GFocal: A Global-Focal Neural Operator for Solving PDEs on Arbitrary Geometries
This work addresses the challenge of accurately modeling multiscale physical systems for applications like aerodynamics, though it appears incremental by building on existing Transformer-based neural operators.
The paper tackles the problem of solving partial differential equations on arbitrary geometries by proposing GFocal, a Transformer-based neural operator that simultaneously learns global and local features, achieving state-of-the-art performance with an average 15.2% relative gain in benchmarks and excelling in industry-scale simulations.
Transformer-based neural operators have emerged as promising surrogate solvers for partial differential equations, by leveraging the effectiveness of Transformers for capturing long-range dependencies and global correlations, profoundly proven in language modeling. However, existing methodologies overlook the coordinated learning of interdependencies between local physical details and global features, which are essential for tackling multiscale problems, preserving physical consistency and numerical stability in long-term rollouts, and accurately capturing transitional dynamics. In this work, we propose GFocal, a Transformer-based neural operator method that enforces simultaneous global and local feature learning and fusion. Global correlations and local features are harnessed through Nyström attention-based \textbf{g}lobal blocks and slices-based \textbf{focal} blocks to generate physics-aware tokens, subsequently modulated and integrated via convolution-based gating blocks, enabling dynamic fusion of multiscale information. GFocal achieves accurate modeling and prediction of physical features given arbitrary geometries and initial conditions. Experiments show that GFocal achieves state-of-the-art performance with an average 15.2\% relative gain in five out of six benchmarks and also excels in industry-scale simulations such as aerodynamics simulation of automotives and airfoils.