# Lie Point Symmetry and Physics Informed Networks

arXiv: https://arxiv.org/abs/2311.04293 · HTML: https://mlsanity.com/paper/2311.04293/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 21 references that are in the map.
- First posted on arXiv 7 Nov 2023.

## Builds on (most cited first, 15 of 15)

- [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)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [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)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [ClimaX: A foundation model for weather and climate](https://mlsanity.com/paper/2301.10343/) (2023)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Pangu-Weather: A 3D High-Resolution Model for Fast and Accurate Global Weather Forecast](https://mlsanity.com/paper/2211.02556/) (2022)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Self-Supervised Learning with Lie Symmetries for Partial Differential Equations](https://mlsanity.com/paper/2307.05432/) (2023)

## Built on by (most cited first, 24 of 24)

- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [Symmetry-Informed Governing Equation Discovery](https://mlsanity.com/paper/2405.16756/) (2024)
- [The Need for Verification in AI-Driven Scientific Discovery](https://mlsanity.com/paper/2509.01398/) (2025)
- [Discovering Symbolic Differential Equations with Symmetry Invariants](https://mlsanity.com/paper/2505.12083/) (2025)
- [Explicit Discovery of Nonlinear Symmetries from Dynamic Data](https://mlsanity.com/paper/2510.01855/) (2025)
- [Latent Mixture of Symmetries for Sample-Efficient Dynamic Learning](https://mlsanity.com/paper/2510.03578/) (2025)
- [Governing Equation Discovery from Data Based on Differential Invariants](https://mlsanity.com/paper/2505.18798/) (2025)
- [Scaling Laws and Symmetry, Evidence from Neural Force Fields](https://mlsanity.com/paper/2510.09768/) (2025)
- [Equivariant Manifold Neural ODEs and Differential Invariants](https://mlsanity.com/paper/2401.14131/) (2024)
- [Feature Mapping in Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2402.06955/) (2024)
- [Unsupervised Representation Learning from Sparse Transformation Analysis](https://mlsanity.com/paper/2410.05564/) (2024)
- [Generalized Lie Symmetries in Physics-Informed Neural Operators](https://mlsanity.com/paper/2502.00373/) (2025)
- [Inverting Data Transformations via Diffusion Sampling](https://mlsanity.com/paper/2602.08267/) (2026)
- [The Use of AI-Robotic Systems for Scientific Discovery](https://mlsanity.com/paper/2406.17835/) (2024)
- [Understanding Generalization in Physics Informed Models through Affine Variety Dimensions](https://mlsanity.com/paper/2501.18879/) (2025)
- [SEAL - A Symmetry EncourAging Loss for High Energy Physics](https://mlsanity.com/paper/2511.01982/) (2025)
- [AdamFLIP: Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN Training](https://mlsanity.com/paper/2605.08408/) (2026)
- [The RAIL Principles for Neurosymbolic AI: Reasoning, Assurances, Interfacing and Learning](https://mlsanity.com/paper/2608.04285/) (2026)
- [Conformal mapping based Physics-informed neural networks for designing neutral inclusions](https://mlsanity.com/paper/2501.07809/) (2025)
- [LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries](https://mlsanity.com/paper/2510.25731/) (2025)
- [Drawback of Enforcing Equivariance and its Compensation via the Lens of Expressive Power](https://mlsanity.com/paper/2512.09673/) (2025)
- [From Data to Laws: Neural Discovery of Conservation Laws Without False Positives](https://mlsanity.com/paper/2603.20474/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (2026)
- [The limits of exactness: On the failure of automatic differentiation in physics-informed machine learning](https://mlsanity.com/paper/2609.33078/) (2026)

## Abstract (arXiv)

Symmetries have been leveraged to improve the generalization of neural networks through different mechanisms from data augmentation to equivariant architectures. However, despite their potential, their integration into neural solvers for partial differential equations (PDEs) remains largely unexplored. We explore the integration of PDE symmetries, known as Lie point symmetries, in a major family of neural solvers known as physics-informed neural networks (PINNs). We propose a loss function that informs the network about Lie point symmetries in the same way that PINN models try to enforce the underlying PDE through a loss function. Intuitively, our symmetry loss ensures that the infinitesimal generators of the Lie group conserve the PDE solutions. Effectively, this means that once the network learns a solution, it also learns the neighbouring solutions generated by Lie point symmetries. Empirical evaluations indicate that the inductive bias introduced by the Lie point symmetries of the PDEs greatly boosts the sample efficiency of PINNs.