Symbolic Algorithm Research Articles

A multi-objective reinforcement learning framework for real-time drilling optimization based on symbolic regression and perception

In the context of the global energy transition, optimizing deep-water oil and gas drilling parameters is crucial for ensuring safety while improving efficiency. Traditional methods face limitations in highly dynamic and nonlinear drilling environments, struggling to balance speed and cost-effectiveness. Furthermore, these methods rely on real-time logging-while-drilling (LWD) data for decision-making, but delays in data collection and processing hinder timely adjustments of drilling parameters, affecting decision accuracy and responsiveness. This paper proposes a multi-objective drilling parameter optimization framework, incorporating symbolic regression, time-series networks, and Markov decision processes to precisely predict ROP, formation conditions, and optimize drilling parameters in real time. Key innovations include a multi-population evolutionary symbolic regression algorithm for constructing empirical equations, the integration of variational mode decomposition (VMD) and sample entropy for data preprocessing, and multi-head self-attention time-series networks to enhance prediction accuracy. Quantile regression further estimates the range of drilling parameter adjustments. Additionally, a drilling parameter optimization deep deterministic policy gradient (DPODDPG) algorithm was developed to automate real-time parameter adjustments. Empirical analysis on the Ledong 10-1 block in the South China Sea demonstrated significant improvements: ROP increased from 54.18 m/hr to 122.17 m/hr, mechanical specific energy (MSE) decreased from 100.82 MPa to 97.78 MPa, and cost per foot reduced from 121.16 × 102 CNY/m to 51.31 × 102 CNY/m. Compared to traditional methods, the proposed framework showed clear advantages in enhancing ROP, reducing MSE, and controlling costs, further validating its superiority in complex drilling environments. This method not only significantly improves drilling efficiency and economic benefits but also adapts to complex and changing drilling conditions, showing broad application potential, particularly in challenging deep-water oil and gas drilling operations, where it can provide more efficient and reliable optimization solutions.

AI-Lorenz: A physics-data-driven framework for Black-Box and Gray-Box identification of chaotic systems with symbolic regression

Discovering mathematical models that characterize the observed behavior of dynamical systems remains a major challenge, especially for systems in a chaotic regime, due to their sensitive dependence on initial conditions and the complex non-linear interactions within the system. The challenge is even greater when the physics underlying such systems is not yet understood, and scientific inquiry must solely rely on empirical data. Despite advancements such as the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, which has shown success in identifying chaotic systems with reasonable accuracy, challenges remain in dealing with noise, sparse data, and the need for models that generalize well across different chaotic systems. Driven by the need to fill this gap, we develop a framework named AI-Lorenz that learns mathematical expressions modeling complex dynamical behaviors by identifying differential equations from noisy and sparse observable data. We train a physics-informed neural network (PINN) with the eXtreme Theory of Functional Connections (X-TFC) algorithm by using data and known physics (when available) to learn the dynamics of a system, its rate of change in time, and missing model terms, which are used as input for a symbolic regression algorithm (PySR) to autonomously distill the explicit mathematical terms. This, in turn, enables us to predict the future evolution of the dynamical behavior. The performance of this framework is validated by recovering the right-hand sides and unknown terms of certain complex, chaotic systems, such as the well-known Lorenz system, a six-dimensional hyperchaotic system, the non-autonomous Sprott chaotic system, and the slow-fast Duffing system, and comparing them with their known analytical expressions and state-of-the-art regression and system identification methods.

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.

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.

