Symbolic regression (SR) is a machine learning-based regression method based on genetic programming principles that integrates techniques and processes from heterogeneous scientific fields and is capable of providing analytical equations purely from data. This remarkable characteristic diminishes the need to incorporate prior knowledge about the investigated system. SR can spot profound and elucidate ambiguous relations that can be generalizable, applicable, explainable and span over most scientific, technological, economical, and social principles. In this review, current state of the art is documented, technical and physical characteristics of SR are presented, the available programming techniques are investigated, fields of application are explored, and future perspectives are discussed.

The viscosity and thermal conductivity coefficients of the Lennard-Jones fluid are extracted through symbolic regression (SR) techniques from data derived from simulations at the atomic scale. This data-oriented approach provides closed form relations that achieve fine accuracy when compared to well-established theoretical, empirical, or approximate equations, fully transparent, with small complexity and high interpretability. The novelty is further outlined by suggesting analytical expressions for estimating fluid transport properties across the whole phase space, from a dilute gas to a dense liquid, by considering only two macroscopic properties (density and temperature). In such expressions, the underlying physical mechanisms are reflected, while, at the same time, it can be a computationally efficient alternative to costly in time and size first principle and/or molecular dynamics simulations.

In this paper, we propose an alternative road to calculate the transport coefficients of fluids and the slip length inside nano-conduits in a Poiseuille-like geometry. These are all computationally demanding properties that depend on dynamic, thermal, and geometrical characteristics of the implied fluid and the wall material. By introducing the genetic programming-based method of symbolic regression, we are able to derive interpretable data-based mathematical expressions based on previous molecular dynamics simulation data. Emphasis is placed on the physical interpretability of the symbolic expressions. The outcome is a set of mathematical equations, with reduced complexity and increased accuracy, that adhere to existing domain knowledge and can be exploited in fluid property interpolation and extrapolation, bypassing timely simulations when possible.

Machine Learning methods are exploited to extract a universal approach for self-diffusion coefficient calculation in molecular fluids. Analytical expressions are derived through symbolic regression for fluids both in bulk and confined nanochannels. The symbolic regression framework is trained on simulation data from molecular dynamics and correlates the values of the self-diffusion coefficients with macroscopic properties, such as density, temperature, and the width of confinement. New expressions are derived for nine different molecular fluids, while an all-fluid universal equation is extracted to capture molecular behavior as well. In such a way, a highly computationally demanding property is predicted by easy-to-define macroscopic parameters, bypassing traditional numerical methods based on mean squared displacement and autocorrelation functions at the atomistic level. To achieve generalizability and interpretability, simple symbolic expressions are selected from a pool of genetic programming-derived equations. The obtained expressions present physical consistency, and they are discussed in terms of explainability. The accurate prediction of the self-diffusion coefficient both in bulk and confined systems is important for advancing the fundamental understanding of fluid behavior and leading the design of nanoscale confinement devices containing real molecular fluids.

A machine learning approach, namely symbolic regression (SR), is applied in the stellarator Wendelstein 7-X (W7-X), to investigate the effect of six plasma parameters (line integrated electron density, heating power, toroidal plasma current, fraction of radiated power, core and edge ion temperatures) on the sub-divertor neutral gas pressure. Based on the data from the OP1.2b experimental campaign, closed-form expressions of the neutral gas pressure in terms of the plasma parameters are deduced for the standard, high iota and high mirror magnetic configurations at three different ports of the exhaust system. While common regression schemes assume a predetermined functional form, SR autonomously discovers, via genetic programming, the functional structure of the model, purely from data. In all cases, the optimized data driven SR framework clearly points out that, in estimating the neutral gas pressure, the most dominant parameters are the electron density and the heating power, while the remaining plasma parameters have minor impact, at least from the statistical point of view and may not be included in the correlations. Balancing model generality, complexity(COMP) and accuracy for all considered magnetic configurations and ports, the proposed closed form expressions contain only the product of electron density and heating power raised at some powers, times a constant. The proposed two-parameter symbolic expressions, exhibiting low COMP and excellent accuracy metrics, provide a practical and analytical tool for the acceleration of the neutral gas pressure calculations, that are otherwise computationally very expensive and for the overall performance assessment of the W7-X exhaust system. They may also contribute to more efficient experimental design and operation. performance assessment of the W7-X exhaust system. They may also contribute to more efficient experimental design and operation.