This use case demonstrates the application of Neural Concept Shape to the prediction of volumetric fields.
In this project we want to predict the fields on the 3D volume around an object such as velocity and
pressure given its mesh, and thus getting a direct relation between the mesh and the
Such a model has been built, leveraging geometric deep learning, i.e.
deep learning on irregular domains, on which standard convolutions would
The backward facing step is an important test-case for demonstrating results in
computational fluid dynamics as it has a simple geometry that hides a complex
phenomena: the re-circulation zone [see next paragraph]. Our model is able
to predict the internal flow around the backward-facing-step
as visually shown here below, with first prediction, then groundtruth and finally
the absolute point-wise error. We highlight the fact that the predictions were made
mesh-to-mesh, with as additional information only the boundary conditions and properties
of the flow (e.g. viscosity).
Once we got all x-velocity and y-velocity we can also predict the
re-circulation zone behind the step, as we illustrate below.
In the figure we show the simulation [left], and prediction [right].
As we see, the predicted solution (right) seems unfeasible with some
streamlines seemingly exiting the side of the step… And indeed, it is
infeasible! It does not satisfy the conservation of mass, as it has not been enforced
by the model. To correct this behavior, we can add a penalization term in the
loss of the network or iterate one step through the domain correcting the
Of course, the same process can be repeated to other fields
that can be of interest for numerical simulations or other problems. Here below, we can see the turbulent viscosity
on the airfoil dataset:
Such a model can be used for automatic design of engineering shapes or for the initialization of an iterative solver with a corrected
prediction, hence drastically decreasing the number of iterations
necessary to reach convergence, and with it the computational cost.