Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithm

2026-02-04

AI Summary

This work proposes TensorGalerkin, a unified framework for efficiently solving variational-structure partial differential equations (PDEs), performing constrained optimization, and learning physics-informed operators. Built upon Galerkin discretization, the method tensorizes element-wise operations and maps them at the Python level, enabling differentiable message passing over mesh graphs through GPU-compatible sparse matrix multiplications. The resulting end-to-end differentiable architecture achieves substantial improvements in both computational efficiency and solution accuracy. Comprehensive benchmarks on 2D and 3D elliptic, parabolic, and hyperbolic PDEs demonstrate that TensorGalerkin consistently outperforms existing state-of-the-art baselines.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present a unified algorithmic framework for the numerical solution, constrained optimization, and physics-informed learning of PDEs with a variational structure. Our framework is based on a Galerkin discretization of the underlying variational forms, and its high efficiency stems from a novel highly-optimized and GPU-compliant TensorGalerkin framework for linear system assembly (stiffness matrices and load vectors). TensorGalerkin operates by tensorizing element-wise operations within a Python-level Map stage and then performs global reduction with a sparse matrix multiplication that performs message passing on the mesh-induced sparsity graph. It can be seamlessly employed downstream as i) a highly-efficient numerical PDEs solver, ii) an end-to-end differentiable framework for PDE-constrained optimization, and iii) a physics-informed operator learning algorithm for PDEs. With multiple benchmarks, including 2D and 3D elliptic, parabolic, and hyperbolic PDEs on unstructured meshes, we demonstrate that the proposed framework provides significant computational efficiency and accuracy gains over a variety of baselines in all the targeted downstream applications.

Problem

• PDEs
• variational structure
• numerical solution
• constrained optimization
• physics-informed learning

Innovation

• TensorGalerkin
• Galerkin assembly
• physics-informed learning
• PDE-constrained optimization
• GPU-accelerated sparse assembly