Model order reduction for parametrized variational inequalities: application to crowd motion

TL;DR

This paper develops a nonlinear model order reduction method combining linear models and deep learning for parametrized variational inequalities, demonstrated on a complex crowd contact problem.

Contribution

It introduces the first application of model order reduction to discrete contact problems with nonlinear constraints, using a hybrid linear-deep learning approach.

Findings

The proposed method effectively reduces model complexity in contact problems.

Hyper-reduction techniques EIM and EQ are compared in terms of accuracy and computational cost.

The approach is validated on a highly congested crowd scenario.

Abstract

This work investigates model order reduction for time-dependent parametrized variational inequalities, with a focus on discrete contact problems. As a prototypical example, we consider an agent-based crowd model [Maury et al., 2011] in which agent velocities are obtained at each time step from a constrained least-squares problem. Geometric parameter variations induce significant variability in both agent positions and contact forces, leading to a slowly decaying Kolmogorov $n$-width of the solution manifold. We propose a nonlinear approach that combines a linear reduced-order model with a deep-learning-based correction. The method utilizes a greedy index selection (gIS) algorithm for compressing Lagrange multipliers and Proper Orthogonal Decomposition (POD) applied to velocity snapshots. Additionally, we explore hyper-reduction techniques, comparing the Empirical Interpolation Method (EIM) and the Empirical Quadrature (EQ) procedure from both computational complexity and accuracy perspectives. Finally, we demonstrate the applicability of the methodology in a complex scenario involving many agents in a highly congested geometric configuration. This work represents the first attempt to apply model order reduction to a discrete contact problem of the type introduced in [Maury et al., 2011] and paves the way for future advancements in nonlinear MOR specifically for this class of problems.