We present the Unified Form Language (UFL), which is a domain-specific
language for representing weak formulations of partial differential equations
with a view to numerical approximation. Features of UFL include support for
variational forms and functionals, automatic differentiation of forms and
expressions, arbitrary function space hierarchies for multi-field problems,
general differential operators and flexible tensor algebra. With these
features, UFL has been used to effortlessly express finite element methods for
complex systems of partial differential equations in near-mathematical
notation, resulting in compact, intuitive and readable programs. We present in
this work the language and its construction. An implementation of UFL is freely
available as an open-source software library. The library generates abstract
syntax tree representations of variational problems, which are used by other
software libraries to generate concrete low-level implementations. Some
application examples are presented and libraries that support UFL are
highlighted.

介绍了一个用于表示偏微分方程弱形式的领域特定语言 Unified Form Language（UFL），支持变分形式和泛函，以及自动微分和任意功能空间等特性，并介绍了该语言及其构造，以及一些应用实例和支持 UFL 的库。