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Symbolic regression is the task of finding tractable mathematical expressions that best fit an input dataset. Deep symbolic regression (DSR) searches the space of tractable mathematical expressions by using a recurrent neural network that generates mathematical expressions. The network is trained using reinforcement learning.
AI/ML informed Symbolic Regression is the next stage of scientific modeling. We utilize a highly customizable symbolic regression package “PySR” to model the x and t dependence of the flavor isovector combination Hu-d(x, t, ) at = 0. These PySR models were trained on GPD results provided by both Lattice QCD and phenomenological sources GGL, GK, and VGG. We demonstrate, for the first time, the consistency and systematic convergence of Symbolic Regression by quantifying the disparate models through their Taylor expansion coefficients. In addition to PySR penalizing models with higher complexity and mean-squared error, we implement schemes that test specific physics hypotheses, including force-factorized x and t dependence and Regge behavior in PySR GPDs. We show that PySR can identify factorizing GPD sources based on their response to the Force-Factorized model. Here, knowing the precise behavior of the GPDs, and their uncertainties in a wide range in x and t, crucially impacts our ability to concretely and quantitatively predict hadronic spatial distributions and their derived quantities.
When performing symbolic regression using genetic programming, overfitting and bloat can negatively impact generalizability and interpretability of the resulting equations as well as increase computation times. A Bayesian fitness metric is introduced and its impact on bloat and overfitting during population evolution is studied and compared to common alternatives in the literature. The proposed approach was found to be more robust to noise and data sparsity in numerical experiments, guiding evolution to a level of complexity appropriate to the dataset. Further evolution of the population resulted not in overfitting or bloat, but rather in slight simplifications in model form. The ability to identify an equation of complexity appropriate to the scale of noise in the training data was also demonstrated. In general, the Bayesian model selection algorithm was shown to be an effective means of regularization which resulted in less bloat and overfitting when any amount of noise was present in the training data.
When performing symbolic regression using genetic programming, overfitting and bloat can negatively impact generalizability and interpretability of the resulting equations as well as increase computation times. A Bayesian fitness metric is introduced and its impact on bloat and overfitting during population evolution is studied and compared to common alternatives in the literature. The proposed approach was found to be more robust to noise and data sparsity in numerical experiments, guiding evolution to a level of complexity appropriate to the dataset. Further evolution of the population resulted not in overfitting or bloat, but rather in slight simplifications in model form. The ability to identify an equation of complexity appropriate to the scale of noise in the training data was also demonstrated. In general, the Bayesian model selection algorithm was shown to be an effective means of regularization which resulted in less bloat and overfitting when any amount of noise was present in the training data.
Symbolic regression searches for analytic expressions that accurately describe studied phenomena. The main promise of this approach is that it may return an interpretable model that can be insightful to users, while maintaining high accuracy. The current standard for benchmarking these algorithms is SRBench, which evaluates methods on hundreds of datasets that are a mix of real-world and simulated processes spanning multiple domains. At present, the ability of SRBench to evaluate interpretability is limited to measuring the size of expressions on real-world data, and the exactness of model forms on synthetic data. In practice, model size is only one of many factors used by subject experts to determine how interpretable a model truly is. Furthermore, SRBench does not characterize algorithm performance on specific, challenging sub-tasks of regression such as feature selection and evasion of local minima. In this work, we propose and evaluate an approach to benchmarking SR algorithms that addresses these limitations of SRBench by 1) incorporating expert evaluations of interpretability on a domain-specific task, and 2) evaluating algorithms over distinct properties of data science tasks. We evaluate 12 modern symbolic regression algorithms on these benchmarks and present an in-depth analysis of the results, discuss current challenges of symbolic regression algorithms and highlight possible improvements for the benchmark itself.
Symbolic regression methods simultaneously determine the model functional form and the regression parameter values by generating expression trees. Symbolic regression can capture the complexity of real–world phenomena but the use of deterministic optimization for symbolic regression has been limited due to the complexity of the search space of existing formulations. Herein we present a novel deterministic mixed–integer nonlinear programming formulation for symbolic regression that incorporates derivative constraints through auxiliary expression trees. By applying the chain rule to mathematical operations, binary expression trees are capable of representing the calculation of first and second derivatives. We apply this formulation to illustrative examples using derivative information to show increased model discrimination capability. In addition, we perform a case study of a thermodynamic equation of state to gain insight on valid functional forms with thermodynamics–based constraints on the first and second derivatives.
The slides summarize recent advances in symbolic regression developed as part of the PrOMMiS project over the past year. It describes the use of various regularization metrics in the context of symbolic regression, and it includes computational experiments performed to build simple and accurate symbolic regression models.
