mlsanityData to 7 Oct 2026

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

arXiv 2602.05052 ↗First posted 4 Feb 2026cs.LGcited by 3

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Abstract (arXiv)

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.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 24 of its 47 references that are in the map. First posted on arXiv 4 Feb 2026.

Builds on

24 of 47 references.

References · 24

  1. 2017Inductive Representation Learning on Large Graphscited by 6,284
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,214
  4. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  5. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  6. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  7. 2022Message Passing Neural PDE Solverscited by 368
  8. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 321
  9. 2024Transolver: A Fast Transformer Solver for PDEs on General Geometriescited by 299
  10. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  11. 2024Poseidon: Efficient Foundation Models for PDEscited by 186
  12. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
12 more
  1. 2021Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problemscited by 100
  2. 2022MultiScale MeshGraphNetscited by 94
  3. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
  4. 2023Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learningcited by 57
  5. 2025Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domainscited by 56
  6. 2022JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data sciencecited by 40
  7. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  8. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  9. 2025RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domainscited by 22
  10. 2021Variational Physics Informed Neural Networks: the role of quadratures and test functionscited by 17
  11. 2025HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutionscited by 8
  12. 2026torch-sla: Differentiable Sparse Linear Algebra with Adjoint Solvers and Sparse Tensor Parallelism for PyTorchcited by 3

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 3.

3 papers.

  1. 2026KernelOnet: An Interpretable Neural Operator Based on Kernel Functionscited by 0
  2. 2026Preconditioned Physics-Informed Neural Operator Trainingcited by 0
  3. 2026Transferability of Learned States in Neural PDE Solverscited by 0