Geometric Operator Learning with Optimal Transport

2025-07-26

AI Summary

This work addresses the limited geometric representational capacity in learning partial differential equation (PDE) operators on complex geometries. To this end, we systematically integrate optimal transport (OT) into neural operator frameworks. Our core method generalizes discrete grids to grid density functions and models them via instance-dependent, differentiable OT maps from each density to a uniform reference density—yielding geometry-aware embeddings. Subsequently, 3D surfaces are parameterized onto a 2D latent manifold, where PDE operator learning is performed directly. To our knowledge, this is the first systematic application of OT for geometry-aware operator learning. Evaluated on ShapeNet-Car, DrivAerNet-Car, and FlowBench, our model achieves significantly higher accuracy and superior geometric generalization while incurring substantially lower time and memory overhead compared to state-of-the-art methods.

Technology Category

• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Integrating optimal transport into PDE operator learning
• Generalizing mesh discretization via density-based geometry embedding
• Enhancing computational efficiency with 2D latent space representation

Innovation

• Integrates optimal transport into PDE operator learning
• Generalizes meshes to density functions via OT
• Embeds 3D surfaces into 2D latent space efficiently