# A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries

**Authors:** Reza Najian Asl, Yusuke Yamazaki, Kianoosh Taghikhani, Mayu Muramatsu, Markus Apel, Shahed Rezaei  
**arXiv:** [2504.02459](https://arxiv.org/abs/2504.02459)  
**Date:** 2025-04-03  
**Summary page:** https://www.summarizepaper.com/en/paper/2504.02459v1/

## Key points

- Introduction of implicit Finite Operator Learning (iFOL) for solving PDEs on complex geometries
- Utilization of physics-informed encoder-decoder network connecting parameter and solution spaces
- Unique loss formulation eliminating traditional encode-process-decode pipeline
- Ability to provide accurate parametric and continuous fields with solution-to-parameter gradients
- Capability to capture sharp discontinuities in solutions and zero-shot super-resolution capability
- Application to arbitrary geometries without constraints on geometry or mesh
- Minimization of physics-informed loss function using discrete residuals from numerical PDE methods
- Incorporation of backpropagation of discrete residuals during training and inference processes
- Evaluation of generalization capabilities across stationary and transient PDEs

## Summary

The paper "A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries" introduces a novel approach called implicit Finite Operator Learning (iFOL) for solving partial differential equations (PDEs) on complex geometries. The key innovation is a physics-informed encoder-decoder network that connects the continuous parameter and solution spaces. By utilizing an implicit neural field network conditioned on latent codes derived through second-order meta-learning techniques, the decoder constructs parametric solution fields. iFOL stands out with its unique loss formulation that eliminates the traditional encode-process-decode pipeline used in operator learning with conditional neural fields for PDEs. This framework not only provides accurate parametric and continuous fields but also offers solution-to-parameter gradients without additional loss terms or sensitivity analysis. Additionally, iFOL can capture sharp discontinuities in solutions and has zero-shot super-resolution capability, making it applicable to arbitrary geometries without constraints on geometry or mesh. The proposed method minimizes a physics-informed loss function using discrete residuals obtained from standard numerical PDE methods during both training and inference stages. Notably, iFOL incorporates backpropagation of discrete residuals during training and inference processes. The authors critically evaluate these features and assess the network's generalization capabilities across stationary and transient PDEs. Overall, iFOL shows promise in addressing challenging computational mechanics problems due to its ability to handle arbitrary geometries effectively. This innovative meta-learning framework opens up new possibilities for solving parametric PDEs with enhanced accuracy and efficiency while providing insights into broader applications within computational science and engineering domains.