Discover physical concepts and equations with machine learning

TL;DR

This work tackles the challenge of discovering physical concepts and their governing equations directly from data. It extends SciNet by coupling a beta-VAE for disentangled latent representations with Neural ODEs to model continuous-time dynamics, enabling simultaneous concept and law discovery. Four history-inspired case studies (Copernicus, Newton, Schrödinger, Pauli) show that latent factors align with known physical ideas and that learned equations reproduce correct dynamics as combinations of derivatives. The results suggest AI can infer core physical theories from data with minimal prior knowledge, highlighting a path toward autonomous, data-driven theory development.

Abstract

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, Schrödinger's wave mechanics, and Pauli's spin-magnetic formulation. The results demonstrate that the correct physical theories can emerge in the neural network.

Figures (48)

• Figure 1: Neural network architecture. Our model consists of two components: the VAE and the Neural ODE. The observed data $x(t)$ are processed by the encoder $\Psi_\phi$, which outputs the distribution parameters $\mu$ and $\sigma$. Then, an initial latent state $h(t_0)$ is sampled using the reparameterization trick as $h(t_0) = \mu + \sigma\cdot \varepsilon$, where $\varepsilon$ is an auxiliary parameter. The latent state $h(t_0)$ is used as the initial condition of the Neural ODE. The Neural ODE is characterized by the governing function $f$, which is a neural network with parameter $\zeta$ that depends on the state $h(t)$ and an external control variable $V$. Using a numerical ODE solver, the Neural ODE outputs a series of latent states at different moments $h(t_i)$. Using these latent states the decoder $\Phi_\theta$ reconstructs the corresponding observed data $\hat{x}(t_i)$.
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