Sheaf and duality methods for analyzing multi-model systems

TL;DR

This paper introduces a sheaf-theoretic framework for analyzing multi-model systems, emphasizing topology and local-to-global information integration to build complex models from simpler components.

Contribution

It proposes modeling multi-model systems using sheaves based on their topology, enabling systematic construction and analysis of complex models from local pieces.

Findings

Sheaf theory effectively manages local information in multi-model systems.

The approach applies to dynamical systems, PDEs, and probabilistic models.

Complex models can be assembled from simpler, local models.

Abstract

There is an interplay between models, specified by variables and equations, and their connections to one another. This dichotomy should be reflected in the abstract as well. Without referring to the models directly -- only that a model consists of spaces and maps between them -- the most readily apparent feature of a multi-model system is its topology. We propose that this topology should be modeled first, and then the spaces and maps of the individual models be specified in accordance with the topology. Axiomatically, this construction leads to sheaves. Sheaf theory provides a toolbox for constructing predictive models described by systems of equations. Sheaves are mathematical objects that manage the combination of 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.