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.

翻译：我们提出了一个统一的算法框架，用于数值求解、约束优化以及具有变分结构的偏微分方程（PDE）的物理信息学习。该框架基于对底层变分形式的伽辽金离散化，其高效性源于一种新颖的、高度优化且兼容GPU的TensorGalerkin框架，用于线性系统（刚度矩阵与载荷向量）的组装。TensorGalerkin通过在Python层面的Map阶段将单元操作张量化来运行，然后利用稀疏矩阵乘法执行全局归约，该乘法在网格诱导的稀疏图上进行消息传递。它可以无缝地应用于下游任务：i) 作为高效的数值PDE求解器，ii) 作为PDE约束优化的端到端可微框架，以及iii) 作为PDE的物理信息算子学习算法。通过多个基准测试，包括非结构网格上的二维和三维椭圆型、抛物型和双曲型PDE，我们证明了所提框架在所有目标下游应用中，相较于多种基线方法，均能提供显著的计算效率与精度提升。