# Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equations

arXiv: https://arxiv.org/abs/2212.04100 · HTML: https://mlsanity.com/paper/2212.04100/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 35 references that are in the map.
- First posted on arXiv 8 Dec 2022.

## Builds on (most cited first, 25 of 25)

- [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)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [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)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Learning to Optimize Multigrid PDE Solvers](https://mlsanity.com/paper/1902.10248/) (2019)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Physics-based Deep Learning](https://mlsanity.com/paper/2109.05237/) (2021)
- [The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?](https://mlsanity.com/paper/2103.09655/) (2021)
- [Physical Symmetries Embedded in Neural Networks](https://mlsanity.com/paper/1904.08991/) (2019)
- [Fast PDE-constrained optimization via self-supervised operator learning](https://mlsanity.com/paper/2110.13297/) (2021)
- [Deep Eikonal Solvers](https://mlsanity.com/paper/1903.07973/) (2019)
- [PhyAug: Physics-Directed Data Augmentation for Deep Sensing Model Transfer in Cyber-Physical Systems](https://mlsanity.com/paper/2104.01160/) (2021)

## Built on by (most cited first, 5 of 5)

- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [Explicit Discovery of Nonlinear Symmetries from Dynamic Data](https://mlsanity.com/paper/2510.01855/) (2025)
- [Governing Equation Discovery from Data Based on Differential Invariants](https://mlsanity.com/paper/2505.18798/) (2025)
- [Generalized Lie Symmetries in Physics-Informed Neural Operators](https://mlsanity.com/paper/2502.00373/) (2025)
- [Optimal scenario design for climate emulation](https://mlsanity.com/paper/2606.19302/) (2026)

## Abstract (arXiv)

Neural networks, especially the recent proposed neural operator models, are increasingly being used to find the solution operator of differential equations. Compared to traditional numerical solvers, they are much faster and more efficient in practical applications. However, one critical issue is that training neural operator models require large amount of ground truth data, which usually comes from the slow numerical solvers. In this paper, we propose a physics-guided data augmentation (PGDA) method to improve the accuracy and generalization of neural operator models. Training data is augmented naturally through the physical properties of differential equations such as linearity and translation. We demonstrate the advantage of PGDA on a variety of linear differential equations, showing that PGDA can improve the sample complexity and is robust to distributional shift.