# Physics informed WNO

arXiv: https://arxiv.org/abs/2302.05925 · HTML: https://mlsanity.com/paper/2302.05925/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 50 references that are in the map.
- First posted on arXiv 12 Feb 2023.

## Builds on (most cited first, 16 of 16)

- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [Machine Learning for semi linear PDEs](https://mlsanity.com/paper/1809.07609/) (2018)
- [Stochastic projection based approach for gradient free physics informed learning](https://mlsanity.com/paper/2209.13724/) (2022)
- [Multi-fidelity wavelet neural operator with application to uncertainty quantification](https://mlsanity.com/paper/2208.05606/) (2022)

## Built on by (most cited first, 27 of 27)

- [Physics-Informed Geometry-Aware Neural Operator](https://mlsanity.com/paper/2408.01600/) (2024)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [A foundational neural operator that continuously learns without forgetting](https://mlsanity.com/paper/2310.18885/) (2023)
- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [Physics-informed Discretization-independent Deep Compositional Operator Network](https://mlsanity.com/paper/2404.13646/) (2024)
- [Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations](https://mlsanity.com/paper/2410.02151/) (2024)
- [DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modeling](https://mlsanity.com/paper/2502.06250/) (2025)
- [Smooth and Sparse Latent Dynamics in Operator Learning with Jerk Regularization](https://mlsanity.com/paper/2402.15636/) (2024)
- [Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics](https://mlsanity.com/paper/2409.10972/) (2024)
- [Learning Generic Solutions for Multiphase Transport in Porous Media via the Flux Functions Operator](https://mlsanity.com/paper/2307.01354/) (2023)
- [How important are specialized transforms in Neural Operators?](https://mlsanity.com/paper/2308.09293/) (2023)
- [DPA-WNO: A gray box model for a class of stochastic mechanics problem](https://mlsanity.com/paper/2309.15128/) (2023)
- [Waveformer for modelling dynamical systems](https://mlsanity.com/paper/2310.04990/) (2023)
- [Generative flow induced neural architecture search: Towards discovering optimal architecture in wavelet neural operator](https://mlsanity.com/paper/2405.06910/) (2024)
- [M2M: Learning controllable Multi of experts and multi-scale operators are the Partial Differential Equations need](https://mlsanity.com/paper/2410.11617/) (2024)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [A unified physics-informed generative operator framework for general inverse problems](https://mlsanity.com/paper/2511.03241/) (2025)
- [Geometric Laplace Neural Operator](https://mlsanity.com/paper/2512.16409/) (2025)
- [Neuroscience inspired scientific machine learning (Part-2): Variable spiking wavelet neural operator](https://mlsanity.com/paper/2311.14710/) (2023)
- [Harnessing physics-informed operators for high-dimensional reliability analysis problems](https://mlsanity.com/paper/2409.04708/) (2024)
- [Event-driven physics-informed operator learning for reliability analysis](https://mlsanity.com/paper/2511.06083/) (2025)
- [Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations](https://mlsanity.com/paper/2602.12706/) (2026)
- [LE-PDE++: Mamba for accelerating PDEs Simulations](https://mlsanity.com/paper/2411.01897/) (2024)
- [Deep Muscle EMG construction using A Physics-Integrated Deep Learning approach](https://mlsanity.com/paper/2503.05201/) (2025)
- [From Local Interactions to Global Operators: Scalable Gaussian Process Operator for Physical Systems](https://mlsanity.com/paper/2506.15906/) (2025)
- [Multi-Fidelity Learning with Shallow Recurrent Decoders for Multi-Physics Applications](https://mlsanity.com/paper/2606.05202/) (2026)
- [Feature tracking in physics-informed neural networks via joint optimization of nonlinear deformation manifolds: application to shocks](https://mlsanity.com/paper/2610.02230/) (2026)

## Abstract (arXiv)

Deep neural operators are recognized as an effective tool for learning solution operators of complex partial differential equations (PDEs). As compared to laborious analytical and computational tools, a single neural operator can predict solutions of PDEs for varying initial or boundary conditions and different inputs. A recently proposed Wavelet Neural Operator (WNO) is one such operator that harnesses the advantage of time-frequency localization of wavelets to capture the manifolds in the spatial domain effectively. While WNO has proven to be a promising method for operator learning, the data-hungry nature of the framework is a major shortcoming. In this work, we propose a physics-informed WNO for learning the solution operators of families of parametric PDEs without labeled training data. The efficacy of the framework is validated and illustrated with four nonlinear spatiotemporal systems relevant to various fields of engineering and science.