Lessons on Datasets and Paradigms in Machine Learning for Symbolic Computation: A Case Study on CAD

Symbolic Computation algorithms and their implementation in computer algebra systems often contain choices which do not affect the correctness of the output but can significantly impact the resources required: such choices can benefit from having them made separately for each problem via a machine learning model. This study reports lessons on such use of machine learning in symbolic computation, in particular on the importance of analysing datasets prior to machine learning and on the different machine learning paradigms that may be utilised. We present results for a particular case study, the selection of variable ordering for cylindrical algebraic decomposition, but expect that the lessons learned are applicable to other decisions in symbolic computation. We utilise an existing dataset of examples derived from applications which was found to be imbalanced with respect to the variable ordering decision. We introduce an augmentation technique for polynomial systems problems that allows us to balance and further augment the dataset, improving the machine learning results by 28% and 38% on average, respectively. We then demonstrate how the existing machine learning methodology used for the problem—classification—might be recast into the regression paradigm. While this does not have a radical change on the performance, it does widen the scope in which the methodology can be applied to make choices.

A Symbolic Algorithm to Obtain Low or High Degree Splines from Discrete Fourier Transforms

Obtaining high-degree splines with the use of traditional spline interpolation methods is not an easy task; therefore, traditional spline interpolation is typically limited to cubic splines. In this paper, we present a symbolic and numeric algorithm to obtain splines of any degree, while providing detailed procedures and examples of how to use this algorithm so that it is immediately useful for an interested user. This method, which was initially developed by Beaudoin and Beauchemin [2, 3], works for splines of any degree and yields very accurate results when the boundary conditions are chosen wisely. It also provides approximations of higher order derivatives, something that is not available with the use of cubic splines. This paper presents formulas that can be used in a straightforward manner to obtain interpolation splines of first degree (linear splines), second degree (parabolic splines) and third degree (cubic splines). For splines of higher degree, a short but complete symbolic algorithm to compute the formulas is presented. The resulting formulas can be used in the same manner as those presented for splines of lower degree. A complete numerical example is included to show how the results are obtained and a link to the complete Maple source code is given.

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.

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.

Pioneering the plethora of soliton for the (3+1)-dimensional fractional heisenberg ferromagnetic spin chain equation

Abstract In this scholarly article, we investigate the complex structured (3+1)-dimensional Fractional Heisenberg Ferromagnetic Spin Chain equation (FHFSCE) with conformable fractional derivatives. We develop a diverse glut of soliton solutions using an improved version of the G ′ G -expansion method, namely the r + G ′ G -expansion method. The constraints for the existence of these solutions are painstakingly explained. Our findings are clearly communicated through a number of 3D and 2D graphical representations displaying periodic, multiple periodic, kink, and shock soliton solutions. These soliton solutions are expressed in several mathematical function forms, such as hyperbolic, trigonometric, and rational functions. Our findings support the suggested method’s efficacy as a powerful symbolic algorithm for discovering innovative soliton solutions within nonlinear evolution systems.

The data-driven discovery of partial differential equations by symbolic genetic algorithm

The data-driven discovery of partial differential equations by symbolic genetic algorithm

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.

Boosting optimal symbolic planning: Operator-potential heuristics

Heuristic search guides the exploration of states via heuristic functions h estimating remaining cost. Symbolic search instead replaces the exploration of individual states with that of state sets, compactly represented using binary decision diagrams (BDDs). In cost-optimal planning, heuristic explicit search performs best overall, but symbolic search performs best in many individual domains, so both approaches together constitute the state of the art. Yet combinations of the two have so far not been an unqualified success, because (i) h must be applicable to sets of states rather than individual ones, and (ii) the different state partitioning induced by h may be detrimental for BDD size. Many competitive heuristic functions in planning do not qualify for (i), and it has been shown that even extremely informed heuristics can deteriorate search performance due to (ii).Here we show how to achieve (i) for a state-of-the-art family of heuristic functions, namely potential heuristics. These assign a fixed potential value to each state-variable/value pair, ensuring by LP constraints that the sum over these values, for any state, yields an admissible and consistent heuristic function. Our key observation is that we can express potential heuristics through fixed potential values for operators instead, capturing the change of heuristic value induced by each operator. These reformulated heuristics satisfy (i) because we can express the heuristic value change as part of the BDD transition relation in symbolic search steps. We run exhaustive experiments on IPC benchmarks, evaluating several different instantiations of potential heuristics in forward, backward, and bi-directional symbolic search. Our operator-potential heuristics turn out to be highly beneficial, in particular they hardly ever suffer from (ii). Our best configurations soundly beat previous optimal symbolic planning algorithms, bringing them on par with the state of the art in optimal heuristic explicit search planning in overall performance.

