Symbolic Regression Algorithm Research Articles

Plastic Limit Pressure and Stress Intensity Factor for Cracked Elbow Containing Axial Semi-Elliptical Part-Through Crack

The aim of this study is to provide a solution for the plastic limit pressure and stress intensity factor of the elbows containing a part-through axial semi-elliptical crack by considering various crack sizes. The supporting system and loading conditions of the pipeline are described. The critical part of the observed pipeline was isolated for analysis and subjected to various sizes of semi-elliptical cracks. By performing numerical analysis, results were obtained for crack dimension ratios of c/a, and depth/thickness ratios of a/t. The obtained results include plastic limit pressure and stress intensity factor. The results were analyzed with a symbolic regression algorithm, and closed-form solutions for the limit pressure and stress intensity factor were proposed. To validate pipeline integrity, the Structural Integrity Assessment Procedure (SINTAP) was applied, and the FAD (Failure Assessment Diagram) was generated for cracks below the FAD function. The failure pressure was calculated by determining the points where the loading paths intersect the FAD function.

Modeling plasticity-mediated void growth at the single crystal scale: A physics-informed machine learning approach

Modeling the evolution of voids during plastic flow as well as their effects on plastic dissipation is critical for both component manufacturing and lifetime estimation purposes. To this end, we propose a rate-dependent constitutive model to homogenize the effects of semi-randomly distributed voids on single crystal plasticity whilst capturing void interaction and plastic anisotropy. The present work focuses on the case of face centered cubic crystals to introduce an anisotropic gauge function applicable within the crystal plasticity formalism. The approach combines analytical methods to describe the micromechanics of the system in combination with symbolic regression to capture analytically intractable mechanisms from data. The hybrid framework uses a physics-informed genetic programming-based symbolic regression algorithm to solve a multiform optimization problem simultaneously producing a new gauge function and a new strain rate equation. This is also a multi-objective optimization problem with many competing objectives. A new search and selection step is introduced to the genetic algorithm that promotes convergence toward a global solution that better satisfies all the objectives. Overall, the symbolic equations produced leverage data-driven methods to achieve greater accuracy than comparable alternatives on an analytically intractable problem while maintaining model transparency.

Automated Data-Driven Discovery of Material Models Based on Symbolic Regression: A Case Study on the Human Brain Cortex

We introduce a data-driven framework to automatically identify interpretable and physically meaningful hyperelastic constitutive models from sparse data. Leveraging symbolic regression, our approach generates elegant hyperelastic models that achieve accurate data fitting with parsimonious mathematic formulas, while strictly adhering to hyperelasticity constraints such as polyconvexity/ellipticity. Our investigation spans three distinct hyperelastic models—invariant-based, principal stretch-based, and normal strain-based—and highlights the versatility of symbolic regression. We validate our new approach using synthetic data from five classic hyperelastic models and experimental data from the human brain cortex to demonstrate algorithmic efficacy. Our results suggest that our symbolic regression algorithms robustly discover accurate models with succinct mathematic expressions in invariant-based, stretch-based, and strain-based scenarios. Strikingly, the strain-based model exhibits superior accuracy, while both stretch-based and strain-based models effectively capture the nonlinearity and tension-compression asymmetry inherent to the human brain tissue. Polyconvexity/ellipticity assessment affirm the rigorous adherence to convexity requirements both within and beyond the training regime. However, the stretch-based models raise concerns regarding potential convexity loss under large deformations. The evaluation of predictive capabilities demonstrates remarkable interpolation capabilities for all three models and acceptable extrapolation performance for stretch-based and strain-based models. Finally, robustness tests on noise-embedded data underscore the reliability of our symbolic regression algorithms. Our study confirms the applicability and accuracy of symbolic regression in the automated discovery of isotropic hyperelastic models for the human brain and gives rise to a wide variety of applications in other soft matter systems. Statement of SignificanceOur research introduces a pioneering data-driven framework that revolutionizes the automated identification of hyperelastic constitutive models, particularly in the context of soft matter systems such as the human brain. By harnessing the power of symbolic regression, we have unlocked the ability to distill intricate physical phenomena into elegant and interpretable mathematical expressions. Our approach not only ensures accurate fitting to sparse data but also upholds crucial hyperelasticity constraints, including polyconvexity, essential for maintaining physical relevance.