This research pioneers advancements in computational mechanics by integrating Bayesian-based uncertainty quantification into symbolic regression, specifically focusing on the critical task of accurately estimating the fractional Bayes factor for selecting arbitrary equations. In our exploration, we rigorously study two prominent methods—sequential Monte Carlo and the Laplace approximation—employed for computing the fractional Bayes factor. Our findings underscore the limitations of the Laplace approximation, revealing its diminished accuracy in nonlinear and multimodal scenarios. Specifically, the Laplace approximation is shown to underpredict fractional Bayes factor on a wide set of equations associated with a symbolic regression benchmark. This comparative analysis sheds light on the nuanced performance of these techniques, guiding researchers toward more informed choices in uncertainty quantification within symbolic regression. Furthermore, we showcase the practical utility of these enhanced symbolic regression tools through their application to a real-world problem in fatigue crack growth modeling, emphasizing their efficacy in capturing the complexities of mechanical systems.
The poster summarizes recent advances in symbolic regression developed as part of the PrOMMiS project over the past year. In particular, it describes the comparison of surrogates for critical minerals (CM) & rare earth element (REE) recovery flowsheets obtained via symbolic regression and ALAMO. It also compares the predictive ability and solvability of optimization models that incorporate these surrogates.
In this paper, we introduce Bingo, a flexible and customizable yet performant Python framework for symbolic regression with genetic programming. Bingo maintains a modular code structure for simple abstraction and easily swappable components. Fitness functions, selection methods, and constant optimization methods allow for easy problem-specific customization. Bingo also maintains several features for increased efficiency such as parallelism, equation simplification, and a C++ backend. We compare Bingo’s performance to other genetic programming for symbolic regression (GPSR) methods to show that it is both competitive and flexible.
Misfolded tau proteins play a critical role in the progression and pathology of Alzheimer's disease. Recent studies suggest that the spatio-temporal pattern of misfolded tau follows a reaction-diffusion type equation. However, the precise mathematical model and parameters that characterize the progression of misfolded protein across the brain remain incompletely understood. Here, we use deep learning and artificial intelligence to discover a mathematical model for the progression of Alzheimer's disease using longitudinal tau positron emission tomography from the Alzheimer's Disease Neuroimaging Initiative database. Specifically, we integrate physics informed neural networks (PINNs) and symbolic regression to discover a reaction-diffusion type partial differential equation for tau protein misfolding and spreading. First, we demonstrate the potential of our model and parameter discovery on synthetic data. Then, we apply our method to discover the best model and parameters to explain tau imaging data from 46 individuals who are likely to develop Alzheimer's disease and 30 healthy controls. Our symbolic regression discovers different misfolding models f(c) for two groups, with a faster misfolding for the Alzheimer's group, f(c) = 0.23c 3 – 1.34c 2 + 1.11c, than for the healthy control group, f(c) = –c 3 + 0.62c 2 + 0.39c. Our results suggest that PINNs, supplemented by symbolic regression, can discover a reaction-diffusion type model to explain misfolded tau protein concentrations in Alzheimer's disease. Furthermore, we expect our study to be the starting point for a more holistic analysis to provide image-based technologies for early diagnosis, and ideally early treatment of neurodegeneration in Alzheimer's disease and possibly other misfolding-protein based neurodegenerative disorders.
We introduce SymbolFit (API: https://github.com/hftsoi/symbolfit), a framework that automates parametric modeling by using symbolic regression to perform a machine-search for functions that fit the data while simultaneously providing uncertainty estimates in a single run. Traditionally, constructing a parametric model to accurately describe binned data has been a manual and iterative process, requiring an adequate functional form to be determined before the fit can be performed. The main challenge arises when the appropriate functional forms cannot be derived from first principles, especially when there is no underlying true closed-form function for the distribution. In this work, we develop a framework that automates and streamlines the process by utilizing symbolic regression, a machine learning technique that explores a vast space of candidate functions without requiring a predefined functional form because the functional form itself is treated as a trainable parameter, making the process far more efficient and effortless than traditional regression methods. We demonstrate the framework in high-energy physics experiments at the CERN Large Hadron Collider (LHC) using five real proton-proton collision datasets from new physics searches, including background modeling in resonance searches for high-mass dijet, trijet, paired-dijet, diphoton, and dimuon events. We show that our framework can flexibly and efficiently generate a wide range of candidate functions that fit a nontrivial distribution well using a simple fit configuration that varies only by random seed, and that the same fit configuration, which defines a vast function space, can also be applied to distributions of different shapes, whereas achieving a comparable result with traditional methods would have required extensive manual effort.
In this paper, we introduce Bingo, a flexible and customizable yet performant Python framework for symbolic regression with genetic programming. Bingo maintains a modular code structure for simple abstraction and easily swappable components. Fitness functions, selection methods, and constant optimization methods allow for easy problem-specific customization. Bingo also maintains several features for increased efficiency such as parallelism, equation simplification, and a C++ backend. We compare Bingo’s performance to other genetic programming for symbolic regression (GPSR) methods to show that it is both competitive and flexible.
