Data to 7 Oct 2026

Learning Neural PDE Solvers with Convergence Guarantees

arXiv 1906.01200 ↗First posted 4 Jun 2019math.NA cs.NA stat.COcited by 77

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Abstract (arXiv)

Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be sub-optimal for specific classes of problems. In contrast to existing hand-crafted solutions, we propose an approach to learn a fast iterative solver tailored to a specific domain. We achieve this goal by learning to modify the updates of an existing solver using a deep neural network. Crucially, our approach is proven to preserve strong correctness and convergence guarantees. After training on a single geometry, our model generalizes to a wide variety of geometries and boundary conditions, and achieves 2-3 times speedup compared to state-of-the-art solvers.

Key facts. Cited by 77 papers in the map (data through 7 Oct 2026). First posted on arXiv 4 Jun 2019.

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Citing papers per year: 2021: 12; 2022: 15; 2023: 16; 2024: 15; 2025: 10; 2026: 9.

77 papers.

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2022Message Passing Neural PDE Solverscited by 368
2022Transformer for Partial Differential Equations' Operator Learningcited by 270
2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
2022Clifford Neural Layers for PDE Modelingcited by 91
2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 76
2022Blending Neural Operators and Relaxation Methods in PDE Numerical Solverscited by 53
2021Reinforcement Learning for Adaptive Mesh Refinementcited by 37
2023Learning Neural PDE Solvers with Parameter-Guided Channel Attentioncited by 37
2024UPS: Efficiently Building Foundation Models for PDE Solving via Cross-Modal Adaptationcited by 27
2021Optimization-Based Algebraic Multigrid Coarsening Using Reinforcement Learningcited by 21

38 more

2023Deep Learning in Deterministic Computational Mechanicscited by 21
2025DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic predictioncited by 21
2023Deep Equilibrium Based Neural Operators for Steady-State PDEscited by 19
2022Meta-PDE: Learning to Solve PDEs Quickly Without a Meshcited by 18
2021Learning optimal multigrid smoothers via neural networkscited by 17
2022Learning the Solution Operator of Boundary Value Problems using Graph Neural Networkscited by 17
2023Latent Traversals in Generative Models as Potential Flowscited by 17
2022Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspectivecited by 15
2024Accelerating PDE Data Generation via Differential Operator Action in Solution Spacecited by 15
2022A Deep Conjugate Direction Method for Iteratively Solving Linear Systemscited by 12
2024Accelerating Data Generation for Neural Operators via Krylov Subspace Recyclingcited by 12
2021Convergent Graph Solverscited by 11
2021Physics-Augmented Learning: A New Paradigm Beyond Physics-Informed Learningcited by 11
2023NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decompositioncited by 11
2023Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layerscited by 11
2023Flow Factorized Representation Learningcited by 11
2022LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Datacited by 10
2024Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methodscited by 10
2025NOWS: Neural Operator Warm Starts for Accelerating Iterative Solverscited by 9
2021Performance and accuracy assessments of an incompressible fluid solver coupled with a deep Convolutional Neural Networkcited by 8
2023Invariant preservation in machine learned PDE solvers via error correctioncited by 7
2021An Interpretive Constrained Linear Model for ResNet and MgNetcited by 6
2022Side Effects of Learning from Low-dimensional Data Embedded in a Euclidean Spacecited by 6
2022Principled Acceleration of Iterative Numerical Methods Using Machine Learningcited by 6
2024M2NO: An Efficient Multi-Resolution Operator Framework for Dynamic Multi-Scale PDE Solverscited by 6
2024UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEscited by 6
2025Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Methodcited by 6
2021Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEscited by 5
2022Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equationscited by 5
2023RANS-PINN based Simulation Surrogates for Predicting Turbulent Flowscited by 5
2024Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equationscited by 5
2025Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogatescited by 5
2025Accurate and scalable deep Maxwell solvers using multilevel iterative methodscited by 5
2025SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioningcited by 5
2026OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inferencecited by 4
2022Leveraging Stochastic Predictions of Bayesian Neural Networks for Fluid Simulationscited by 3
2023An Implicit GNN Solver for Poisson-like problemscited by 3
2024Stabilizing Backpropagation Through Time to Learn Complex Physicscited by 3

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