  Partial differential equations (PDEs) see widespread use in sciences and
engineering to describe simulation of physical processes as scalar and vector
fields interacting and coevolving over time. Due to the computationally
expensive nature of their standard solution methods, neural PDE surrogates have
become an active research topic to accelerate these simulations. However,
current methods do not explicitly take into account the relationship between
different fields and their internal components, which are often correlated.
Viewing the time evolution of such correlated fields through the lens of
multivector fields allows us to overcome these limitations. Multivector fields
consist of scalar, vector, as well as higher-order components, such as
bivectors and trivectors. Their algebraic properties, such as multiplication,
addition and other arithmetic operations can be described by Clifford algebras.
To our knowledge, this paper presents the first usage of such multivector
representations together with Clifford convolutions and Clifford Fourier
transforms in the context of deep learning. The resulting Clifford neural
layers are universally applicable and will find direct use in the areas of
fluid dynamics, weather forecasting, and the modeling of physical systems in
general. We empirically evaluate the benefit of Clifford neural layers by
replacing convolution and Fourier operations in common neural PDE surrogates by
their Clifford counterparts on 2D Navier-Stokes and weather modeling tasks, as
well as 3D Maxwell equations. For similar parameter count, Clifford neural
layers consistently improve generalization capabilities of the tested neural
PDE surrogates. Source code for our PyTorch implementation is available at
https://microsoft.github.io/cliffordlayers/.
