribution. The global existence of strong solutions to the intermediate and the local diffusion systems is proved for sufficiently small initial data. Furthermore, numerical simulations on the particle level are presented.

Large Deviations and Gradient Flows for the Brownian One-Dimensional Hard-Rod System

We study a system of hard rods of finite size in one space dimension, which move by Brownian noise while avoiding overlap. We consider a scaling in which the number of particles tends to infinity while the volume fraction of the rods remains constant; in this limit the empirical measure of the rod positions converges almost surely to a deterministic limit evolution. We prove a large-deviation principle on path space for the empirical measure, by exploiting a one-to-one mapping between the hard-rod system and a system of non-interacting particles on a contracted domain. The large-deviation principle naturally identifies a gradient-flow structure for the limit evolution, with clear interpretations for both the driving functional (an ‘entropy’) and the dissipation, which in this case is the Wasserstein dissipation. This study is inspired by recent developments in the continuum modelling of multiple-species interacting particle systems with finite-size effects; for such systems many different modelling choices appear in the literature, raising the question how one can understand such choices in terms of more microscopic models. The results of this paper give a clear answer to this question, albeit for the simpler one-dimensional hard-rod system. For this specific system this result provides a clear understanding of the value and interpretation of different modelling choices, while giving hints for more general systems.