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

**Authors:** Tomoharu Iwata, Yusuke Tanaka, Naonori Ueda  
**arXiv:** [2310.13270](https://arxiv.org/abs/2310.13270)  
**Date:** 2023-10-20  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2310.13270v1/

## Key points

- Authors propose a neural network-based meta-learning method for solving PDE problems
- Method encodes PDE problems into a problem representation using neural networks
- Problem representation includes coefficients of a polynomial function for governing equations and point-condition pairs for boundary conditions
- Neural network predicts solutions based on the problem representation without updating model parameters
- Model is trained by minimizing expected error using physics-informed neural network framework
- Proposed method outperforms existing methods in predicting solutions for various PDE problems
- Paper presents an innovative approach with potential for broader applications in solving complex mathematical equations

## Summary

In their paper titled "Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given PDEs," authors Tomoharu Iwata, Yusuke Tanaka, and Naonori Ueda propose a novel neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method aims to meta-learn how to solve a wide variety of PDE problems and utilize this knowledge to solve newly given PDE problems. To achieve this, the authors encode a given PDE problem into a problem representation using neural networks. In this representation, the governing equations are represented by coefficients of a polynomial function of partial derivatives, while boundary conditions are represented by a set of point-condition pairs. This problem representation serves as an input for a neural network that predicts solutions. One key advantage of their approach is that it enables efficient prediction of problem-specific solutions without updating model parameters. By leveraging the forwarding process of the neural network, the authors can predict solutions quickly and accurately. To train their model, the authors minimize the expected error when adapted to a specific PDE problem based on the physics-informed neural network framework. This framework allows them to evaluate the error even when solutions are unknown. The authors demonstrate the effectiveness of their proposed method by comparing it with existing methods in predicting solutions for various PDE problems. Their method consistently outperforms these existing methods, highlighting its potential for efficient and accurate solution prediction in PDE problems. Overall, this paper presents an innovative meta-learning approach that leverages neural networks to efficiently solve PDE problems. The results indicate its superiority over existing methods and suggest its potential for broader applications in solving complex mathematical equations.