# Meta-learning Loss Functions of Parametric Partial Differential Equations Using Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2412.00225 · HTML: https://mlsanity.com/paper/2412.00225/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 26 references that are in the map.
- First posted on arXiv 29 Nov 2024.

## Builds on (most cited first, 14 of 14)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Meta-Learning for Semi-Supervised Few-Shot Classification](https://mlsanity.com/paper/1803.00676/) (2018)
- [Learning to Adapt in Dynamic, Real-World Environments Through Meta-Reinforcement Learning](https://mlsanity.com/paper/1803.11347/) (2018)
- [Meta-Learning for Low-Resource Neural Machine Translation](https://mlsanity.com/paper/1808.08437/) (2018)
- [GAMI-Net: An Explainable Neural Network based on Generalized Additive Models with Structured Interactions](https://mlsanity.com/paper/2003.07132/) (2020)
- [On Loss Functions for Deep Neural Networks in Classification](https://mlsanity.com/paper/1702.05659/) (2017)
- [Meta-learning PINN loss functions](https://mlsanity.com/paper/2107.05544/) (2021)
- [A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs](https://mlsanity.com/paper/2110.13361/) (2021)
- [PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data](https://mlsanity.com/paper/2212.04971/) (2022)
- [GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEs](https://mlsanity.com/paper/2303.14878/) (2023)
- [A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data](https://mlsanity.com/paper/2305.10706/) (2023)
- [A survey and taxonomy of loss functions in machine learning](https://mlsanity.com/paper/2301.05579/) (2023)
- [GAM(e) changer or not? An evaluation of interpretable machine learning models based on additive model constraints](https://mlsanity.com/paper/2204.09123/) (2022)

## Built on by (most cited first, 2 of 2)

- [ReBaNO: Reduced Basis Neural Operator Mitigating Generalization Gaps and Achieving Discretization Invariance](https://mlsanity.com/paper/2509.09611/) (2025)
- [IP-Basis PINNs: Efficient Multi-Query Inverse Parameter Estimation](https://mlsanity.com/paper/2509.07245/) (2025)

## Abstract (arXiv)

This paper proposes a new way to learn Physics-Informed Neural Network loss functions using Generalized Additive Models. We apply our method by meta-learning parametric partial differential equations, PDEs, on Burger's and 2D Heat Equations. The goal is to learn a new loss function for each parametric PDE using meta-learning. The derived loss function replaces the traditional data loss, allowing us to learn each parametric PDE more efficiently, improving the meta-learner's performance and convergence.