Deep Symbolic Regression for Physics Guided by Units Constraints: Toward the Automated Discovery of Physical Laws

Abstract

Symbolic regression (SR) is the study of algorithms that automate the search for analytic expressions that fit data. While recent advances in deep learning have generated renewed interest in such approaches, the development of SR methods has not been focused on physics, where we have important additional constraints due to the units associated with our data. Here we present Φ-SO, a physical symbolic optimization framework for recovering analytical symbolic expressions from physics data using deep reinforcement learning techniques by learning units constraints. Our system is built, from the ground up, to propose solutions where the physical units are consistent by construction. This is useful not only in eliminating physically impossible solutions but also because the grammatical rules of dimensional analysis enormously restrict the freedom of the equation generator, thus vastly improving performance. The algorithm can be used to fit noiseless data, which can be useful, for instance, when attempting to derive an analytical property of a physical model, and it can also be used to obtain analytical approximations of noisy data. We test our machinery on a standard benchmark of equations from the Feynman Lectures on Physics and other physics textbooks, achieving state-of-the-art performance in the presence of noise (exceeding 0.1%) and show that it is robust even in the presence of substantial (10%) noise. We showcase its abilities on a panel of examples from astrophysics.

From inflation to dark matter halo profiles: the impact of primordial non-Gaussianities on the central density cusp

“…In the future, we will need to extend our analysis to a large suite of hydrodynamical simulations at various redshifts, as well as consider larger boxes with scale-dependent PNG and/or with the full PDF. We will explore whether Machine Learning methods lead to the same result concerning the non-universality of NFW in the presence of PNG [99][100][101][102]. Since we have mostly concentrated here on the inner structure of halos, it will also be interesting to consider the effect on the outer slope, splashback radius, and large-scale triaxility of the halos.…”

Analysis of Strong Coupling Constant with Machine Learning and Its Application

“…Furthermore, researchers at the University of Strasbourg have combined a recurrent neural network (RNN) with a symbolic regression algorithm to create a comprehensive algorithmic framework called Φ-SO. [61] This framework primarily utilizes recurrent neural networks and reinforcement learning strategies to identify and analyze optimal physical expressions of symbolic data and unit rules. Undoubtedly, these integrated algorithm frameworks offer tremendous potential for the study of physical problems using machine learning approach.…”

“…In this work, we present an application of the Φ-SO algorithmic framework [61] to perform machine learning research on the intricate functional relationship that governs the strong coupling constant over a global energy scale. This study represents the first implementation of the algorithmic framework in this context, and we believe it serves as a significant advance and exemplary demonstration of the potential of machine learning to address concrete physics problems, particularly in the field of particle physics.…”

“…To achieve this purpose, we employ the Φ-SO framework recently developed by a team at Strasbourg University. [61] Φ-SO framework can be used to search for analytic properties of noiseless data and to obtain analytical approximations of noisy data. As shown in Fig.…”

“…[55,56] In recent years, with the continuous advancement of various algorithms, particularly artificial neural networks and regression algorithms, which are fundamental in the field of artificial intelligence, there has been rapid progress in the development of integrated algorithms. Notably, AI Poincaré, [57,58] AI Feynman, [59,60] and Φ-SO [61] are noteworthy examples. These algorithms, based on popular frameworks such as TensorFlow or PyTorch, [62] offer significant advantages in applying machine learning techniques to tackle physical problems.…”