# Discover physical concepts and equations with machine learning

arXiv: https://arxiv.org/abs/2412.12161 · HTML: https://mlsanity.com/paper/2412.12161/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 74 references that are in the map.
- First posted on arXiv 11 Dec 2024.

## Builds on (most cited first, 26 of 26)

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Learning Differential Equations that are Easy to Solve](https://mlsanity.com/paper/2007.04504/) (2020)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [On scientific understanding with artificial intelligence](https://mlsanity.com/paper/2204.01467/) (2022)
- [STEER: Simple Temporal Regularization For Neural ODEs](https://mlsanity.com/paper/2006.10711/) (2020)
- [Heavy Ball Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.04840/) (2021)
- ["Hey, that's not an ODE": Faster ODE Adjoints via Seminorms](https://mlsanity.com/paper/2009.09457/) (2020)
- [Noether Networks: Meta-Learning Useful Conserved Quantities](https://mlsanity.com/paper/2112.03321/) (2021)
- [A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning](https://mlsanity.com/paper/2311.00212/) (2023)
- [AI Poincar'e 2.0: Machine Learning Conservation Laws from Differential Equations](https://mlsanity.com/paper/2203.12610/) (2022)
- [Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert](https://mlsanity.com/paper/2308.09474/) (2023)
- [Latent Neural ODEs with Sparse Bayesian Multiple Shooting](https://mlsanity.com/paper/2210.03466/) (2022)
- [Operationally meaningful representations of physical systems in neural networks](https://mlsanity.com/paper/2001.00593/) (2020)
- [Forecasting the outcome of spintronic experiments with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2108.02318/) (2021)
- [Explainable Representation Learning of Small Quantum States](https://mlsanity.com/paper/2306.05694/) (2023)
- [Metalearning generalizable dynamics from trajectories](https://mlsanity.com/paper/2301.00957/) (2023)
- [Learning minimal representations of stochastic processes with variational autoencoders](https://mlsanity.com/paper/2307.11608/) (2023)
- [Neural modal ordinary differential equations: Integrating physics-based modeling with neural ordinary differential equations for modeling high-dimensional monitored structures](https://mlsanity.com/paper/2207.07883/) (2022)
- [Application of Neural Ordinary Differential Equations for Tokamak Plasma Dynamics Analysis](https://mlsanity.com/paper/2403.01635/) (2024)
- [Optical Neural Ordinary Differential Equations](https://mlsanity.com/paper/2209.12898/) (2022)

## Built on by (most cited first, 1 of 1)

- [AI-Newton: A Concept-Driven Physical Law Discovery System without Prior Physical Knowledge](https://mlsanity.com/paper/2504.01538/) (2025)

## Abstract (arXiv)

Machine learning can uncover physical concepts or physical equations when prior knowledge from the other is available. However, these two aspects are often intertwined and cannot be discovered independently. We extend SciNet, which is a neural network architecture that simulates the human physical reasoning process for physics discovery, by proposing a model that combines Variational Autoencoders (VAE) with Neural Ordinary Differential Equations (Neural ODEs). This allows us to simultaneously discover physical concepts and governing equations from simulated experimental data across various physical systems. We apply the model to several examples inspired by the history of physics, including Copernicus' heliocentrism, Newton's law of gravity, Schr"odinger's wave mechanics, and Pauli's spin-magnetic formulation. The results demonstrate that the correct physical theories can emerge in the neural network.