# Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given PDEs

arXiv: https://arxiv.org/abs/2310.13270 · HTML: https://mlsanity.com/paper/2310.13270/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 21 references that are in the map.
- First posted on arXiv 20 Oct 2023.

## Builds on (most cited first, 12 of 12)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks](https://mlsanity.com/paper/2111.01008/) (2021)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [Meta-Auto-Decoder for Solving Parametric Partial Differential Equations](https://mlsanity.com/paper/2111.08823/) (2021)
- [MetaFun: Meta-Learning with Iterative Functional Updates](https://mlsanity.com/paper/1912.02738/) (2019)
- [Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh](https://mlsanity.com/paper/2211.01604/) (2022)

## Built on by (most cited first, 5 of 5)

- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions](https://mlsanity.com/paper/2509.05117/) (2025)
- [Adaptive Physics-informed Neural Networks: A Survey](https://mlsanity.com/paper/2503.18181/) (2025)
- [Compositional Meta-Learning for Mitigating Task Heterogeneity in Physics-Informed Neural Networks](https://mlsanity.com/paper/2604.26999/) (2026)
- [Amortizing Physics-Informed Neural Solvers via Graph Hypernetworks](https://mlsanity.com/paper/2609.19915/) (2026)

## Abstract (arXiv)

We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for solving newly given PDE problems. We encode a PDE problem into a problem representation using neural networks, where governing equations are represented by coefficients of a polynomial function of partial derivatives, and boundary conditions are represented by a set of point-condition pairs. We use the problem representation as an input of a neural network for predicting solutions, which enables us to efficiently predict problem-specific solutions by the forwarding process of the neural network without updating model parameters. To train our model, we minimize the expected error when adapted to a PDE problem based on the physics-informed neural network framework, by which we can evaluate the error even when solutions are unknown. We demonstrate that our proposed method outperforms existing methods in predicting solutions of PDE problems.