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Partial differential equations (PDEs) play a dominant role in themathematical modeling of many complex dynamical processes. Solving these PDEsoften requires prohibitively high computational costs, especially when multipleevaluations must be made for different parameters or conditions. Aftertraining, neural operators can provide PDEs solutions significantly faster thantraditional PDE solvers. In this work, invariance properties and computationalcomplexity of two neural operators are examined for transport PDE of a scalarquantity. Neural operator based on graph kernel network (GKN) operates ongraph-structured data to incorporate nonlocal dependencies. Here we propose amodified formulation of GKN to achieve frame invariance. Vector cloud neuralnetwork (VCNN) is an alternate neural operator with embedded frame invariancewhich operates on point cloud data. GKN-based neural operator demonstratesslightly better predictive performance compared to VCNN. However, GKN requiresan excessively high computational cost that increases quadratically with theincreasing number of discretized objects as compared to a linear increase forVCNN.

Frame invariance and scalability of neural operators for partial differential equations