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Abstract:Symbolic regression is a machine learning method with the goal to produce interpretable results. Unlike other machine learning methods such as, e.g. random forests or neural networks, which are opaque, symbolic regression aims to model and map data in a way that can be understood by scientists. Recent advancements, have attempted to bridge the gap between these two fields; new methodologies attempt to fuse the mapping power of neural networks and deep learning techniques with the explanatory power of symbolic regression. In this paper, we examine these new emerging systems and test the performance of an end-to-end transformer model for symbolic regression versus the reigning traditional methods based on genetic programming that have spearheaded symbolic regression throughout the years. We compare these systems on novel datasets to avoid bias to older methods who were improved on well-known benchmark datasets. Our results show that traditional GP methods as implemented e.g., by Operon still remain superior to two recently published symbolic regression methods.

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Authors:Gabriel Kronberger, Fabricio Olivetti de Franca, Harry Desmond, Deaglan J. Bartlett, Lukas Kammerer

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Abstract:We analyse the search behaviour of genetic programming for symbolic regression in practically relevant but limited settings, allowing exhaustive enumeration of all solutions. This enables us to quantify the success probability of finding the best possible expressions, and to compare the search efficiency of genetic programming to random search in the space of semantically unique expressions. This analysis is made possible by improved algorithms for equality saturation, which we use to improve the Exhaustive Symbolic Regression algorithm; this produces the set of semantically unique expression structures, orders of magnitude smaller than the full symbolic regression search space. We compare the efficiency of random search in the set of unique expressions and genetic programming. For our experiments we use two real-world datasets where symbolic regression has been used to produce well-fitting univariate expressions: the Nikuradse dataset of flow in rough pipes and the Radial Acceleration Relation of galaxy dynamics. The results show that genetic programming in such limited settings explores only a small fraction of all unique expressions, and evaluates expressions repeatedly that are congruent to already visited expressions.

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Authors:Deaglan J. Bartlett, Lukas Kammerer, Gabriel Kronberger, Harry Desmond, Pedro G. Ferreira, Benjamin D. Wandelt, Bogdan Burlacu, David Alonso, Matteo Zennaro

Abstract:Computing the matter power spectrum, $P(k)$, as a function of cosmological parameters can be prohibitively slow in cosmological analyses, hence emulating this calculation is desirable. Previous analytic approximations are insufficiently accurate for modern applications, so black-box, uninterpretable emulators are often used. We utilise an efficient genetic programming based symbolic regression framework to explore the space of potential mathematical expressions which can approximate the power spectrum and $\sigma_8$. We learn the ratio between an existing low-accuracy fitting function for $P(k)$ and that obtained by solving the Boltzmann equations and thus still incorporate the physics which motivated this earlier approximation. We obtain an analytic approximation to the linear power spectrum with a root mean squared fractional error of 0.2% between $k = 9\times10^{-3} - 9 \, h{\rm \, Mpc^{-1}}$ and across a wide range of cosmological parameters, and we provide physical interpretations for various terms in the expression. We also provide a simple analytic approximation for $\sigma_8$ with a similar accuracy, with a root mean squared fractional error of just 0.4% when evaluated across the same range of cosmologies. This function is easily invertible to obtain $A_{\rm s}$ as a function of $\sigma_8$ and the other cosmological parameters, if preferred. It is possible to obtain symbolic approximations to a seemingly complex function at a precision required for current and future cosmological analyses without resorting to deep-learning techniques, thus avoiding their black-box nature and large number of parameters. Our emulator will be usable long after the codes on which numerical approximations are built become outdated.

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Abstract:People with diabetes must carefully monitor their blood glucose levels, especially after eating. Blood glucose regulation requires a proper combination of food intake and insulin boluses. Glucose prediction is vital to avoid dangerous post-meal complications in treating individuals with diabetes. Although traditional methods, such as artificial neural networks, have shown high accuracy rates, sometimes they are not suitable for developing personalised treatments by physicians due to their lack of interpretability. In this study, we propose a novel glucose prediction method emphasising interpretability: Interpretable Sparse Identification by Grammatical Evolution. Combined with a previous clustering stage, our approach provides finite difference equations to predict postprandial glucose levels up to two hours after meals. We divide the dataset into four-hour segments and perform clustering based on blood glucose values for the twohour window before the meal. Prediction models are trained for each cluster for the two-hour windows after meals, allowing predictions in 15-minute steps, yielding up to eight predictions at different time horizons. Prediction safety was evaluated based on Parkes Error Grid regions. Our technique produces safe predictions through explainable expressions, avoiding zones D (0.2% average) and E (0%) and reducing predictions on zone C (6.2%). In addition, our proposal has slightly better accuracy than other techniques, including sparse identification of non-linear dynamics and artificial neural networks. The results demonstrate that our proposal provides interpretable solutions without sacrificing prediction accuracy, offering a promising approach to glucose prediction in diabetes management that balances accuracy, interpretability, and computational efficiency.

