Nonparametric Boundary Geometry in Physics Informed Deep Learning

Scott Alexander Cameron, Arnu Pretorius, Stephen J. Roberts

Keywords: PINNs, physics informed neural networks, geometric deep learning, neural operator, PDEs

Abstract:

Engineering design problems frequently require solving systems of
partial differential equations with boundary conditions specified on
object geometries in the form of a triangular mesh. These boundary
geometries are provided by a designer and are problem dependent.
The efficiency of the design process greatly benefits from fast turnaround
times when repeatedly solving PDEs on various geometries. However,
most current work that uses machine learning to speed up the solution
process relies heavily on a fixed parameterization of the geometry, which
cannot be changed after training. This severely limits the possibility of
reusing a trained model across a variety of design problems.
In this work, we propose a novel neural operator architecture which accepts
boundary geometry, in the form of triangular meshes, as input and produces an
approximate solution to a given PDE as output. Once trained, the model can be
used to rapidly estimate the PDE solution over a new geometry, without the need for
retraining or representation of the geometry to a pre-specified parameterization.

Submission Number: 13273