Neural networks' expressiveness comes at the cost of complex, black-box models that often extrapolate poorly beyond the domain of the training dataset, conflicting with the goal of finding compact analytic expressions to describe scientific data. We introduce OccamNet, a neural network model that finds interpretable, compact, and sparse symbolic fits to data, à la Occam's razor. Our model defines a probability distribution over functions with efficient sampling and function evaluation. We train by sampling functions and biasing the probability mass toward better fitting solutions, backpropagating using cross-entropy matching in a reinforcement-learning loss. OccamNet can identify symbolic fits for a variety of problems, including analytic and non-analytic functions, implicit functions, and simple image classification, and can outperform state-of-the-art symbolic regression methods on real-world regression datasets. Our method requires a minimal memory footprint, fits complicated functions in minutes on a single CPU, and scales on a GPU.
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Discovering the underlying mathematical expressions describing a dataset is a core challenge for artificial intelligence. This is the problem of $\textit{symbolic regression}$. Despite recent advances in training neural networks to solve complex tasks, deep learning approaches to symbolic regression are underexplored. We propose a framework that leverages deep learning for symbolic regression via a simple idea: use a large model to search the space of small models. Specifically, we use a recurrent neural network to emit a distribution over tractable mathematical expressions and employ a novel …
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Symbolic regression is the process of identifying mathematical expressions that fit observed output from a black-box process. It is a discrete optimization problem generally believed to be NP-hard. Prior approaches to solving the problem include neural-guided search (e.g. using reinforcement learning) and genetic programming. In this work, we introduce a hybrid neural-guided/genetic programming approach to symbolic regression and other combinatorial optimization problems. We propose a neural-guided component used to seed the starting population of a random restart genetic programming component…
Here we present Symplectically Integrated Symbolic Regression (SISR), a novel technique for learning physical governing equations from data. SISR employs a deep symbolic regression approach, using a multi-layer LSTM-RNN with mutation to probabilistically sample Hamiltonian symbolic expressions. Using symplectic neural networks, we develop a model-agnostic approach for extracting meaningful physical priors from the data that can be imposed on-the-fly into the RNN output, limiting its search space. Hamiltonians generated by the RNN are optimized and assessed using a fourth-order symplectic integ…
Université de Lorraine（洛林大学）；CNRS（法国国家科学研究中心）；LORIA（洛林计算机科学与应用数学实验室）；Réseau de Transport d’Electricité (Rte) R&D（法国电力传输网络研发部）
In recent years, symbolic regression has been of wide interest to provide an interpretable symbolic representation of potentially large data relationships. Initially circled to genetic algorithms, symbolic regression methods now include a variety of Deep Learning based alternatives. However, these methods still do not generalize well to real-world data, mainly because they hardly include domain knowledge nor consider physical relationships between variables such as known equations and units. Regarding these issues, we propose a Reinforcement-Based Grammar-Guided Symbolic Regression (RBG2-SR) m…
Kahlert School of Computing（卡勒特计算学院）；Scientific Computing and Imaging Institute（科学计算与成像研究所）；Department of Mechanical Engineering（机械工程系）；University of Utah（犹他大学）
In this paper we provide a broad benchmarking of recent genetic programming approaches to symbolic regression in the context of state of the art machine learning approaches. We use a set of nearly 100 regression benchmark problems culled from open source repositories across the web. We conduct a rigorous benchmarking of four recent symbolic regression approaches as well as nine machine learning approaches from scikit-learn. The results suggest that symbolic regression performs strongly compared to state-of-the-art gradient boosting algorithms, although in terms of running times is among the sl…
The Interaction-Transformation (IT) is a new representation for Symbolic Regression that restricts the search space into simpler, but expressive, function forms. This representation has the advantage of creating a smoother search space unlike the space generated by Expression Trees, the common representation used in Genetic Programming. This paper introduces an Evolutionary Algorithm capable of evolving a population of IT expressions supported only by the mutation operator. The results show that this representation is capable of finding better approximations to real-world data sets when compar…
Ben-Gurion University（本-古里安大学）；University of Pennsylvania（宾夕法尼亚大学）
Modifying standard gradient boosting by replacing the embedded weak learner in favor of a strong(er) one, we present SyRBo: Symbolic-Regression Boosting. Experiments over 98 regression datasets show that by adding a small number of boosting stages -- between 2--5 -- to a symbolic regressor, statistically significant improvements can often be attained. We note that coding SyRBo on top of any symbolic regressor is straightforward, and the added cost is simply a few more evolutionary rounds. SyRBo is essentially a simple add-on that can be readily added to an extant symbolic regressor, often with…
Josef Ressel Centre for Symbolic Regression（约瑟夫·雷塞尔符号回归中心）；University of Applied Sciences Upper Austria（上奥地利应用科学大学）；Johannes Kepler University（约翰内斯·开普勒大学）
We introduce in this paper a runtime-efficient tree hashing algorithm for the identification of isomorphic subtrees, with two important applications in genetic programming for symbolic regression: fast, online calculation of population diversity and algebraic simplification of symbolic expression trees. Based on this hashing approach, we propose a simple diversity-preservation mechanism with promising results on a collection of symbolic regression benchmark problems.