Axial compressive capacity prediction and optimal design of circular UHPC-filled steel tube based on Hybrid Symbolic Regression - Neural Network model

This study presents two methodologies for estimating the ultimate bearing capacity of ultra-high-performance concrete-filled circular steel tube (UHPCFST) columns and offers design optimization guidance: one employing a Symbolic Regression algorithm and the other a Hybrid Symbolic Regression - Neural Network (SR-NN) model. The models were trained, tested, and validated using experimental data from 498 samples sourced from existing literature. The optimal model, SR-NN_14_6, was identified from a pool of 480 models using a trial-and-error approach. The formula and model exhibited enhanced stability and precision, as indicated by their mean absolute error of 8.601 % and 8.051 %, respectively, marking an improvement exceeding 30 % relative to the best existing AIJ code. Using the validated model, a parametric analysis was conducted to assess the influence of the confinement effect coefficient (ξ) and steel ratio (α) on the strength index (SI) of UHPCFST members, leading to the identification of two optimal parameter ranges. The optimal ranges of ξ and α were determined to be 0.84 to 1.29 % and 20 % to 30 %, respectively, offering valuable guidelines for the design of UHPCFST members.

A generalized grey model with symbolic regression algorithm and its application in predicting aircraft remaining useful life

As a sparse data analysis method, a grey model faces challenges in interpretability for its effective application in uncertain systems. This study proposes a generalized grey model (GGM) based on symbolic regression, designed to improve the intelligence and adaptability of grey models. The GGM serves as a unified framework, integrating various grey model families and addresses regression challenges to determine the model structure. Symbolic regression in the GGM identifies symbolic input-output relationships, offering an interpretable approach for structure determination. By leveraging the non-uniqueness principle in grey system theory and employing structural penalty parameters, the model balances complexity and interpretability. A comparative analysis between GGM and conventional grey function models is conducted focusing on the differences in modeling, structure identification, and parameter optimization. Validation on the M3 competition dataset demonstrated the GGM's superior performance, achieving a significant reduction in prediction error compared to other grey forecasting models. Additionally, a rigorous analysis of aircraft lifespan data underscored the robustness and accuracy of GGM in practical engineering applications.

Automated discovery of symbolic laws governing skill acquisition from naturally occurring data.

Skill acquisition is a key area of research in cognitive psychology as it encompasses multiple psychological processes. The laws discovered under experimental paradigms are controversial and lack generalizability. This paper aims to unearth the laws of skill learning from large-scale training log data. A two-stage algorithm was developed to tackle the issues of unobservable cognitive states and an algorithmic explosion in searching. A deep learning model is initially employed to determine the learner's cognitive state and assess the feature importance. Symbolic regression algorithms are then used to parse the neural network model into algebraic equations. Experimental results show that the algorithm can accurately restore preset laws within a noise range in continuous feedback settings. When applied to Lumosity training data, the method outperforms traditional and recent models in fitness terms. The study reveals two new forms of skill acquisition laws and reaffirms some previous findings.

Equivalent method for assessment of bearings at operational types of loadings

To investigate the effect of various factors on bearing stress response, Huber-Hencky-von Mises stress serves as a bridge, the equivalent interrelation between radial loading, axial loading, and temperature of bearing is studied using finite element method (FEM). Symbolic regression (SR) algorithm is employed to analyze simulation results, establishing a functional expression between independent and dependent variables by optimizing combinations of variables, constants, and functional forms. The results showed that within the specified force and temperature values, the curved surface of the equivalent correlation function, trained using the SR algorithm, demonstrates smoothness. Both training and validation data exhibit a strong correlation with this curved surface. Among the three factors, temperature exerts the greatest influence on bearing stress values, followed by radial loading, and axial loading components had the smallest impact.