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Abstract:We describe an approach for empirical modeling of steel phase kinetics based on symbolic regression and genetic programming. The algorithm takes processed data gathered from dilatometer measurements and produces a system of differential equations that models the phase kinetics. Our initial results demonstrate that the proposed approach allows to identify compact differential equations that fit the data. The model predicts ferrite, pearlite and bainite formation for a single steel type. Martensite is not yet included in the model. Future work shall incorporate martensite and generalize to multiple steel types with different chemical compositions.

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Abstract:Describing dynamic medical systems using machine learning is a challenging topic with a wide range of applications. In this work, the possibility of modeling the blood glucose level of diabetic patients purely on the basis of measured data is described. A combination of the influencing variables insulin and calories are used to find an interpretable model. The absorption speed of external substances in the human body depends strongly on external influences, which is why time-shifts are added for the influencing variables. The focus is put on identifying the best timeshifts that provide robust models with good prediction accuracy that are independent of other unknown external influences. The modeling is based purely on the measured data using Sparse Identification of Nonlinear Dynamics. A differential equation is determined which, starting from an initial value, simulates blood glucose dynamics. By applying the best model to test data, we can show that it is possible to simulate the long-term blood glucose dynamics using differential equations and few, influencing variables.

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Abstract:Fast Function Extraction (FFX) is a deterministic algorithm for solving symbolic regression problems. We improve the accuracy of FFX by adding parameters to the arguments of nonlinear functions. Instead of only optimizing linear parameters, we optimize these additional nonlinear parameters with separable nonlinear least squared optimization using a variable projection algorithm. Both FFX and our new algorithm is applied on the PennML benchmark suite. We show that the proposed extensions of FFX leads to higher accuracy while providing models of similar length and with only a small increase in runtime on the given data. Our results are compared to a large set of regression methods that were already published for the given benchmark suite.

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Abstract:Industrial and scientific applications handle large volumes of data that render manual validation by humans infeasible. Therefore, we require automated data validation approaches that are able to consider the prior knowledge of domain experts to produce dependable, trustworthy assessments of data quality. Prior knowledge is often available as rules that describe interactions of inputs with regard to the target e.g. the target must be monotonically decreasing and convex over increasing input values. Domain experts are able to validate multiple such interactions at a glance. However, existing rule-based data validation approaches are unable to consider these constraints. In this work, we compare different shape-constrained regression algorithms for the purpose of data validation based on their classification accuracy and runtime performance.

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Abstract:Symbolic regression is a nonlinear regression method which is commonly performed by an evolutionary computation method such as genetic programming. Quantification of uncertainty of regression models is important for the interpretation of models and for decision making. The linear approximation and so-called likelihood profiles are well-known possibilities for the calculation of confidence and prediction intervals for nonlinear regression models. These simple and effective techniques have been completely ignored so far in the genetic programming literature. In this work we describe the calculation of likelihood profiles in details and also provide some illustrative examples with models created with three different symbolic regression algorithms on two different datasets. The examples highlight the importance of the likelihood profiles to understand the limitations of symbolic regression models and to help the user taking an informed post-prediction decision.

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Authors:Gabriel Kronberger

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Abstract:Gradient-based local optimization has been shown to improve results of genetic programming (GP) for symbolic regression. Several state-of-the-art GP implementations use iterative nonlinear least squares (NLS) algorithms such as the Levenberg-Marquardt algorithm for local optimization. The effectiveness of NLS algorithms depends on appropriate scaling and conditioning of the optimization problem. This has so far been ignored in symbolic regression and GP literature. In this study we use a singular value decomposition of NLS Jacobian matrices to determine the numeric rank and the condition number. We perform experiments with a GP implementation and six different benchmark datasets. Our results show that rank-deficient and ill-conditioned Jacobian matrices occur frequently and for all datasets. The issue is less extreme when restricting GP tree size and when using many non-linear functions in the function set.

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