# Physical Symmetries Embedded in Neural Networks

arXiv: https://arxiv.org/abs/1904.08991 · HTML: https://mlsanity.com/paper/1904.08991/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Apr 2019.

## Built on by (most cited first, 30 of 40)

- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
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- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Physics-Informed Neural Networks for Quantum Eigenvalue Problems](https://mlsanity.com/paper/2203.00451/) (2022)
- [Exact conservation laws for neural network integrators of dynamical systems](https://mlsanity.com/paper/2209.11661/) (2022)
- [Transfer Learning with Physics-Informed Neural Networks for Efficient Simulation of Branched Flows](https://mlsanity.com/paper/2211.00214/) (2022)
- [Physics-Driven Learning for Inverse Problems in Quantum Chromodynamics](https://mlsanity.com/paper/2501.05580/) (2025)
- [A simple equivariant machine learning method for dynamics based on scalars](https://mlsanity.com/paper/2110.03761/) (2021)
- [Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equations](https://mlsanity.com/paper/2212.04100/) (2022)
- [Characteristics-Informed Neural Networks for Forward and Inverse Hyperbolic Problems](https://mlsanity.com/paper/2212.14012/) (2022)
- [Combining complex Langevin dynamics with score-based and energy-based diffusion models](https://mlsanity.com/paper/2510.01328/) (2025)
- [DEQGAN: Learning the Loss Function for PINNs with Generative Adversarial Networks](https://mlsanity.com/paper/2209.07081/) (2022)
- [Error-Aware B-PINNs: Improving Uncertainty Quantification in Bayesian Physics-Informed Neural Networks](https://mlsanity.com/paper/2212.06965/) (2022)
- [Fine-Tune Language Models as Multi-Modal Differential Equation Solvers](https://mlsanity.com/paper/2308.05061/) (2023)
- [Enforcing governing equation constraints in neural PDE solvers via training-free projections](https://mlsanity.com/paper/2511.17258/) (2025)
- [Why self-attention is Natural for Sequence-to-Sequence Problems? A Perspective from Symmetries](https://mlsanity.com/paper/2210.06741/) (2022)
- [One-Shot Transfer Learning for Nonlinear ODEs](https://mlsanity.com/paper/2311.14931/) (2023)
- [Physics and geometry informed neural operator network with application to acoustic scattering](https://mlsanity.com/paper/2406.03407/) (2024)
- [Disentangled Representation Learning for Parametric Partial Differential Equations](https://mlsanity.com/paper/2410.02136/) (2024)
- [Joint Parameter Discovery and Generative Modeling of Dynamic Systems](https://mlsanity.com/paper/2103.10905/) (2021)
- [Beyond permutation equivariance in graph networks](https://mlsanity.com/paper/2103.14066/) (2021)
- [A connection between probability, physics and neural networks](https://mlsanity.com/paper/2209.12737/) (2022)
- [On the Compatibility between Neural Networks and Partial Differential Equations for Physics-informed Learning](https://mlsanity.com/paper/2212.00270/) (2022)
- [Physical Constraint Embedded Neural Networks for inference and noise regulation](https://mlsanity.com/paper/2105.09146/) (2021)
- [Symmetry-driven graph neural networks](https://mlsanity.com/paper/2105.14058/) (2021)
- [Encoding Involutory Invariances in Neural Networks](https://mlsanity.com/paper/2106.12891/) (2021)
- [Symplectically Integrated Symbolic Regression of Hamiltonian Dynamical Systems](https://mlsanity.com/paper/2209.01521/) (2022)
- [Pairwise Comparisons Are All You Need](https://mlsanity.com/paper/2403.09746/) (2024)
- [Physics-embedded Fourier Neural Network for Partial Differential Equations](https://mlsanity.com/paper/2407.11158/) (2024)
- [Design of Restricted Normalizing Flow towards Arbitrary Stochastic Policy with Computational Efficiency](https://mlsanity.com/paper/2412.12894/) (2024)

## Abstract (arXiv)

Neural networks are a central technique in machine learning. Recent years have seen a wave of interest in applying neural networks to physical systems for which the governing dynamics are known and expressed through differential equations. Two fundamental challenges facing the development of neural networks in physics applications is their lack of interpretability and their physics-agnostic design. The focus of the present work is to embed physical constraints into the structure of the neural network to address the second fundamental challenge. By constraining tunable parameters (such as weights and biases) and adding special layers to the network, the desired constraints are guaranteed to be satisfied without the need for explicit regularization terms. This is demonstrated on upervised and unsupervised networks for two basic symmetries: even/odd symmetry of a function and energy conservation. In the supervised case, the network with embedded constraints is shown to perform well on regression problems while simultaneously obeying the desired constraints whereas a traditional network fits the data but violates the underlying constraints. Finally, a new unsupervised neural network is proposed that guarantees energy conservation through an embedded symplectic structure. The symplectic neural network is used to solve a system of energy-conserving differential equations and out-performs an unsupervised, non-symplectic neural network.