Federal University of ABC（ABC联邦大学）；University of Applied Sciences Upper Austria（上奥地利应用科学大学）
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 calculat…
Symbolic regression is a type of discrete optimization problem that involves searching expressions that fit given data points. In many cases, other mathematical constraints about the unknown expression not only provide more information beyond just values at some inputs, but also effectively constrain the search space. We identify the asymptotic constraints of leading polynomial powers as the function approaches zero and infinity as useful constraints and create a system to use them for symbolic regression. The first part of the system is a conditional production rule generating neural network …
We present three evolutionary symbolic regression-based classification algorithms for binary and multinomial datasets: GPLearnClf, CartesianClf, and ClaSyCo. Tested over 162 datasets and compared to three state-of-the-art machine learning algorithms -- XGBoost, LightGBM, and a deep neural network -- we find our algorithms to be competitive. Further, we demonstrate how to find the best method for one's dataset automatically, through the use of a state-of-the-art hyperparameter optimizer.
University of Applied Sciences Upper Austria（上奥地利应用科学大学）
The typical methods for symbolic regression produce rather abrupt changes in solution candidates. In this work, we have tried to transform symbolic regression from an optimization problem, with a landscape that is so rugged that typical analysis methods do not produce meaningful results, to one that can be compared to typical and very smooth real-valued problems. While the ruggedness might not interfere with the performance of optimization, it restricts the possibilities of analysis. Here, we have explored different aspects of a transformation and propose a simple procedure to create real-valu…
University of Applied Sciences Upper Austria（上奥地利应用科学大学）；Miba Frictec GmbH（米巴摩擦技术有限公司）
Friction systems are mechanical systems wherein friction is used for force transmission (e.g. mechanical braking systems or automatic gearboxes). For finding optimal and safe design parameters, engineers have to predict friction system performance. This is especially difficult in real-world applications, because it is affected by many parameters. We have used symbolic regression and genetic programming for finding accurate and trustworthy prediction models for this task. However, it is not straight-forward how nominal variables can be included. In particular, a one-hot-encoding is unsatisfacto…
LLM-Meta-SR: In-Context Learning for Evolving Selection Operators in Symbolic Regression
Centre for Data Science and Artificial Intelligence & School of Engineering and Computer Science, Victoria University of Wellington（惠灵顿维多利亚大学数据科学与人工智能中心及工程与计算机科学学院）；Department of Computer Science and Engineering, Michigan State University（密歇根州立大学计算机科学与工程系）
Discovering equations from data: symbolic regression in dynamical systems
Institute of Sciences Mathematics and Computation, Universidade de São Paulo-Campus de São Carlos（数学与计算科学研究院，圣保罗大学-萨o卡洛斯校区）；Statistic Department, State University of Maringá（统计系，马林加州立大学）；Department of Mathematics Applied and Statistics, Institute of Sciences Mathematics and Computation, Universidade de São Paulo-Campus de São Carlos（应用数学与统计系，数学与计算科学研究院，圣保罗大学-萨o卡洛斯校区）
University of Applied Sciences Upper Austria（上奥地利应用科学大学）；MCL Forschungs GmbH（MCL研究有限责任公司）
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 multi…
In symbolic regression, the search for analytic models is typically driven purely by the prediction error observed on the training data samples. However, when the data samples do not sufficiently cover the input space, the prediction error does not provide sufficient guidance toward desired models. Standard symbolic regression techniques then yield models that are partially incorrect, for instance, in terms of their steady-state characteristics or local behavior. If these properties were considered already during the search process, more accurate and relevant models could be produced. We propo…
Université de Paris（巴黎大学）；Ochanomizu University（御茶水女子大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
