# MetaNO: How to Transfer Your Knowledge on Learning Hidden Physics

arXiv: https://arxiv.org/abs/2301.12095 · HTML: https://mlsanity.com/paper/2301.12095/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 64 references that are in the map.
- First posted on arXiv 28 Jan 2023.

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

- [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)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Generalizing from a Few Examples: A Survey on Few-Shot Learning](https://mlsanity.com/paper/1904.05046/) (2019)
- [Rapid Learning or Feature Reuse? Towards Understanding the Effectiveness of MAML](https://mlsanity.com/paper/1909.09157/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Meta-Learning: A Survey](https://mlsanity.com/paper/1810.03548/) (2018)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Bayesian Model-Agnostic Meta-Learning](https://mlsanity.com/paper/1806.03836/) (2018)
- [Provable Meta-Learning of Linear Representations](https://mlsanity.com/paper/2002.11684/) (2020)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [LEADS: Learning Dynamical Systems that Generalize Across Environments](https://mlsanity.com/paper/2106.04546/) (2021)
- [Meta-Learning Dynamics Forecasting Using Task Inference](https://mlsanity.com/paper/2102.10271/) (2021)
- [Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural network](https://mlsanity.com/paper/2108.11985/) (2021)
- [MetaFun: Meta-Learning with Iterative Functional Updates](https://mlsanity.com/paper/1912.02738/) (2019)
- [One-shot learning for solution operators of partial differential equations](https://mlsanity.com/paper/2104.05512/) (2021)
- [MAML and ANIL Provably Learn Representations](https://mlsanity.com/paper/2202.03483/) (2022)
- [Reliable and Explainable Machine Learning Methods for Accelerated Material Discovery](https://mlsanity.com/paper/1901.02717/) (2019)
- [IAE-Net: Integral Autoencoders for Discretization-Invariant Learning](https://mlsanity.com/paper/2203.05142/) (2022)
- [Deep learning of material transport in complex neurite networks](https://mlsanity.com/paper/2103.08654/) (2021)
- [Stochastic Deep Networks with Linear Competing Units for Model-Agnostic Meta-Learning](https://mlsanity.com/paper/2208.01573/) (2022)
- [MetaNOR: A Meta-Learnt Nonlocal Operator Regression Approach for Metamaterial Modeling](https://mlsanity.com/paper/2206.02040/) (2022)

## Built on by (most cited first, 10 of 10)

- [GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEs](https://mlsanity.com/paper/2303.14878/) (2023)
- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Masked Autoencoders are PDE Learners](https://mlsanity.com/paper/2403.17728/) (2024)
- [Heterogeneous Peridynamic Neural Operators: Discover Biotissue Constitutive Law and Microstructure From Digital Image Correlation Measurements](https://mlsanity.com/paper/2403.18597/) (2024)
- [Disentangled Representation Learning for Parametric Partial Differential Equations](https://mlsanity.com/paper/2410.02136/) (2024)
- [MD-PNOP: Equation-Recast Neural Operators for Minimal-Data Extrapolation and PDE Solver Acceleration](https://mlsanity.com/paper/2509.01416/) (2025)
- [ReBaNO: Reduced Basis Neural Operator Mitigating Generalization Gaps and Achieving Discretization Invariance](https://mlsanity.com/paper/2509.09611/) (2025)
- [Neural Operator Processes for Probabilistic Operator Learning under Partial Observations](https://mlsanity.com/paper/2606.22946/) (2026)
- [Equation Recast for Canonical Operator Learning Across Parametric PDEs](https://mlsanity.com/paper/2609.02982/) (2026)
- [Transfer Operator Learning with Fusion Frame](https://mlsanity.com/paper/2408.10458/) (2024)

## Abstract (arXiv)

Gradient-based meta-learning methods have primarily been applied to classical machine learning tasks such as image classification. Recently, PDE-solving deep learning methods, such as neural operators, are starting to make an important impact on learning and predicting the response of a complex physical system directly from observational data. Since the data acquisition in this context is commonly challenging and costly, the call of utilization and transfer of existing knowledge to new and unseen physical systems is even more acute. Herein, we propose a novel meta-learning approach for neural operators, which can be seen as transferring the knowledge of solution operators between governing (unknown) PDEs with varying parameter fields. Our approach is a provably universal solution operator for multiple PDE solving tasks, with a key theoretical observation that underlying parameter fields can be captured in the first layer of neural operator models, in contrast to typical final-layer transfer in existing meta-learning methods. As applications, we demonstrate the efficacy of our proposed approach on PDE-based datasets and a real-world material modeling problem, illustrating that our method can handle complex and nonlinear physical response learning tasks while greatly improving the sampling efficiency in unseen tasks.