---
author:
- Michael Robinson
bibliography:
- 'multimodel\_bib.bib'
title: 'Sheaf and duality methods for analyzing multi-model systems'
---

Introduction
============

Complex predictive models are notoriously hard to construct and to study. Sheaf theory provides a toolbox for constructing predictive models described by systems of equations. Sheaves are mathematical objects that combine bits of local information into a consistent whole. The power of this approach is that complex models can be assembled from smaller, easier-to-construct models. The models discussed in this chapter span the study of continuous dynamical systems, partial differential equations, probabilistic graphical models, and discrete approximations of these models. Shadows of the sheaf theoretic perspective are apparent in a variety of disciplines, for instance in the construction of volume meshers (which construct pullbacks and pushforwards of sheaves of functions), finite element solvers (which construct the space of global sections of a sheaf), and loopy belief propagation (which iteratively determines individual global sections).

Multi-model systems can be constructed from diagrams of individual models. It is therefore helpful to abstract this idea into a convenient formalism, in which the basic features are captured. There is an interplay between the models themselves and their connections to one another. This dichotomy should be 