By combining Genetic Programming, MAP-Elites and Covariance Matrix Adaptation Evolution Strategy, we demonstrate very high success rates in Symbolic Regression problems. MAP-Elites is used to improve exploration while preserving diversity and avoiding premature convergence and bloat. Then, a Covariance Matrix Adaptation-Evolution Strategy is used to evaluate free scalars through a non-gradient-based black-box optimizer. Although this evaluation approach is not computationally scalable to high dimensional problems, our algorithm is able to find exactly most of the $31$ targets extracted from th…
Symbolic regression is the task of identifying a mathematical expression that best fits a provided dataset of input and output values. Due to the richness of the space of mathematical expressions, symbolic regression is generally a challenging problem. While conventional approaches based on genetic evolution algorithms have been used for decades, deep learning-based methods are relatively new and an active research area. In this work, we present SymbolicGPT, a novel transformer-based language model for symbolic regression. This model exploits the advantages of probabilistic language models lik…
Evaluation of Population Initialization Methods for Genetic Programming-based Symbolic Regression
Heuristic and Evolutionary Algorithms Laboratory, University of Applied Sciences Upper Austria（上奥地利应用科学大学启发式与进化算法实验室）；Astrophysics, University of Oxford（牛津大学天体物理学系）；Institute of Cosmology & Gravitation, University of Portsmouth（朴茨茅斯大学宇宙学与引力研究所）
Massachusetts Institute of Technology（麻省理工学院）；University of Zagreb（萨格勒布大学）
Symbolic regression is a powerful technique that can discover analytical equations that describe data, which can lead to explainable models and generalizability outside of the training data set. In contrast, neural networks have achieved amazing levels of accuracy on image recognition and natural language processing tasks, but are often seen as black-box models that are difficult to interpret and typically extrapolate poorly. Here we use a neural network-based architecture for symbolic regression called the Equation Learner (EQL) network and integrate it with other deep learning architectures …
University of Applied Sciences Upper Austria（上奥地利应用科学大学）；Johannes Kepler University Linz（林茨约翰·开普勒大学）
Multi-objective symbolic regression has the advantage that while the accuracy of the learned models is maximized, the complexity is automatically adapted and need not be specified a-priori. The result of the optimization is not a single solution anymore, but a whole Pareto-front describing the trade-off between accuracy and complexity. In this contribution we study which complexity measures are most appropriately used in symbolic regression when performing multi- objective optimization with NSGA-II. Furthermore, we present a novel complexity measure that includes semantic information based on …
NEC Corporation（日本电气株式会社）；National Institute of Advanced Industrial Science and Technology (AIST)（日本国立研究开发法人产业技术综合研究所）
Discovery of new natural laws has for a long time relied on the inspiration of some genius. Recently, however, machine learning technologies, which analyze big data without human prejudice and bias, are expected to find novel natural laws. Here we demonstrate that our proposed machine learning, recursive-LASSO-based symbolic (RLS) regression, enables data-driven formulation of natural laws from noisy data. The RLS regression recurrently repeats feature generation and feature selection, eventually constructing a data-driven model with highly nonlinear features. This data-driven formulation meth…
ETH（苏黎世联邦理工学院）；CSEM SA（瑞士电子与微技术中心）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Symbolic equations are at the core of scientific discovery. The task of discovering the underlying equation from a set of input-output pairs is called symbolic regression. Traditionally, symbolic regression methods use hand-designed strategies that do not improve with experience. In this paper, we introduce the first symbolic regression method that leverages large scale pre-training. We procedurally generate an unbounded set of equations, and simultaneously pre-train a Transformer to predict the symbolic equation from a corresponding set of input-output-pairs. At test time, we query the model …
Symbolic-Diffusion: Deep Learning Based Symbolic Regression with D3PM Discrete Token Diffusion
Department of Mechatronics Engineering University of Waterloo（机械工程系大学水力学院）；Department of Computer Science University of Waterloo（计算机科学系大学水力学院）；Department of Statistics and Actuarial Science University of Waterloo（统计与精算科学系大学水力学院）