Symbolic regression (SR) with a multi-gene genetic program has been used to elucidate new empirical equations describing diffusion in Lennard-Jones (LJ) fluids. Some examples include equations to predict self-diffusion in pure LJ fluids and equations describing the finite-size correction for self-diffusion in binary LJ fluids. The performance of the SR-obtained equations was compared to that of both the existing empirical equations in the literature and to the results from artificial neural net (ANN) models recently reported. It is found that the SR equations have improved predictive performance in comparison to the existing empirical equations, even though employing a smaller number of adjustable parameters, but show an overall reduced performance in comparison to more extensive ANNs.
Symbolic regression (SR) is an approach of interpretable machine learning for building mathematical formulas that best fit certain datasets. In this work, SR is used to guide the design of new oxide perovskite catalysts with improved oxygen evolution reaction (OER) activities. A simple descriptor, μ/t, where μ and t are the octahedral and tolerance factors, respectively, is identified, which accelerates the discovery of a series of new oxide perovskite catalysts with improved OER activity. We successfully synthesise five new oxide perovskites and characterise their OER activities. Remarkably, four of them, Cs 0.4 La 0.6 Mn 0.25 Co 0.75 O 3 , Cs 0.3 La 0.7 NiO 3 , SrNi 0.75 Co 0.25 O 3 , and Sr 0.25 Ba 0.75 NiO 3 , are among the oxide perovskite catalysts with the highest intrinsic activities. Our results demonstrate the potential of SR for accelerating the data-driven design and discovery of new materials with improved properties.
A data-driven computational method is introduced to extract chemical reaction mechanisms from time series chemical concentration data. It is realized through the use of dynamic symbolic regression in which a sparse analytical form for a dynamical system is discoverable from the underlying data. We specifically develop the stoichiometrically-informed symbolic regression (SISR) method to address a standing challenge in complex chemical reaction networks: given a time-series dataset of concentrations of several components, what is the mechanism and the associated rate constants? SISR finds the optimal mechanism, kinetic equations and rate constants by combining differential optimization with a genetic optimization approach that searches a symbolic space of possible reaction mechanisms. Use of SISR in several paradigmatic examples spanning linear and nonlinear reaction schemes results in excellent agreement between true and predicted mechanisms, including when the method is applied to noisy data. The advantages of a stoichiometrically-informed approach such as SISR to address reaction discovery is illustrated through comparison with the use of generic state-of-the-art data-driven approaches.
Abstract Compact symbolic expressions have been shown to be more efficient than neural network (NN) models in terms of resource consumption and inference speed when implemented on custom hardware such as field-programmable gate arrays (FPGAs), while maintaining comparable accuracy (Tsoi et al 2024 EPJ Web Conf. 295 09036). These capabilities are highly valuable in environments with stringent computational resource constraints, such as high-energy physics experiments at the CERN Large Hadron Collider. However, finding compact expressions for high-dimensional datasets remains challenging due to the inherent limitations of genetic programming (GP), the search algorithm of most symbolic regression (SR) methods. Contrary to GP, the NN approach to SR offers scalability to high-dimensional inputs and leverages gradient methods for faster equation searching. Common ways of constraining expression complexity often involve multistage pruning with fine-tuning, which can result in significant performance loss. In this work, we propose S y m b o l N e t , a NN approach to SR specifically designed as a model compression technique, aimed at enabling low-latency inference for high-dimensional inputs on custom hardware such as FPGAs. This framework allows dynamic pruning of model weights, input features, and mathematical operators in a single training process, where both training loss and expression complexity are optimized simultaneously. We introduce a sparsity regularization term for each pruning type, which can adaptively adjust its strength, leading to convergence at a target sparsity ratio. Unlike most existing SR methods that struggle with datasets containing more than O ( 10 ) inputs, we demonstrate the effectiveness of our model on the LHC jet tagging task (16 inputs), MNIST (784 inputs), and SVHN (3072 inputs).
Here, an interpretable machine learning method, physics-informed genetic programming-based symbolic regression (P-GPSR), is integrated into a continuum thermodynamic approach to developing constitutive models. The proposed strategy for combining a thermodynamic analysis with P-GPSR is demonstrated by generating a yield function for an idealized material with voids, i.e., the Gurson yield function. First, a thermodynamic-based analysis is used to derive model requirements that are exploited in a custom P-GPSR implementation as fitness criteria or are strongly enforced in the solution. The P-GPSR implementation improved accuracy, generalizability, and training time compared to the same GPSR code without physics-informed fitness criteria. The yield function generated through the P-GPSR framework is in the form of a composite function that describes a class of materials and is characteristically more interpretable than GPSR-derived equations. The physical significance of the input functions learned by P-GPSR within the composite function is acquired from the thermodynamic analysis. Fundamental explanations of why the implemented P-GPSR capabilities improve results over a conventional GPSR algorithm are provided.