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High level domain specific languages for the finite element method underpinhigh productivity programming environments for simulations based on partialdifferential equations (PDE) while employing automatic code generation toachieve high performance. However, a limitation of this approach is that itdoes not support operators that are not directly expressible in the vectorcalculus. This is critical in applications where PDEs are not enough toaccurately describe the physical problem of interest. The use of deep learningtechniques have become increasingly popular in filling this knowledge gap, forexample to include features not represented in the differential equations, orclosures for unresolved spatiotemporal scales. We introduce an interface withinthe Firedrake finite element system that enables a seamless interface with deeplearning models. This new feature composes with the automatic differentiationcapabilities of Firedrake, enabling the automated solution of inverse problems.Our implementation interfaces with PyTorch and can be extended to other machinelearning libraries. The resulting framework supports complex models couplingPDEs and deep learning whilst maintaining separation of concerns betweenapplication scientists and software experts.

Escaping the abstraction: a foreign function interface for the Unified Form Language [UFL]