# Applications of physics informed neural operators

arXiv: https://arxiv.org/abs/2203.12634 · HTML: https://mlsanity.com/paper/2203.12634/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 67 references that are in the map.
- First posted on arXiv 23 Mar 2022.

## Builds on (most cited first, 23 of 23)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Towards A Rigorous Science of Interpretable Machine Learning](https://mlsanity.com/paper/1702.08608/) (2017)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control](https://mlsanity.com/paper/2105.14094/) (2021)
- [Accelerated, Scalable and Reproducible AI-driven Gravitational Wave Detection](https://mlsanity.com/paper/2012.08545/) (2020)
- [Enabling real-time multi-messenger astrophysics discoveries with deep learning](https://mlsanity.com/paper/1911.11779/) (2019)
- [DLHub: Model and Data Serving for Science](https://mlsanity.com/paper/1811.11213/) (2018)
- [A FAIR and AI-ready Higgs boson decay dataset](https://mlsanity.com/paper/2108.02214/) (2021)
- [Advances in Machine and Deep Learning for Modeling and Real-time Detection of Multi-Messenger Sources](https://mlsanity.com/paper/2105.06479/) (2021)
- [Convergence of Artificial Intelligence and High Performance Computing on NSF-supported Cyberinfrastructure](https://mlsanity.com/paper/2003.08394/) (2020)
- [Interpretable AI forecasting for numerical relativity waveforms of quasi-circular, spinning, non-precessing binary black hole mergers](https://mlsanity.com/paper/2110.06968/) (2021)
- [Inference-optimized AI and high performance computing for gravitational wave detection at scale](https://mlsanity.com/paper/2201.11133/) (2022)
- [Interpreting a Machine Learning Model for Detecting Gravitational Waves](https://mlsanity.com/paper/2202.07399/) (2022)

## Built on by (most cited first, 30 of 33)

- [Plasma Surrogate Modelling using Fourier Neural Operators](https://mlsanity.com/paper/2311.05967/) (2023)
- [Neural Spectral Methods: Self-supervised learning in the spectral domain](https://mlsanity.com/paper/2312.05225/) (2023)
- [Masked Autoencoders are PDE Learners](https://mlsanity.com/paper/2403.17728/) (2024)
- [TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation](https://mlsanity.com/paper/2505.17341/) (2025)
- [FAIR for AI: An interdisciplinary and international community building perspective](https://mlsanity.com/paper/2210.08973/) (2022)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applications](https://mlsanity.com/paper/2406.03569/) (2024)
- [Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML](https://mlsanity.com/paper/2301.13331/) (2023)
- [PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations](https://mlsanity.com/paper/2507.02227/) (2025)
- [Magnetohydrodynamics with Physics Informed Neural Operators](https://mlsanity.com/paper/2302.08332/) (2023)
- [Component Fourier Neural Operator for Singularly Perturbed Differential Equations](https://mlsanity.com/paper/2409.04779/) (2024)
- [Enabling Automatic Differentiation with Mollified Graph Neural Operators](https://mlsanity.com/paper/2504.08277/) (2025)
- [Large-scale flood modeling and forecasting with FloodCast](https://mlsanity.com/paper/2403.12226/) (2024)
- [Physics-Informed DeepONet Coupled with FEM for Convective Transport in Porous Media with Sharp Gaussian Sources](https://mlsanity.com/paper/2508.19847/) (2025)
- [Generalized Lie Symmetries in Physics-Informed Neural Operators](https://mlsanity.com/paper/2502.00373/) (2025)
- [An effective physics-informed neural operator framework for predicting wavefields](https://mlsanity.com/paper/2507.16431/) (2025)
- [Data-Efficient Time-Dependent PDE Surrogates: Graph Neural Simulators vs. Neural Operators](https://mlsanity.com/paper/2509.06154/) (2025)
- [Learning Physical Operators using Neural Operators](https://mlsanity.com/paper/2602.23113/) (2026)
- [Chaos into Order: Neural Framework for Expected Value Estimation of Stochastic Partial Differential Equations](https://mlsanity.com/paper/2502.03670/) (2025)
- [Bridging ocean wave physics and deep learning: Physics-informed neural operators for nonlinear wavefield reconstruction in real-time](https://mlsanity.com/paper/2508.03315/) (2025)
- [Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations](https://mlsanity.com/paper/2602.12706/) (2026)
- [Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution](https://mlsanity.com/paper/2603.09693/) (2026)
- [Debiased neural operators for estimating functionals](https://mlsanity.com/paper/2604.19296/) (2026)
- [On the training of physics-informed neural operators for solving parametric partial differential equations](https://mlsanity.com/paper/2606.06164/) (2026)
- [Number Theoretic Accelerated Learning of Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.13869/) (2023)
- [Equation identification for fluid flows via physics-informed neural networks](https://mlsanity.com/paper/2408.17271/) (2024)
- [LE-PDE++: Mamba for accelerating PDEs Simulations](https://mlsanity.com/paper/2411.01897/) (2024)
- [How to Re-enable PDE Loss for Physical Systems Modeling Under Partial Observation](https://mlsanity.com/paper/2412.09116/) (2024)
- [Pseudo-Physics-Informed Neural Operators: Enhancing Operator Learning from Limited Data](https://mlsanity.com/paper/2502.02682/) (2025)
- [ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marching](https://mlsanity.com/paper/2512.19643/) (2025)

## Abstract (arXiv)

We present an end-to-end framework to learn partial differential equations that brings together initial data production, selection of boundary conditions, and the use of physics-informed neural operators to solve partial differential equations that are ubiquitous in the study and modeling of physics phenomena. We first demonstrate that our methods reproduce the accuracy and performance of other neural operators published elsewhere in the literature to learn the 1D wave equation and the 1D Burgers equation. Thereafter, we apply our physics-informed neural operators to learn new types of equations, including the 2D Burgers equation in the scalar, inviscid and vector types. Finally, we show that our approach is also applicable to learn the physics of the 2D linear and nonlinear shallow water equations, which involve three coupled partial differential equations. We release our artificial intelligence surrogates and scientific software to produce initial data and boundary conditions to study a broad range of physically motivated scenarios. We provide the source code, an interactive website to visualize the predictions of our physics informed neural operators, and a tutorial for their use at the Data and Learning Hub for Science.