Nonparametric Boundary Geometry in Physics Informed Deep Learning

Engineering design problems frequently require solving systems ofpartial differential equations with boundary conditions specified onobject geometries in the form of a triangular mesh. These boundarygeometries are provided by a designer and are problem dependent.The efficiency of the design process greatly benefits from fast turnaroundtimes when repeatedly solving PDEs on various geometries. However,most current work that uses machine learning to speed up the solutionprocess relies heavily on a fixed parameterization of the geometry, whichcannot be changed after training. This severely limits the possibility ofreusing a trained model across a variety of design problems.In this work, we propose a novel neural operator architecture which acceptsboundary geometry, in the form of triangular meshes, as input and produces anapproximate solution to a given PDE as output. Once trained, the model can beused to rapidly estimate the PDE solution over a new geometry, without the need forretraining or representation of the geometry to a pre-specified parameterization.