Geometric Operator Learning with Optimal Transport

TL;DR

This paper introduces an optimal transport-based neural operator framework for PDEs on complex geometries, improving flexibility, accuracy, and computational efficiency over traditional mesh-based methods.

Contribution

It formulates geometry embedding as an OT problem, enabling instance-dependent deformation and efficient computations on surface manifolds for PDE simulations.

Findings

Achieves better accuracy than existing models on RANS datasets.

Reduces computational time and memory usage.

Improves accuracy on highly variable geometries in FlowBench.

Abstract

We propose integrating optimal transport (OT) into operator learning for partial differential equations (PDEs) on complex geometries. Classical geometric learning methods typically represent domains as meshes, graphs, or point clouds. Our approach generalizes discretized meshes to mesh density functions, formulating geometry embedding as an OT problem that maps these functions to a uniform density in a reference space. Compared to previous methods relying on interpolation or shared deformation, our OT-based method employs instance-dependent deformation, offering enhanced flexibility and effectiveness. For 3D simulations focused on surfaces, our OT-based neural operator embeds the surface geometry into a 2D parameterized latent space. By performing computations directly on this 2D representation of the surface manifold, it achieves significant computational efficiency gains compared to volumetric simulation. Experiments with Reynolds-averaged Navier-Stokes equations (RANS) on the ShapeNet-Car and DrivAerNet-Car datasets show that our method achieves better accuracy and also reduces computational expenses in terms of both time and memory usage compared to existing machine learning models. Additionally, our model demonstrates significantly improved accuracy on the FlowBench dataset, underscoring the benefits of employing instance-dependent deformation for datasets with highly variable geometries.