A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge

Abstract

Discovering a meaningful symbolic expression that explains experimental data is a fundamental challenge in many scientific fields. We present a novel, open-source computational framework called Scientist-Machine Equation Detector (SciMED), which integrates scientific discipline wisdom in a scientist-in-the-loop approach, with state-of-the-art symbolic regression (SR) methods. SciMED combines a wrapper selection method, that is based on a genetic algorithm, with automatic machine learning and two levels of SR methods. We test SciMED on five configurations of a settling sphere, with and without aerodynamic non-linear drag force, and with excessive noise in the measurements. We show that SciMED is sufficiently robust to discover the correct physically meaningful symbolic expressions from the data, and demonstrate how the integration of domain knowledge enhances its performance. Our results indicate better performance on these tasks than the state-of-the-art SR software packages , even in cases where no knowledge is integrated. Moreover, we demonstrate how SciMED can alert the user about possible missing features, unlike the majority of current SR systems.

Deep symbolic regression for physics guided by units constraints: toward the automated discovery of physical laws

“…It could be argued that we could have tackled the physical units validity of expressions in SR by using the Buckingham Π theorem (Buckingham 1914), with vari-ables and constants rendered dimensionless by means of multiplicative operations amongst them (such an approach has recently been proposed by Matchev et al 2022;Keren et al 2023). However, working on so called Π groups can make SR much more difficult in practice as it makes the dimensionless solutions often more complex.…”

“…SR has traditionally been tackled using genetic algorithms where a population of candidate mathematical expressions are iteratively improved through operations inspired by natural evolution such as natural selection, crossover, and mutation. This type of approach includes the well known Eureqa software (Schmidt & Lipson 2009) (see Graham et al 2013 for a benchmark of Eureqa's capabilities on astrophysical test cases), as well as more recent works (Cranmer 2020;Virgolin & Bosman 2022;Stephens 2015;Kommenda et al 2020;Keren et al 2023). In addition, SR has been implemented using various methods ranging from brute force to (un-)guided Monte-Carlo, all the way to probabilistic searches (Mc-Conaghy 2011;Kammerer et al 2020;Bartlett et al 2022;Brence et al 2021;Jin et al 2019), as well as through problem simplification algorithms (Luo et al 2017;Tohme et al 2022).…”

“…Given the great successes of deep learning techniques in many other fields, it is not surprising that they have now been applied to symbolic regression, and now challenge the reign of Eureqa-type approaches (La Cava et al 2021;Matsubara et al 2022). Multiple methods for incorporating neural networks into SR have been developed, ranging from powerful problem simplification schemes (Udrescu & Tegmark 2020;Cranmer et al 2020;Keren et al 2023), to end-to-end symbolic regression methods where a neural network is trained in a supervised manner to map the relationship between datasets and their corresponding symbolic func-tions (Kamienny et al 2022;Vastl et al 2022;d'Ascoli et al 2022;Becker et al 2022;Biggio et al 2020;Alnuqaydan et al 2022;Aréchiga et al 2021), all the way to incorporating symbols into neural networks in place of activation functions to enable interpretability or to recover a mathematical expression (Martius & Lampert 2016;Zheng et al 2022;Sahoo et al 2018;Valle & Haddadin 2021;Kim et al 2020;Panju & Ghodsi 2020). See La Cava et al (2021) & Makke & Chawla (2022 for recent reviews of symbolic regression algorithms.…”

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

“…In the present study, we develop a foundational symbolic embedding for physics that enables the entire expression tree graph to be tackled, as well as local units constraints. Unlike previous attempts to consider units in which data sets were rendered dimensionless before applying standard SR techniques (Udrescu & Tegmark 2020;Matchev et al 2022;Keren et al 2023), our approach allows us to anticipate the required units for the subsequent symbol to be generated in a partially composed mathematical expression. By adopting this approach, we not only focus on training a neural network to generate increasingly precise expressions, as in Petersen et al (2021a), but we also generate labels of the necessary units and actively train our neural network to adhere to such constraints.…”

“…Indeed, it could be argued that we could have tackled physical units validity of expressions in SR by taking advantage of the Buckingham Π theorem (Buckingham 1914), with variables and constants rendered dimensionless by means of multiplicative operations among them. Such an approach can actually be adopted as a preliminary step in conjunction with any SR framework (see, e.g., Matchev et al 2022;Keren et al 2023). However, although working with so-called Π groups ensures the generation of physically valid expressions (since all terms become dimensionless), it simultaneously removes constraints imposed by dimensional analysis, complicating the SR process.…”