An Interpretable Approach to the Solutions of High-Dimensional Partial Differential Equations

In recent years, machine learning algorithms, especially deep learning, have shown promising prospects in solving Partial Differential Equations (PDEs). However, as the dimension increases, the relationship and interaction between variables become more complex, and existing methods are difficult to provide fast and interpretable solutions for high-dimensional PDEs. To address this issue, we propose a genetic programming symbolic regression algorithm based on transfer learning and automatic differentiation to solve PDEs. This method uses genetic programming to search for a mathematically understandable expression and combines automatic differentiation to determine whether the search result satisfies the PDE and boundary conditions to be solved. To overcome the problem of slow solution speed caused by large search space, we propose a transfer learning mechanism that transfers the structure of one-dimensional PDE analytical solution to the form of high-dimensional PDE solution. We tested three representative types of PDEs, and the results showed that our proposed method can obtain reliable and human-understandable real solutions or algebraic equivalent solutions of PDEs, and the convergence speed is better than the compared methods. Code of this project is at https://github.com/grassdeerdeer/HD-TLGP.

Heterojunction of MXenes and MN4–graphene: Machine learning to accelerate the design of bifunctional oxygen electrocatalysts

Oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) are essential for the development of excellent bifunctional electrocatalysts, which are key functions in clean energy production. The emphasis of this study lies in the rapid design and investigation of 153 MN4–graphene (Gra)/ MXene (M2NO) electrocatalysts for ORR/OER catalytic activity using machine learning (ML) and density functional theory (DFT). The DFT results indicated that CoN4–Gra/Ti2NO had both good ORR (0.37 V) and OER (0.30 V) overpotentials, while TiN4–Gra/M2NO and MN4–Gra/Cr2NO had high overpotentials. Our research further indicated orbital spin polarization and d-band centers far from the Fermi energy level, affecting the adsorption energy of oxygen-containing intermediates and thus reducing the catalytic activity. The ML results showed that the gradient boosting regression (GBR) model successfully predicted the overpotentials of the monofunctional catalysts RhN4–Gra/Ti2NO (ORR, 0.39 V) and RuN4–Gra/W2NO (OER, 0.45 V) as well as the overpotentials of the bifunctional catalyst RuN4–Gra/W2NO (ORR, 0.39 V; OER, 0.45 V). The symbolic regression (SR) algorithm was used to construct the overpotential descriptors without environmental variable features to accelerate the catalyst screening and shorten the trial-and-error costs from the source, providing a reliable theoretical basis for the experimental synthesis of MXene heterostructures.

Comparative study of models predicting punching shear capacity of strengthened corroded RC slab-column joints

ABSTRACT Degradation in reinforced concrete (RC) members associated with rebar corrosion in old buildings is one major concern that can significantly affect the strength and serviceability of RC structures. Strengthening existing structures using FRP can comprise complex assessment and design processes. This study addresses the critical issue of modeling the behavior associated with steel reinforcement corrosion in RC slab-column joints (SCJs). Three modeling methods, namely artificial neural network (ANN), Eureqa’s symbolic regression algorithm, and analytical modeling, are proposed to predict the punching shear (PS) capacity of corroded RC SCJs by accounting for the changes in material properties due to corrosion and the strengthening provided by different types of FRP. The accuracy of the proposed models is validated through comparisons with experimental results and a verified finite element (FE) model that considers material deterioration and bond degradation. The results demonstrate that the proposed models offer more accurate PS capacity predictions than existing design codes. Among the considered prediction models, the ANN model exhibits superior performance and is recommended for PS capacity prediction. A practical design tool in the form of a graphical user interface (GUI) was created to facilitate the assessment and prediction of the PS behavior of corroded RC SCJs. Engineers can conveniently use this GUI to evaluate and predict the performance of such joints.

Exploring approximate analytical expression for magnetic skyrmion structure based on symbolic regression method

Magnetic skyrmion is a kind of nontrivial topological magnetic structure, which can exist stably in chiral magnet with Dzyaloshinskii-Moriya (DM) interaction, and its static and dynamic properties are closely related to its structural characteristics. However, there are no general analytical expressions for skyrmion profiles. Therefore, many researchers have provided approximate solutions. In this paper, a new approach to exploring magnetic skyrmion structures is introduced by using a symbolic regression approach. Considering the influence of DM interaction and external magnetic field on magnetic skyrmion structure, two suitable approximate expressions are obtained through symbolic regression algorithms. The applicability of these expressions depends on the dominant interaction. The research results in this work validate the powerful capability of symbolic regression algorithms in exploring the magnetic skyrmion profiles. So, the present study provides a new method for finding the analytical expressions for magnetic structure.

