Geometric Operator Learning with Optimal Transport

We propose integrating optimal transport (OT) into operator learning for partial differential equations (PDEs) on complex geometries, for applications in fluid dynamics. 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. Unlike prior methods that use shared deformation, our OT-based method employs instance-dependent deformation, providing enhanced flexibility and effectiveness. For 3D surface simulations, our neural operator embeds the geometry into a 2D parameterized latent space. By performing computations directly on this 2D representation, the method achieves significant computational efficiency gains over volumetric simulation. Experiments with RANS on the ShapeNet-Car and DrivAerNet-Car datasets show improved accuracy and reduced computational expenses in both time and memory. Additionally, our model shows significantly improved accuracy on the FlowBench dataset, highlighting the benefits of instance-dependent deformation for highly variable geometries.