FLARE: Fast Low-rank Attention Routing Engine

The quadratic complexity of self-attention limits its applicability and scalability on large unstructured meshes. We introduce Fast Low-rank Attention Routing Engine (FLARE), a linear complexity self-attention mechanism that routes attention through fixed-length latent sequences. Each attention head performs global communication among $N$ tokens by projecting the input sequence onto a fixed length latent sequence of $M \ll N$ tokens using learnable query tokens. By routing attention through a bottleneck sequence, FLARE learns a low-rank form of attention that can be applied at $O(NM)$ cost. FLARE not only scales to unprecedented problem sizes, but also delivers superior accuracy compared to state-of-the-art neural PDE surrogates across diverse benchmarks. We also release a new additive manufacturing dataset to spur further research. Our code is available at https://github.com/vpuri3/FLARE.py.

DL-Polycube: Deep learning enhanced polycube method for high-quality hexahedral mesh generation and volumetric spline construction

In this paper, we present a novel algorithm that integrates deep learning with the polycube method (DL-Polycube) to generate high-quality hexahedral (hex) meshes, which are then used to construct volumetric splines for isogeometric analysis. Our DL-Polycube algorithm begins by establishing a connection between surface triangular meshes and polycube structures. We employ deep neural network to classify surface triangular meshes into their corresponding polycube structures. Following this, we combine the acquired polycube structural information with unsupervised learning to perform surface segmentation of triangular meshes. This step addresses the issue of segmentation not corresponding to a polycube while reducing manual intervention. Quality hex meshes are then generated from the polycube structures, with employing octree subdivision, parametric mapping and quality improvement techniques. The incorporation of deep learning for creating polycube structures, combined with unsupervised learning for segmentation of surface triangular meshes, substantially accelerates hex mesh generation. Finally, truncated hierarchical B-splines are constructed on the generated hex meshes. We extract trivariate B\'ezier elements from these splines and apply them directly in isogeometric analysis. We offer several examples to demonstrate the robustness of our DL-Polycube algorithm.

SRL-Assisted AFM: Generating Planar Unstructured Quadrilateral Meshes with Supervised and Reinforcement Learning-Assisted Advancing Front Method

High-quality mesh generation is the foundation of accurate finite element analysis. Due to the vast interior vertices search space and complex initial boundaries, mesh generation for complicated domains requires substantial manual processing and has long been considered the most challenging and time-consuming bottleneck of the entire modeling and analysis process. In this paper, we present a novel computational framework named ``SRL-assisted AFM" for meshing planar geometries by combining the advancing front method with neural networks that select reference vertices and update the front boundary using ``policy networks." These deep neural networks are trained using a unique pipeline that combines supervised learning with reinforcement learning to iteratively improve mesh quality. First, we generate different initial boundaries by randomly sampling points in a square domain and connecting them sequentially. These boundaries are used for obtaining input meshes and extracting training datasets in the supervised learning module. We then iteratively improve the reinforcement learning model performance with reward functions designed for special requirements, such as improving the mesh quality and controlling the number and distribution of extraordinary points. Our proposed supervised learning neural networks achieve an accuracy higher than 98% on predicting commercial software. The final reinforcement learning neural networks automatically generate high-quality quadrilateral meshes for complex planar domains with sharp features and boundary layers.