GFocal: A Global-Focal Neural Operator for Solving PDEs on Arbitrary Geometries

• URL: http://arxiv.org/abs/2508.04463v1
• Date: Wed, 06 Aug 2025 14:02:39 GMT
• Title: GFocal: A Global-Focal Neural Operator for Solving PDEs on Arbitrary Geometries
• Authors: Fangzhi Fei, Jiaxin Hu, Qiaofeng Li, Zhenyu Liu,
• Abstract summary: Transformer-based neural operators have emerged as promising surrogate solvers for partial differential equations.<n>We propose GFocal, a method that enforces simultaneous global and local feature learning and fusion.<n>Experiments show that GFocal achieves state-of-the-art performance with an average 15.2% relative gain in five out of six benchmarks.
• Score: 5.323843026995587
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 Nystr\"{o}m 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.

Related papers

• Geometry aware inference of steady state PDEs using Equivariant Neural Fields representations [0.0]
  We introduce enf2enf, an encoder--decoder methodology for predicting steady-state Partial Differential Equations.<n>Our method supports real time inference and zero-shot super-resolution, enabling efficient training on low-resolution meshes.
  arXiv  Detail & Related papers  (2025-04-24T08:30:32Z)
• Integrating Locality-Aware Attention with Transformers for General Geometry PDEs [24.336598771550157]
  We propose the Locality-Aware Attention Transformer (LA2Former) for learning mappings governed by partial differential equations (PDEs)<n>By combining linear attention for efficient global context encoding with pairwise attention for capturing intricate local interactions, LA2Former achieves an optimal balance between computational efficiency and predictive accuracy.<n>This work underscores the critical importance of localized feature learning in advancing Transformer-based neural operators for solving PDEs on complex and irregular domains.
  arXiv  Detail & Related papers  (2025-04-18T05:43:49Z)
• Accelerating Multiscale Modeling with Hybrid Solvers: Coupling FEM and Neural Operators with Domain Decomposition [3.0635300721402228]
  This work introduces a novel hybrid framework that integrates PI-NO with finite element method (FE) through domain decomposition.<n>The framework efficacy has been validated across a range of problems, spanning static, quasi-static, and dynamic regimes.<n>Our study shows that our hybrid solver: (1) maintains solution continuity across subdomain interfaces, (2) reduces computational costs by eliminating fine mesh requirements, (3) mitigates error accumulation in time dependent simulations, and (4) enables automatic adaptation to evolving physical phenomena.
  arXiv  Detail & Related papers  (2025-04-15T16:54:04Z)
• Learnable Multi-Scale Wavelet Transformer: A Novel Alternative to Self-Attention [0.0]
  Learnable Multi-Scale Wavelet Transformer (LMWT) is a novel architecture that replaces the standard dot-product self-attention.<n>We present the detailed mathematical formulation of the learnable Haar wavelet module and its integration into the transformer framework.<n>Our results indicate that the LMWT achieves competitive performance while offering substantial computational advantages.
  arXiv  Detail & Related papers  (2025-04-08T22:16:54Z)
• Efficient Transformed Gaussian Process State-Space Models for Non-Stationary High-Dimensional Dynamical Systems [49.819436680336786]
  We propose an efficient transformed Gaussian process state-space model (ETGPSSM) for scalable and flexible modeling of high-dimensional, non-stationary dynamical systems.<n>Specifically, our ETGPSSM integrates a single shared GP with input-dependent normalizing flows, yielding an expressive implicit process prior that captures complex, non-stationary transition dynamics.<n>Our ETGPSSM outperforms existing GPSSMs and neural network-based SSMs in terms of computational efficiency and accuracy.
  arXiv  Detail & Related papers  (2025-03-24T03:19:45Z)
• A domain decomposition-based autoregressive deep learning model for unsteady and nonlinear partial differential equations [2.7755345520127936]
  We propose a domain-decomposition-based deep learning (DL) framework, named CoMLSim, for accurately modeling unsteady and nonlinear partial differential equations (PDEs)<n>The framework consists of two key components: (a) a convolutional neural network (CNN)-based autoencoder architecture and (b) an autoregressive model composed of fully connected layers.
  arXiv  Detail & Related papers  (2024-08-26T17:50:47Z)
• Dynamical Mean-Field Theory of Self-Attention Neural Networks [0.0]
  Transformer-based models have demonstrated exceptional performance across diverse domains. Little is known about how they operate or what are their expected dynamics. We use methods for the study of asymmetric Hopfield networks in nonequilibrium regimes.
  arXiv  Detail & Related papers  (2024-06-11T13:29:34Z)
• Global-to-Local Modeling for Video-based 3D Human Pose and Shape Estimation [53.04781510348416]
  Video-based 3D human pose and shape estimations are evaluated by intra-frame accuracy and inter-frame smoothness. We propose to structurally decouple the modeling of long-term and short-term correlations in an end-to-end framework, Global-to-Local Transformer (GLoT) Our GLoT surpasses previous state-of-the-art methods with the lowest model parameters on popular benchmarks, i.e., 3DPW, MPI-INF-3DHP, and Human3.6M.
  arXiv  Detail & Related papers  (2023-03-26T14:57:49Z)
• DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion [66.21290235237808]
  We introduce an energy constrained diffusion model which encodes a batch of instances from a dataset into evolutionary states. We provide rigorous theory that implies closed-form optimal estimates for the pairwise diffusion strength among arbitrary instance pairs. Experiments highlight the wide applicability of our model as a general-purpose encoder backbone with superior performance in various tasks.
  arXiv  Detail & Related papers  (2023-01-23T15:18:54Z)
• Multi-fidelity Hierarchical Neural Processes [79.0284780825048]
  Multi-fidelity surrogate modeling reduces the computational cost by fusing different simulation outputs. We propose Multi-fidelity Hierarchical Neural Processes (MF-HNP), a unified neural latent variable model for multi-fidelity surrogate modeling. We evaluate MF-HNP on epidemiology and climate modeling tasks, achieving competitive performance in terms of accuracy and uncertainty estimation.
  arXiv  Detail & Related papers  (2022-06-10T04:54:13Z)
• Equivariant Graph Mechanics Networks with Constraints [83.38709956935095]
  We propose Graph Mechanics Network (GMN) which is efficient, equivariant and constraint-aware. GMN represents, by generalized coordinates, the forward kinematics information (positions and velocities) of a structural object. Extensive experiments support the advantages of GMN compared to the state-of-the-art GNNs in terms of prediction accuracy, constraint satisfaction and data efficiency.
  arXiv  Detail & Related papers  (2022-03-12T14:22:14Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Fast Gravitational Approach for Rigid Point Set Registration with Ordinary Differential Equations [79.71184760864507]
  This article introduces a new physics-based method for rigid point set alignment called Fast Gravitational Approach (FGA) In FGA, the source and target point sets are interpreted as rigid particle swarms with masses interacting in a globally multiply-linked manner while moving in a simulated gravitational force field. We show that the new method class has characteristics not found in previous alignment methods.
  arXiv  Detail & Related papers  (2020-09-28T15:05:39Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.