Detail pages for a few of our papers. The complete lists live on the Research and Publications pages.
Journal of Computational Science, 2024.
Mathematics in Engineering, 2022.
arXiv preprint, 2022.
We present Decapodes, a diagrammatic tool for representing, composing, and solving partial differential equations. Decapodes provides an intuitive diagrammatic representation of the relationships between variables in a system of equations, a method for composing systems of partial differential equations using an operad of wiring diagrams, and an algorithm for deriving solvers using hypergraphs and string diagrams. The string diagrams are in turn compiled into executable programs using the techniques of categorical data migration, graph traversal, and the discrete exterior calculus. The generated solvers produce numerical solutions consistent with state-of-the-art open source tools as demonstrated by benchmark comparisons with SU2. These numerical experiments demonstrate the feasibility of this approach to multiphysics simulation and identify areas requiring further development.
Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm mathematical footing, while also systematizing a broadly applicable framework to reason formally about systems of equations and their solutions. Our main mathematical tools are category-theoretic diagrams, which are well known, and morphisms between diagrams, which have been less appreciated. As an application of the diagrammatic framework, we show how complex, multiphysical systems can be modularly constructed from basic physical principles. A wealth of examples, drawn from electromagnetism, transport phenomena, fluid mechanics, and other fields, is included.
Lifts of categorical diagrams D:π©βπ· against discrete opfibrations Ο:π€βπ· can be interpreted as presenting solutions to systems of equations. With this interpretation in mind, it is natural to ask if there is a notion of equivalence of diagrams DβDβ² that precisely captures the idea of the two diagrams “having the same solutions”. We give such a definition, and then show how the localisation of the category of diagrams in π· along such equivalences is isomorphic to the localisation of the slice category π’πΊπ/π· along the class of initial functors. Finally, we extend this result to the 2-categorical setting, proving the analogous statement for any locally presentable 2-category in place of π’πΊπ.