mlsanityData to 7 Oct 2026

Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh

arXiv 2211.01604 ↗First posted 3 Nov 2022cs.LG cs.AIcited by 18

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Abstract (arXiv)

Partial differential equations (PDEs) are often computationally challenging to solve, and in many settings many related PDEs must be be solved either at every timestep or for a variety of candidate boundary conditions, parameters, or geometric domains. We present a meta-learning based method which learns to rapidly solve problems from a distribution of related PDEs. We use meta-learning (MAML and LEAP) to identify initializations for a neural network representation of the PDE solution such that a residual of the PDE can be quickly minimized on a novel task. We apply our meta-solving approach to a nonlinear Poisson's equation, 1D Burgers' equation, and hyperelasticity equations with varying parameters, geometries, and boundary conditions. The resulting Meta-PDE method finds qualitatively accurate solutions to most problems within a few gradient steps; for the nonlinear Poisson and hyper-elasticity equation this results in an intermediate accuracy approximation up to an order of magnitude faster than a baseline finite element analysis (FEA) solver with equivalent accuracy. In comparison to other learned solvers and surrogate models, this meta-learning approach can be trained without supervision from expensive ground-truth data, does not require a mesh, and can even be used when the geometry and topology varies between tasks.

Key facts. Cited by 18 papers in the map (data through 7 Oct 2026). Builds on 17 of its 30 references that are in the map. First posted on arXiv 3 Nov 2022.

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17 of 30 references.

References · 17

  1. 2017Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networkscited by 4,861
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,214
  4. 2018On First-Order Meta-Learning Algorithmscited by 1,003
  5. 2021Machine learning accelerated computational fluid dynamicscited by 461
  6. 2022Message Passing Neural PDE Solverscited by 368
  7. 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
  8. 2021Learned Coarse Models for Efficient Turbulence Simulationcited by 90
  9. 2019Learning to Optimize Multigrid PDE Solverscited by 79
  10. 2019Learning Neural PDE Solvers with Convergence Guaranteescited by 77
  11. 2021Meta-learning PINN loss functionscited by 51
  12. 2021HyperPINN: Learning parameterized differential equations with physics-informed hypernetworkscited by 38
5 more
  1. 2021A novel meta-learning initialization method for physics-informed neural networkscited by 35
  2. 2020Meshless physics-informed deep learning method for three-dimensional solid mechanicscited by 27
  3. 2020Using Machine Learning to Augment Coarse-Grid Computational Fluid Dynamics Simulationscited by 24
  4. 2018Transferring Knowledge across Learning Processescited by 22
  5. 2020Learning Composable Energy Surrogates for PDE Order Reductioncited by 13

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 6; 2024: 2; 2025: 5; 2026: 5.

18 papers.

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  1. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  2. 2023Efficient physics-informed neural networks using hash encodingcited by 21
  3. 2023GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEscited by 20
  4. 2023Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrologycited by 11
  5. 2023About optimal loss function for training physics-informed neural networks under respecting causalitycited by 6
  6. 2025Enabling Automatic Differentiation with Mollified Graph Neural Operatorscited by 6
  7. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  8. 2023Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given PDEscited by 5
  9. 2025Adaptive Physics-informed Neural Networks: A Surveycited by 5
  10. 2023Meta-Learning for Airflow Simulations with Graph Neural Networkscited by 1
  11. 2024Towards Foundation Model for Chemical Reactor Modeling: Meta-Learning with Physics-Informed Adaptationcited by 1
  12. 2025Physics-Informed Deep B-Spline Networkscited by 1
6 more
  1. 2026Operator Learning Using Weak Supervision from Walk-on-Spherescited by 1
  2. 2026Equation Recast for Canonical Operator Learning Across Parametric PDEscited by 1
  3. 2025Beyond Static Models: Hypernetworks for Adaptive and Generalizable Forecasting in Complex Parametric Dynamical Systemscited by 0
  4. 2025Multi-Operator Few-Shot Learning for Generalization Across PDE Familiescited by 0
  5. 2026LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networkscited by 0
  6. 2026Transferability of Learned States in Neural PDE Solverscited by 0