External Operators in Firedrake

Firedrake is an automated system for the solution of partial differential equations (PDEs). It enables writing down PDE-based problems, like those occurring across the geophysical sciences, in a highly productive way while relying on automatic code generation to achieve high-performance simulations. Many software packages have been built on top of Firedrake, including Thetis, Gusto, and Icepack. One limitation of high level domain-specific languages for describing simulations is that they do not take into account operators that are not directly expressible using vector calculus. This limitation is critical in many applications where PDEs are not enough to accurately describe the physical problem of interest. These applications include nonlinear implicit constitutive laws such as the Glen's flow law for glacier flow, the use of neural networks to include features not represented in the differential equations, or closures for unresolved spatiotemporal scales. Example applications of of neural networks include regularization of inverse problems such as in seismic inversion and subgrid parameterization of atmospheric or oceanographic processes like clouds or turbulence. We present extensions to Firedrake that enable the inclusion of arbitrary external operators. This external operator feature composes seamlessly with the automatic differentiation capabilities of Firedrake.