Learning frame-independent, nonlocal constitutive relations on unstructured meshes with an embedding neural network

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. 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, non-local constitutive model based on an embedding neural network that can be trained with data. The learned constitutive model can predicate the closure variable at a point based on the flow information in its neighborhood. It can take any number of points arbitrarily arranged, and thus it is suitable for unstructured meshes, which are typical for finite-element and finite-volume simulations. The merits of the proposed model are demonstrated on both scalar and tensor transport PDEs on a family of parameterized periodic hill geometries.