Centro de investigación CITIC（CITIC研究中心）；University of A Coruña（拉科鲁尼亚大学）
This paper describes a new method for Symbolic Regression that allows to find mathematical expressions from a dataset. This method has a strong mathematical basis. As opposed to other methods such as Genetic Programming, this method is deterministic, and does not involve the creation of a population of initial solutions. Instead of it, a simple expression is being grown until it fits the data. The experiments performed show that the results are as good as other Machine Learning methods, in a very low computational time. Another advantage of this technique is that the complexity of the expressi…
IBM Research AI（IBM研究院人工智能部门）；University of Minnesota（明尼苏达大学）；IBM Research - Almaden（IBM阿尔马登研究院）
The Symbolic Regression (SR) problem, where the goal is to find a regression function that does not have a pre-specified form but is any function that can be composed of a list of operators, is a hard problem in machine learning, both theoretically and computationally. Genetic programming based methods, that heuristically search over a very large space of functions, are the most commonly used methods to tackle SR problems. An alternative mathematical programming approach, proposed in the last decade, is to express the optimal symbolic expression as the solution of a system of nonlinear equatio…
Recently, several algorithms for symbolic regression (SR) emerged which employ a form of multiple linear regression (LR) to produce generalized linear models. The use of LR allows the algorithms to create models with relatively small error right from the beginning of the search; such algorithms are thus claimed to be (sometimes by orders of magnitude) faster than SR algorithms based on vanilla genetic programming. However, a systematic comparison of these algorithms on a common set of problems is still missing. In this paper we conceptually and experimentally compare several representatives of…
Machine learning is often applied in health science to obtain predictions and new understandings of complex phenomena and relationships, but an availability of sufficient data for model training is a widespread problem. Traditional machine learning techniques, such as random forests and gradient boosting, tend to overfit when working with data sets of only a few hundred observations. This study demonstrates that for small training sets of 250 observations, symbolic regression generalises better to out-of-sample data than traditional machine learning frameworks, as measured by the coefficient o…
Adaptive Protection for Evolutionary Feature Construction in Symbolic Regression with Application to Credit Classification
Centre for Data Science and Artificial Intelligence（数据科学与人工智能中心）；School of Engineering and Computer Science（工程与计算机科学学院）；Victoria University of Wellington（惠灵顿维多利亚大学）；Business School, Sichuan University（四川大学商学院）；Michigan State University（密歇根州立大学）
Enhancing Symbolic Regression and Universal Physics-Informed Neural Networks with Dimensional Analysis
Cheriton School of Computer Science（切尔顿计算机科学学院）；Department of Electrical and Computer Engineering（电气与计算机工程系）；University of Waterloo（滑铁卢大学）；Electrical Engineering and Computer Science Department（电气工程与计算机科学系）；Department of Applied Mathematics（应用数学系）；University of Michigan（密歇根大学）
Neuro-Evolutionary Approach to Physics-Aware Symbolic Regression
Czech Institute of Informatics, Robotics, and Cybernetics CTU in Prague（捷克信息学、机器人学和自动化研究中心CTU布拉格分校）；Dept. of Cognitive Robotics, TU Delft（认知机器人学系代尔夫特理工大学）；CIIRC, CTU in Prague（智能信息研究中心CTU布拉格分校）
University of Utah（犹他大学）；Scientific Computing and Imaging Institute（科学计算与成像研究所）；NASA Langley Research Center（美国国家航空航天局兰利研究中心）
In this paper, we present a machine learning method for the discovery of analytic solutions to differential equations. The method utilizes an inherently interpretable algorithm, genetic programming based symbolic regression. Unlike conventional accuracy measures in machine learning we demonstrate the ability to recover true analytic solutions, as opposed to a numerical approximation. The method is verified by assessing its ability to recover known analytic solutions for two separate differential equations. The developed method is compared to a conventional, purely data-driven genetic programmi…
