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

arXiv: https://arxiv.org/abs/2211.01604 · HTML: https://mlsanity.com/paper/2211.01604/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 30 references that are in the map.
- First posted on arXiv 3 Nov 2022.

## Builds on (most cited first, 17 of 17)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [On First-Order Meta-Learning Algorithms](https://mlsanity.com/paper/1803.02999/) (2018)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Meta-learning PINN loss functions](https://mlsanity.com/paper/2107.05544/) (2021)
- [HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks](https://mlsanity.com/paper/2111.01008/) (2021)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [Meshless physics-informed deep learning method for three-dimensional solid mechanics](https://mlsanity.com/paper/2012.01547/) (2020)
- [Using Machine Learning to Augment Coarse-Grid Computational Fluid Dynamics Simulations](https://mlsanity.com/paper/2010.00072/) (2020)
- [Transferring Knowledge across Learning Processes](https://mlsanity.com/paper/1812.01054/) (2018)
- [Learning Composable Energy Surrogates for PDE Order Reduction](https://mlsanity.com/paper/2005.06549/) (2020)

## Built on by (most cited first, 18 of 18)

- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Efficient physics-informed neural networks using hash encoding](https://mlsanity.com/paper/2302.13397/) (2023)
- [GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEs](https://mlsanity.com/paper/2303.14878/) (2023)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [About optimal loss function for training physics-informed neural networks under respecting causality](https://mlsanity.com/paper/2304.02282/) (2023)
- [Enabling Automatic Differentiation with Mollified Graph Neural Operators](https://mlsanity.com/paper/2504.08277/) (2025)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given PDEs](https://mlsanity.com/paper/2310.13270/) (2023)
- [Adaptive Physics-informed Neural Networks: A Survey](https://mlsanity.com/paper/2503.18181/) (2025)
- [Operator Learning Using Weak Supervision from Walk-on-Spheres](https://mlsanity.com/paper/2603.01193/) (2026)
- [Meta-Learning for Airflow Simulations with Graph Neural Networks](https://mlsanity.com/paper/2306.10624/) (2023)
- [Towards Foundation Model for Chemical Reactor Modeling: Meta-Learning with Physics-Informed Adaptation](https://mlsanity.com/paper/2405.11752/) (2024)
- [Physics-Informed Deep B-Spline Networks](https://mlsanity.com/paper/2503.16777/) (2025)
- [Equation Recast for Canonical Operator Learning Across Parametric PDEs](https://mlsanity.com/paper/2609.02982/) (2026)
- [Beyond Static Models: Hypernetworks for Adaptive and Generalizable Forecasting in Complex Parametric Dynamical Systems](https://mlsanity.com/paper/2506.19609/) (2025)
- [Multi-Operator Few-Shot Learning for Generalization Across PDE Families](https://mlsanity.com/paper/2508.01211/) (2025)
- [LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networks](https://mlsanity.com/paper/2607.14233/) (2026)
- [Transferability of Learned States in Neural PDE Solvers](https://mlsanity.com/paper/2610.10972/) (2026)

## 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.