使用TensorGalerkin学习、求解和优化PDE：一种高效的高性能Galerkin组装算法
Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithm
AI总结：
提出基于Galerkin离散化的统一算法框架，通过张量化元素操作和稀疏矩阵乘法实现O(1)图规模的系统组装，高效求解、约束优化和物理信息学习变分PDE。
摘要：
我们提出了一个统一的算法框架，用于具有变分结构的PDE的数值求解、约束优化和物理信息学习。该框架基于底层变分形式的Galerkin离散化，其高效率源于一种新颖的高度优化且兼容GPU的TensorGalerkin框架，用于线性系统组装（刚度矩阵和载荷向量）。TensorGalerkin通过在Python级Map阶段张量化元素操作，然后使用稀疏矩阵乘法进行全局归约，该乘法在网格诱导的稀疏图上执行消息传递。Map和Reduce阶段在PyTorch的autograd内部协同设计，使得组装图包含O(1)个节点，无论元素数量和局部自由度如何缩放。我们通过将TensorGalerkin部署为i)高效的数值PDE求解器，ii)用于PDE约束优化的端到端可微框架，以及iii)用于PDE的物理信息算子学习算法，验证了这种O(1)图属性。通过多个基准测试，包括非结构化网格上的2D和3D椭圆、抛物线和双曲PDE，我们证明了所提出的框架在所有目标下游应用中相比各种基线提供了显著的计算效率和精度提升。
英文摘要：
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. The Map and Reduce stages are co-designed inside PyTorch's autograd so that the assembly graph contains $O(1)$ nodes regardless of how the number of elements and local DoFs scale. We validate this $O(1)$-graph property by deploying TensorGalerkin 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
