Frame invariance and scalability of neural operators for partial differential equations. (arXiv:2112.14769v1 [cs.LG])

Partial differential equations (PDEs) play a dominant role in the
mathematical modeling of many complex dynamical processes. Solving these PDEs
often requires prohibitively high computational costs, especially when multiple
evaluations must be made for different parameters or conditions. After
training, neural operators can provide PDEs solutions significantly faster than
traditional PDE solvers. In this work, invariance properties and computational
complexity of two neural operators are examined for transport PDE of a scalar
quantity. Neural operator based on graph kernel network (GKN) operates on
graph-structured data to incorporate nonlocal dependencies. Here we propose a
modified formulation of GKN to achieve frame invariance. Vector cloud neural
network (VCNN) is an alternate neural operator with embedded frame invariance
which operates on point cloud data. GKN-based neural operator demonstrates
slightly better predictive performance compared to VCNN. However, GKN requires
an excessively high computational cost that increases quadratically with the
increasing number of discretized objects as compared to a linear increase for



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