# From interacting agents to density-based modeling with stochastic PDEs

## TL;DR

This paper develops a stochastic PDE-based density model from agent-based models to efficiently simulate large populations, with applications demonstrated through innovation spreading examples.

## Contribution

It introduces a novel reduction from agent-based models to stochastic PDEs, enabling scalable simulation of large systems with regularization via finite element discretization.

## Key findings

- SPDE model effectively approximates agent-based models for large populations
- Finite element discretization improves simulation efficiency and regularizes the SPDE
- Illustrative examples demonstrate the model's applicability to innovation spreading

## Abstract

Many real-world processes can naturally be modeled as systems of interacting agents. However, the long-term simulation of such agent-based models is often intractable when the system becomes too large. In this paper, starting from a stochastic spatio-temporal agent-based model (ABM), we present a reduced model in terms of stochastic PDEs that describes the evolution of agent number densities for large populations. We discuss the algorithmic details of both approaches; regarding the SPDE model, we apply Finite Element discretization in space which not only ensures efficient simulation but also serves as a regularization of the SPDE. Illustrative examples for the spreading of an innovation among agents are given and used for comparing ABM and SPDE models.