Multitree Genetic Programming With Feature-Based Transfer Learning for Symbolic Regression on Incomplete Data.

Data incompleteness is a serious challenge in real-world machine-learning tasks. Nevertheless, it has not received enough attention in symbolic regression (SR). Data missingness exacerbates data shortage, especially in domains with limited available data, which in turn limits the learning ability of SR algorithms. Transfer learning (TL), which aims to transfer knowledge across tasks, is a potential solution to solve this issue by making amends for the lack of knowledge. However, this approach has not been adequately investigated in SR. To fill this gap, a multitree genetic programming-based TL method is proposed in this work to transfer knowledge from complete source domains (SDs) to incomplete related target domains (TDs). The proposed method transforms the features from a complete SD to an incomplete TD. However, having many features complicates the transformation process. To mitigate this problem, we integrate a feature selection mechanism to eliminate unnecessary transformations. The method is examined on real-world and synthetic SR tasks with missing values to consider different learning scenarios. The obtained results not only show the effectiveness of the proposed method but also show its training efficiency compared with the existing TL methods. Compared to state-of-the-art methods, the proposed method reduced an average of more than 2.58% and 4% regression error on heterogeneous and homogeneous domains, respectively.

Explainable machine learning identifies multi-omics signatures of muscle response to spaceflight in mice

The adverse effects of microgravity exposure on mammalian physiology during spaceflight necessitate a deep understanding of the underlying mechanisms to develop effective countermeasures. One such concern is muscle atrophy, which is partly attributed to the dysregulation of calcium levels due to abnormalities in SERCA pump functioning. To identify potential biomarkers for this condition, multi-omics data and physiological data available on the NASA Open Science Data Repository (osdr.nasa.gov) were used, and machine learning methods were employed. Specifically, we used multi-omics (transcriptomic, proteomic, and DNA methylation) data and calcium reuptake data collected from C57BL/6 J mouse soleus and tibialis anterior tissues during several 30+ day-long missions on the international space station. The QLattice symbolic regression algorithm was introduced to generate highly explainable models that predict either experimental conditions or calcium reuptake levels based on multi-omics features. The list of candidate models established by QLattice was used to identify key features contributing to the predictive capability of these models, with Acyp1 and Rps7 proteins found to be the most predictive biomarkers related to the resilience of the tibialis anterior muscle in space. These findings could serve as targets for future interventions aiming to reduce the extent of muscle atrophy during space travel.

Solid-liquid phase transition temperature prediction of alloys based on machine learning key feature screening

A machine learning strategy is proposed based on the demand for prediction of solid-liquid phase transition temperature properties of multi-component precious metal alloys. Firstly, the candidate feature set are constructed by mathematical operation of the physical and chemical parameters of the material according to the chemical ratio of the alloy chemical formula. Then, a novel feature selection framework of correlation screening → genetic algorithm screening → feature weight ranking → exhaustive screening is proposed to effectively identify key feature combinations affecting solid-liquid phase transition temperature. Finally, the support vector regression algorithm is used to establish the "key feature combination-solid phase transition temperature" model with an error less than 9.83% and the "key feature combination-liquid phase transition temperature" model with an error less than 9.35%. The proposed new feature framework overcomes the inability of conventional feature selection techniques to simultaneously meet the requirements of interpretability, low computational complexity, and strong feature generality. Moreover, the comprehensive equilibrium complexity and accuracy (R2) constructed "solid temperature-key feature combination" with an error of less than 6.50% and "liquid phase transition temperature-key feature combination" with an error of less than 6.71 % by symbolic regression algorithm, which strengthened the understanding of the influence of key independent variable feature parameters on solid-liquid phase transition temperature prediction. This established machine learning strategy has excellent reliability in the prediction of solid-liquid phase transition temperature of multi-component alloys, and the proposed feature selection framework is expected to develop into a new feature selection technique.

