# A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs

arXiv: https://arxiv.org/abs/2110.13361 · HTML: https://mlsanity.com/paper/2110.13361/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 80 references that are in the map.
- First posted on arXiv 26 Oct 2021.

## Builds on (most cited first, 30 of 37)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [RL2: Fast Reinforcement Learning via Slow Reinforcement Learning](https://mlsanity.com/paper/1611.02779/) (2016)
- [Meta-Learning with Latent Embedding Optimization](https://mlsanity.com/paper/1807.05960/) (2018)
- [TADAM: Task dependent adaptive metric for improved few-shot learning](https://mlsanity.com/paper/1805.10123/) (2018)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Meta-SGD: Learning to Learn Quickly for Few-Shot Learning](https://mlsanity.com/paper/1707.09835/) (2017)
- [Learning to reinforcement learn](https://mlsanity.com/paper/1611.05763/) (2016)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Meta-Dataset: A Dataset of Datasets for Learning to Learn from Few Examples](https://mlsanity.com/paper/1903.03096/) (2019)
- [Meta Networks](https://mlsanity.com/paper/1703.00837/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Probabilistic Model-Agnostic Meta-Learning](https://mlsanity.com/paper/1806.02817/) (2018)
- [Learning to Optimize](https://mlsanity.com/paper/1606.01885/) (2016)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Infinite Mixture Prototypes for Few-Shot Learning](https://mlsanity.com/paper/1902.04552/) (2019)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Meta-learning PINN loss functions](https://mlsanity.com/paper/2107.05544/) (2021)

## Built on by (most cited first, 30 of 32)

- [Generating synthetic data for neural operators](https://mlsanity.com/paper/2401.02398/) (2024)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [Spectrally Adapted Physics-Informed Neural Networks for Solving Unbounded Domain Problems](https://mlsanity.com/paper/2202.02710/) (2022)
- [VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Models](https://mlsanity.com/paper/2506.13754/) (2025)
- [Towards a Physics Foundation Model](https://mlsanity.com/paper/2509.13805/) (2025)
- [Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networks](https://mlsanity.com/paper/2210.12669/) (2022)
- [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)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Advancing Generalizability by the Baldwin Effect](https://mlsanity.com/paper/2312.03243/) (2023)
- [Approximating Families of Sharp Solutions to Fisher's Equation with Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.08313/) (2024)
- [MetaSym: A Symplectic Meta-learning Framework for Physical Intelligence](https://mlsanity.com/paper/2502.16667/) (2025)
- [Adaptive Physics-informed Neural Networks: A Survey](https://mlsanity.com/paper/2503.18181/) (2025)
- [Out-of-Distribution Generalization for Neural Physics Solvers](https://mlsanity.com/paper/2601.19091/) (2026)
- [Controlled physics-informed data generation for deep learning-based remaining useful life prediction under unseen operation conditions](https://mlsanity.com/paper/2304.11702/) (2023)
- [Resolution invariant deep operator network for PDEs with complex geometries](https://mlsanity.com/paper/2402.00825/) (2024)
- [Learning solutions of parametric Navier-Stokes with physics-informed neural networks](https://mlsanity.com/paper/2402.03153/) (2024)
- [iPINNs: Incremental learning for Physics-informed neural networks](https://mlsanity.com/paper/2304.04854/) (2023)
- [Branched Latent Neural Maps](https://mlsanity.com/paper/2308.02599/) (2023)
- [Common pitfalls to avoid while using multiobjective optimization in machine learning](https://mlsanity.com/paper/2405.01480/) (2024)
- [Meta-learning Loss Functions of Parametric Partial Differential Equations Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2412.00225/) (2024)
- [Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metrics](https://mlsanity.com/paper/2402.11126/) (2024)
- [Towards Foundation Model for Chemical Reactor Modeling: Meta-Learning with Physics-Informed Adaptation](https://mlsanity.com/paper/2405.11752/) (2024)
- [Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs](https://mlsanity.com/paper/2510.01039/) (2025)
- [Neuro-Symbolic Multitasking: A Unified Framework for Discovering Generalizable Solutions to PDE Families](https://mlsanity.com/paper/2602.11630/) (2026)
- [Equation Recast for Canonical Operator Learning Across Parametric PDEs](https://mlsanity.com/paper/2609.02982/) (2026)
- [Multi-level datasets training method in Physics-Informed Neural Networks](https://mlsanity.com/paper/2504.21328/) (2025)
- [Learning and Transferring Physical Models through Derivatives](https://mlsanity.com/paper/2505.01391/) (2025)
- [Latent Representation Learning of Multi-scale Thermophysics: Application to Dynamics in Shocked Porous Energetic Material](https://mlsanity.com/paper/2506.12996/) (2025)
- [LFR-PINO: A Layered Fourier Reduced Physics-Informed Neural Operator for Parametric PDEs](https://mlsanity.com/paper/2506.17582/) (2025)
- [SUPN: Shallow Universal Polynomial Networks for Efficient Function Approximation](https://mlsanity.com/paper/2511.21414/) (2025)
- [Meta-Learned Basis Adaptation for Parametric Linear PDEs](https://mlsanity.com/paper/2604.09289/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) as a means of discretizing partial differential equations (PDEs) are garnering much attention in the Computational Science and Engineering (CS&E) world. At least two challenges exist for PINNs at present: an understanding of accuracy and convergence characteristics with respect to tunable parameters and identification of optimization strategies that make PINNs as efficient as other computational science tools. The cost of PINNs training remains a major challenge of Physics-informed Machine Learning (PiML) - and, in fact, machine learning (ML) in general. This paper is meant to move towards addressing the latter through the study of PINNs on new tasks, for which parameterized PDEs provides a good testbed application as tasks can be easily defined in this context. Following the ML world, we introduce metalearning of PINNs with application to parameterized PDEs. By introducing metalearning and transfer learning concepts, we can greatly accelerate the PINNs optimization process. We present a survey of model-agnostic metalearning, and then discuss our model-aware metalearning applied to PINNs as well as implementation considerations and algorithmic complexity. We then test our approach on various canonical forward parameterized PDEs that have been presented in the emerging PINNs literature.