Automated adjoints of coupled PDE-ODE systems

TL;DR

This paper extends the FEniCS software to efficiently solve coupled PDE-ODE systems and automatically derive their adjoints, demonstrated through applications in cardiac electrophysiology and mitochondrial swelling.

Contribution

It introduces an extension to FEniCS that handles coupled PDE-ODE systems with automatic adjoint derivation, enhancing modeling and optimization capabilities.

Findings

Efficient code generation for coupled PDE-ODE systems.

Automatic derivation of adjoints and tangent linearizations.

Successful application to biological systems like cardiac electrophysiology.

Abstract

Mathematical models that couple partial differential equations (PDEs) and spatially distributed ordinary differential equations (ODEs) arise in biology, medicine, chemistry and many other fields. In this paper we discuss an extension to the FEniCS finite element software for expressing and efficiently solving such coupled systems. Given an ODE described using an augmentation of the Unified Form Language (UFL) and a discretisation described by an arbitrary Butcher tableau, efficient code is automatically generated for the parallel solution of the ODE. The high-level description of the solution algorithm also facilitates the automatic derivation of the adjoint and tangent linearization of coupled PDE-ODE solvers. We demonstrate the capabilities of the approach on examples from cardiac electrophysiology and mitochondrial swelling.