Analysis of depressive EEG signals via symbolic phase transfer entropy with an adaptive template method

Depression is a prevalent mental disorder in contemporary society. Symbolic phase transfer entropy can quantify the dynamic interaction and information flow between electroencephalogram (EEG) signals in depressed patients and healthy groups, which can help diagnose and treat depression. However, the traditional symbolization process of symbolic phase transfer entropy adopts the basic template method, which makes the symbolic phase transfer entropy unable to express the characteristics and changes of time series in different time periods in detail. Therefore, this paper proposes an improved symbolic phase transfer entropy algorithm, which adopts the adaptive template method in the symbolization process of the symbolic phase transfer entropy algorithm so that it can capture the dynamic changes of time series more finely. It was verified on the task EEG signals of 40 depressed patients and 40 healthy people. The experimental results show that the improved symbolic phase transfer entropy can more accurately distinguish depressed patients from healthy people in lead F4 and lead O1, which is helpful for the study of the EEG pathological characteristics of depression. The improved symbolic phase transfer entropy algorithm makes up for the shortcomings of the traditional symbolic phase transfer entropy in capturing the dynamic changes of time series and provides help for the study of dynamic changes in complex systems.

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.

Outcome Separation Logic: Local Reasoning for Correctness and Incorrectness with Computational Effects

Separation logic’s compositionality and local reasoning properties have led to significant advances in scalable static analysis. But program analysis has new challenges—many programs display computational effects and, orthogonally, static analyzers must handle incorrectness too. We present Outcome Separation Logic (OSL), a program logic that is sound for both correctness and incorrectness reasoning in programs with varying effects. OSL has a frame rule—just like separation logic—but uses different underlying assumptions that open up local reasoning to a larger class of properties than can be handled by any single existing logic. Building on this foundational theory, we also define symbolic execution algorithms that use bi-abduction to derive specifications for programs with effects. This involves a new tri-abduction procedure to analyze programs whose execution branches due to effects such as nondeterministic or probabilistic choice. This work furthers the compositionality promised by separation logic by opening up the possibility for greater reuse of analysis tools across two dimensions: bug-finding vs verification in programs with varying effects.

On calculating partial sums of multiple numerical series by methods of Computer Algebra

A method to calculate partial sums of some multiple numerical series arising when searching for the resultant of a polynomial and an entire function is proposed. One can apply a symbolic algorithm that uses recurrent Newton formulas to find power sums of roots included in this formula without finding the very roots of the system. The algorithm that implements the proposed approach to calculate partial sums of multiple numerical series is implemented in Maple. Examples of using this algorithm to find partial sums of some classes of multiple numerical series are given.

Identification of Real and Complex Solution Varieties and Their Singularities Defined by Loop Constraints of Linkages Using the Kinematic Tangent Cone

Abstract The configuration space (c-space) of a mechanism is the real-solution variety of a set of loop closure constraints, which is therefore the chief object in kinematic analysis. Singularities of this variety (referred to as c-space singularities) are singular configurations of the mechanism. In addition, a mechanism may exhibit other kinematic singularities that are not visible from the differential geometry of the c-space (referred to as hidden singularities). Such situations were analyzed by investigating the local geometry of the c-space and its corank stratification. It has been shown recently that hidden singularities and shakiness can be attributed to the fact that complex solution branches intersect with the c-space, i.e., with real-solution branches. This paper employs the kinematic tangent cone to identify local solution branches. While the kinematic tangent cone is an established generally applicable concept, which gives rise to a computational (numeric and symbolic) algorithm, it has yet only been applied to analyzing the real-solution set. Application of the method is shown for several examples. Further, the algebraic aspects are briefly elaborated.

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.

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.