  Constitutive models are widely used for modeling 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. However, traditional constitutive models
based on partial differential equations (PDEs) often lack robustness and are
too rigid to accommodate diverse calibration datasets. We propose a
frame-independent, nonlocal constitutive model based on a vector-cloud neural
network that can be learned with data. The model predicts 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,
invariant both to coordinate translation and rotation and to the ordering of
points in the cloud. As such, the network can deal with any number of
arbitrarily arranged grid points and thus is suitable for unstructured meshes
in fluid simulations. The merits of the proposed network are demonstrated for
scalar transport PDEs on a family of parameterized periodic hill geometries.
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.
