Learning Sparse Approximate Inverse Preconditioners for Conjugate Gradient Solvers on GPUs

The conjugate gradient solver (CG) is a prevalent method for solving symmetric and positive definite linear systems Ax=b, where effective preconditioners are crucial for fast convergence. Traditional preconditioners rely on prescribed algorithms to offer rigorous theoretical guarantees, while limiting their ability to exploit optimization from data. Existing learning-based methods often utilize Graph Neural Networks (GNNs) to improve the performance and speed up the construction. However, their reliance on incomplete factorization leads to significant challenges: the associated triangular solve hinders GPU parallelization in practice, and introduces long-range dependencies which are difficult for GNNs to model. To address these issues, we propose a learning-based method to generate GPU-friendly preconditioners, particularly using GNNs to construct Sparse Approximate Inverse (SPAI) preconditioners, which avoids triangular solves and requires only two matrix-vector products at each CG step. The locality of matrix-vector product is compatible with the local propagation mechanism of GNNs. The flexibility of GNNs also allows our approach to be applied in a wide range of scenarios. Furthermore, we introduce a statistics-based scale-invariant loss function. Its design matches CG's property that the convergence rate depends on the condition number, rather than the absolute scale of A, leading to improved performance of the learned preconditioner. Evaluations on three PDE-derived datasets and one synthetic dataset demonstrate that our method outperforms standard preconditioners (Diagonal, IC, and traditional SPAI) and previous learning-based preconditioners on GPUs. We reduce solution time on GPUs by 40%-53% (68%-113% faster), along with better condition numbers and superior generalization performance. Source code available at https://github.com/Adversarr/LearningSparsePreconditioner4GPU

MM-Agent: LLM as Agents for Real-world Mathematical Modeling Problem

Mathematical modeling is a cornerstone of scientific discovery and engineering practice, enabling the translation of real-world problems into formal systems across domains such as physics, biology, and economics. Unlike mathematical reasoning, which assumes a predefined formulation, modeling requires open-ended problem analysis, abstraction, and principled formalization. While Large Language Models (LLMs) have shown strong reasoning capabilities, they fall short in rigorous model construction, limiting their utility in real-world problem-solving. To this end, we formalize the task of LLM-powered real-world mathematical modeling, where agents must analyze problems, construct domain-appropriate formulations, and generate complete end-to-end solutions. We introduce MM-Bench, a curated benchmark of 111 problems from the Mathematical Contest in Modeling (MCM/ICM), spanning the years 2000 to 2025 and across ten diverse domains such as physics, biology, and economics. To tackle this task, we propose MM-Agent, an expert-inspired framework that decomposes mathematical modeling into four stages: open-ended problem analysis, structured model formulation, computational problem solving, and report generation. Experiments on MM-Bench show that MM-Agent significantly outperforms baseline agents, achieving an 11.88\% improvement over human expert solutions while requiring only 15 minutes and \$0.88 per task using GPT-4o. Furthermore, under official MCM/ICM protocols, MM-Agent assisted two undergraduate teams in winning the Finalist Award (\textbf{top 2.0\% among 27,456 teams}) in MCM/ICM 2025, demonstrating its practical effectiveness as a modeling copilot. Our code is available at https://github.com/usail-hkust/LLM-MM-Agent

SUPRA: Subspace Parameterized Attention for Neural Operator on General Domains

Neural operators are efficient surrogate models for solving partial differential equations (PDEs), but their key components face challenges: (1) in order to improve accuracy, attention mechanisms suffer from computational inefficiency on large-scale meshes, and (2) spectral convolutions rely on the Fast Fourier Transform (FFT) on regular grids and assume a flat geometry, which causes accuracy degradation on irregular domains. To tackle these problems, we regard the matrix-vector operations in the standard attention mechanism on vectors in Euclidean space as bilinear forms and linear operators in vector spaces and generalize the attention mechanism to function spaces. This new attention mechanism is fully equivalent to the standard attention but impossible to compute due to the infinite dimensionality of function spaces. To address this, inspired by model reduction techniques, we propose a Subspace Parameterized Attention (SUPRA) neural operator, which approximates the attention mechanism within a finite-dimensional subspace. To construct a subspace on irregular domains for SUPRA, we propose using the Laplacian eigenfunctions, which naturally adapt to domains' geometry and guarantee the optimal approximation for smooth functions. Experiments show that the SUPRA neural operator reduces error rates by up to 33% on various PDE datasets while maintaining state-of-the-art computational efficiency.