Using LLMs and Explainable ML to Analyze Biomarkers at Single-Cell Level for Improved Understanding of Diseases.

Single-cell RNA sequencing (scRNA-seq) technology has significantly advanced our understanding of the diversity of cells and how this diversity is implicated in diseases. Yet, translating these findings across various scRNA-seq datasets poses challenges due to technical variability and dataset-specific biases. To overcome this, we present a novel approach that employs both an LLM-based framework and explainable machine learning to facilitate generalization across single-cell datasets and identify gene signatures to capture disease-driven transcriptional changes. Our approach uses scBERT, which harnesses shared transcriptomic features among cell types to establish consistent cell-type annotations across multiple scRNA-seq datasets. Additionally, we employed a symbolic regression algorithm to pinpoint highly relevant, yet minimally redundant models and features for inferring a cell type's disease state based on its transcriptomic profile. We ascertained the versatility of these cell-specific gene signatures across datasets, showcasing their resilience as molecular markers to pinpoint and characterize disease-associated cell types. The validation was carried out using four publicly available scRNA-seq datasets from both healthy individuals and those suffering from ulcerative colitis (UC). This demonstrates our approach's efficacy in bridging disparities specific to different datasets, fostering comparative analyses. Notably, the simplicity and symbolic nature of the retrieved gene signatures facilitate their interpretability, allowing us to elucidate underlying molecular disease mechanisms using these models.

Prediction of urban airflow fields around isolated high-rise buildings using data-driven non-linear correction models

When it comes to predicting urban airflow, steady Reynolds-averaged Navier-Stokes (SRANS) models that rely on Reynolds stress often face a challenge called the closure problem. This problem involves unresolved structural flaws and uncertainties in the closure coefficients used in the models. Previous attempts to recalibrate coefficients for specific urban flows without breaking the linear constitutive relation have resulted in simulation results constrained by the baseline turbulence model. Therefore, this study aims to enhance the performance of SRANS models by addressing these structural flaws. To achieve this, a novel data-driven framework is proposed. It leverages the deterministic symbolic regression algorithm to discover explicit algebraic expressions for a non-linear Reynolds stress correction model. The robustness of the correction model is ensured by maintaining the linear eddy viscosity model for iterative calculations while keeping the non-linear component frozen. The proposed framework is evaluated using three isolated building cases with varying geometric configurations and inflow boundary conditions. Findings demonstrate that computational fluid dynamics (CFD) predictions incorporating the data-driven non-linear correction model consistently align closer to wind tunnel experimental results compared to both standard and non-linear versions of the k-ε turbulence model. This improvement is reflected in reduced reattachment lengths and more accurate mean velocity distributions in the wake of buildings. However, it should be noted that there is a possibility of overpredicting wind velocity in the windward area. This study introduces valuable insights and additional strategies to enhance the prediction accuracy of SRANS models in urban airflow simulations.

Accelerating the Discovery of Hybrid Perovskites with Targeted Band Gaps via Interpretable Machine Learning.

The band gap of hybrid organic-inorganic perovskites (HOIPs) is a key factor affecting the light absorption characteristics and thus the performance of perovskite solar cells (PSCs). However, band gap engineering, using experimental trial and error and high-throughput density functional theory calculations, is blind and costly. Therefore, it is critical to statistically identify the multiple factors influencing band gaps and to rationally design perovskites with targeted band gaps. This study combined feature engineering, the gradient-boosted regression tree (GBRT) algorithm, and the genetic algorithm-based symbolic regression (GASR) algorithm to develop an interpretable machine learning (ML) strategy for predicting the band gap of HOIPs accurately and quantitatively interpreting the factors affecting the band gap. Seven best physical features were selected to construct a GBRT model with a root-mean-square error of less than 0.060 eV, and the most important feature is the electronegativity difference between the B-site and the X-site (χB-X). Further, a mathematical formula (Eg = χB-X2 + 0.881χB-X) was constructed with GASR for a quantitative interpretation of the band gap influence patterns. According to the ML model, the HOIP MA0.23FA0.02Cs0.75Pb0.59Sn0.41Br0.24I2.76 was obtained, with a suitable band gap of 1.39 eV. Our proposed interpretable ML strategy provides an effective approach for developing HOIP structures with targeted band gaps, which can also be applied to other material fields.

