Koopman-Based Surrogate Models for Multi-Objective Optimization of Agent-Based Systems

TL;DR

Data-driven reduced models based on the Koopman generator are considered to enable the efficient solution of multi-objective optimization problems involving ABMs and it is demonstrated that the surrogate models effectively approximate the Pareto-optimal points of the ABM dynamics.

Abstract

Agent-based models (ABMs) provide an intuitive and powerful framework for studying social dynamics by modeling the interactions of individuals from the perspective of each individual. In addition to simulating and forecasting the dynamics of ABMs, the demand to solve optimization problems to support, for example, decision-making processes naturally arises. Most ABMs, however, are non-deterministic, high-dimensional dynamical systems, so objectives defined in terms of their behavior are computationally expensive. In particular, if the number of agents is large, evaluating the objective functions often becomes prohibitively time-consuming. We consider data-driven reduced models based on the Koopman generator to enable the efficient solution of multi-objective optimization problems involving ABMs. In a first step, we show how to obtain data-driven reduced models of non-deterministic dynamical systems (such as ABMs) that depend potentially nonlinearly on control inputs. We then use them in the second step as surrogate models to solve multi-objective optimal control problems. We first illustrate our approach using the example of a voter model, where we compute optimal controls to steer the agents to a predetermined majority, and then using the example of an epidemic ABM, where we compute optimal containment strategies in a prototypical situation. We demonstrate that the surrogate models effectively approximate the Pareto-optimal points of the ABM dynamics by comparing the surrogate-based results with test points, where the objectives are evaluated using the ABM. Our results show that when objectives are defined by the dynamic behavior of ABMs, data-driven surrogate models support or even enable the solution of multi-objective optimization problems.

Figures (8)

• Figure 1: (a) Objective functions $f_1$ and $f_2$ on the interval $\mathcal{R} = [-0.5, 2.5]$ with Pareto set (gray shaded) and (b) Pareto front (dotted and dashed). The dotted line segment shows that only points in $[1.5, 2]$ are globally Pareto-optimal since they clearly dominate all points in $[0, 1]$.
• Figure 2: First four iterations of the sampling algorithm demonstrated using Example \ref{['expl:pareto']}. The dashed lines represent the boundaries of each box. In this example, each box is split in half before a non-dominance test is performed. The plots on the right show the images of the current box collections (gray/shaded) covering the true global Pareto front (black/solid).
• Figure 3: (a) Trajectories of susceptible and infected agents as a fraction of the population for the SIR model with (dashed) and without (solid) control $u$ and (b) control $u(t)$ applied to SIR model.
• Figure 4: Aggregated trajectories of (a) the voter model and (b) the GERDA model for the parameters given in Table \ref{['tab:parameters']}.
• Figure 5: Pareto set covering (light blue boxes) after 12 iterations plotted on (a) objective function \ref{['eq:VM_expensive']} and (b) objective function \ref{['eq:VM_cheap']} at time $t=10$ depending on the control $[u_\text{push}, u_\text{pull}]^\top \in \mathcal{R}$. The yellow line in (a) marks the area of interest where the majority switches from $S_1$ to $S_2$.

Theorems & Definitions (10)

• Remark 2.1
• Example 2.2
• Definition 3.1
• Theorem 3.2
• Example 3.3
• Example 3.4
• Remark 4.1
• Remark 4.2
• Remark 4.3
• proof