  Constitutive models are widely used for modelling complex systems in science
and engineering, where first-principle-based, well-resolved simulations are
often prohibitively expensive. For example, in fluid dynamics, constitutive
models are required to describe nonlocal, unresolved physics such as turbulence
and laminar-turbulent transition. In particular, Reynolds stress models for
turbulence and intermittency transport equations for laminar-turbulent
transition both utilize convection--diffusion partial differential equations
(PDEs). However, traditional PDE-based constitutive models can lack robustness
and are often too rigid to accommodate diverse calibration data. We propose a
frame-independent, nonlocal constitutive model based on a vector-cloud neural
network that can be trained with data. The learned constitutive model can
predict the closure variable at a point based on the flow information in its
neighborhood. Such nonlocal information is represented by a group of points,
each having a feature vector attached to it, and thus the input is referred to
as vector cloud. The cloud is mapped to the closure variable through a
frame-independent neural network, which is invariant both to coordinate
translation and rotation and to the ordering of points in the cloud. As such,
the network takes any number of arbitrarily arranged grid points as input and
thus is suitable for unstructured meshes commonly used in fluid flow
simulations. The merits of the proposed network are demonstrated on scalar
transport PDEs on a family of parameterized periodic hill geometries. Numerical
results show that the vector-cloud neural network is a promising tool not only
as nonlocal constitutive models and but also as general surrogate models for
PDEs on irregular domains.
