Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domains

2025-05-24

AI Summary

To address the low accuracy, poor generalization, and inefficiency in learning solution operators for partial differential equations (PDEs) on arbitrary geometric domains, this paper proposes a geometry-aware multi-scale graph attention neural operator framework. The method encodes domain geometry into learnable embeddings and integrates graph neural operators with vision Transformer-based processors within an end-to-end encoder–decoder architecture, enabling strong cross-geometry and cross-resolution generalization while preserving computational efficiency. Innovatively combining multi-scale attention mechanisms with geometric prior modeling, it significantly enhances the expressivity and robustness of operator learning. Evaluated on diverse PDE benchmarks and a large-scale 3D industrial computational fluid dynamics (CFD) dataset, the approach consistently outperforms existing state-of-the-art methods, achieving substantial speedups in both training and inference, and demonstrating excellent scalability.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Deep Neural Architectures and Foundation Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The very challenging task of learning solution operators of PDEs on arbitrary domains accurately and efficiently is of vital importance to engineering and industrial simulations. Despite the existence of many operator learning algorithms to approximate such PDEs, we find that accurate models are not necessarily computationally efficient and vice versa. We address this issue by proposing a geometry aware operator transformer (GAOT) for learning PDEs on arbitrary domains. GAOT combines novel multiscale attentional graph neural operator encoders and decoders, together with geometry embeddings and (vision) transformer processors to accurately map information about the domain and the inputs into a robust approximation of the PDE solution. Multiple innovations in the implementation of GAOT also ensure computational efficiency and scalability. We demonstrate this significant gain in both accuracy and efficiency of GAOT over several baselines on a large number of learning tasks from a diverse set of PDEs, including achieving state of the art performance on a large scale three-dimensional industrial CFD dataset.

Problem

• Learning PDE solution operators on arbitrary domains accurately and efficiently
• Balancing accuracy and computational efficiency in neural PDE surrogates
• Handling diverse PDEs with geometry-aware transformer architectures

Innovation

• Geometry aware operator transformer for PDEs
• Multiscale attentional graph neural operators
• Geometry embeddings with transformer processors