How wavelength affects hydrodynamic performance of two accelerating mirror-symmetric undulating hydrofoils

Fish schools are capable of simultaneous linear acceleration. To reveal the underlying hydrodynamic mechanism, we numerically investigate how Reynolds number Re=1000–2000, Strouhal number St=0.2–0.7, and wavelength λ=0.5–2 affect the mean net thrust and net propulsive efficiency of two side-by-side hydrofoils undulating in anti-phase. In total, 550 cases are simulated using immersed boundary method. The thrust increases significantly with the wavelength and the Strouhal number, yet only slightly with the Reynolds number. We apply a symbolic regression algorithm to formulate this relationship. Furthermore, we find that mirror-symmetric schooling can achieve a net thrust more than ten times that of a single swimmer, especially at low Reynolds numbers. The highest efficiency is obtained at St=0.5 and λ=1.2, where St is consistent with that observed in the linear-accelerating natural swimmers, e.g., Crevalle jack. Six distinct flow structures are identified. The highest thrust corresponds to an asymmetric flow pattern, whereas the highest efficiency occurs when the flow is symmetric with converging vortex streets.

Co-inversion of island leaf area index combination morphological and spectral parameters based on UAV multi-source remote sensing data

The precise and rapid estimation of leaf area index (LAI) is crucial for quantitative ecological monitoring and assessment, as it serves as an important ecological indicator of plant growth state and canopy structure. The acquisition, integration, and application of multiple remote sensing data from Unmanned Aerial Vehicle (UAV) platforms in quantitative ecological research is a current research hotspot. This study aims to explore the potential of combining morphological and spectral parameters using the symbolic regression (SR) algorithm with allometric models to improve the estimation of Leaf Area Index (LAI) at a regional scale of island ecosystem using UAV multi-source remote sensing data. The Canopy Height Model (CHM), Vegetation Density (VD), and Vegetation Indices (VIs), derived from UAV multi-platforms, were used as independent variables in the development of the retrieval models. Subsequently, regression equations were used to construct the Singlet, Duplet, and Triplet models. The precise assessments indicated that the triplet model demonstrates the highest accuracy, with a mean absolute error (MAE) of 9.48% and root mean square error (RMSE) of 0.2345. This improvement was observed compared to the singlet model (MAE = 19.55%, RMSE = 0.4091) and duplet model (MAE = 15.88%, RMSE = 0.3450), which can be attributed to incorporating morphological parameters to reduce the impact of tree height and leaf density. The analysis also showed that the estimated LAI accuracy is generally higher for shrubs and herbs compared to trees, and that lower plants with high vegetation coverage are better retrieved. The methodology advaneced in this study demonstrates significant potential and implications for ecological monitoring and assessment of island ecosystems, thereby expanding the research scope in the field of island ecology. Furthermore, it provides scientific evidence and technical support for decision-makers and scholars to propose and implement various measures for the conservation and restoration of island ecosystems.

Cosmic backreaction and the mean redshift drift from symbolic regression

The possibility of obtaining symbolic expressions for cosmic backreaction is explored through a case study of so-called 2-region models. By using the publicly available symbolic regression algorithm AI Feynman, it is shown that the kinematical backreaction from a single 2-region model can be well described as a function of the mean redshift (or, equivalently, the volume averaged scale factor). A single expression depending on the redshift/scale factor as well as a model parameter, $f$, that can accurately describe the backreaction for a significant range of models is naturally more complicated but is also achieved with percent-level accuracy. \newline\indent Data sets of redshift drift in the 2-region models are also considered. Again utilizing AI Feynman, expressions for the redshift drift are found. In particular, an expression for the difference between the mean redshift drift and the drift of the mean redshift in terms of the kinematical backreaction is easily obtained for a single 2-region model. An accurate symbolic expression that describes this difference for an array of 2-region models is achieved by using the redshift as a feature instead of the kinematical backreaction.