Symbolic regression, i.e. predicting a function from the observation of its values, is well-known to be a challenging task. In this paper, we train Transformers to infer the function or recurrence relation underlying sequences of integers or floats, a typical task in human IQ tests which has hardly been tackled in the machine learning literature. We evaluate our integer model on a subset of OEIS sequences, and show that it outperforms built-in Mathematica functions for recurrence prediction. We also demonstrate that our float model is able to yield informative approximations of out-of-vocabulary functions and constants, e.g. $\operatorname{bessel0}(x)\approx \frac{\sin(x)+\cos(x)}{\sqrt{πx}}$ and $1.644934\approx π^2/6$. An interactive demonstration of our models is provided at https://symbolicregression.metademolab.com.
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…
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 …
T-SHRED: Symbolic Regression for Regularization and Model Discovery with Transformer Shallow Recurrent Decoders
Electrical and Computer Engineering, University of Washington（华盛顿大学电气与计算机工程系）；Applied Mathematics, University of Washington（华盛顿大学应用数学系）；Computer Science & Engineering, University of Washington（华盛顿大学计算机科学与工程系）
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）
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…
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）
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…
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…
State Key Lab of Processors, Institute of Computing Technology, CAS（中国科学院计算技术研究所处理器国家重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）；Cambricon Technologies（寒武纪科技）
Symbolic regression, the task of extracting mathematical expressions from the observed data $\{ \vx_i, y_i \}$, plays a crucial role in scientific discovery. Despite the promising performance of existing methods, most of them conduct symbolic regression in an \textit{offline} setting. That is, they treat the observed data points as given ones that are simply sampled from uniform distributions without exploring the expressive potential of data. However, for real-world scientific problems, the data used for symbolic regression are usually actively obtained by doing experiments, which is an \text…
In this paper we introduce an active learning method for symbolic regression using StackGP. The approach begins with a small number of data points for StackGP to model. To improve the model the system incrementally adds a data point such that the new point maximizes prediction uncertainty as measured by the model ensemble. Symbolic regression is re-run with the larger data set. This cycle continues until the system satisfies a termination criterion. We use the Feynman AI benchmark set of equations to examine the ability of our method to find appropriate models using fewer data points. The appr…
University of Applied Sciences Upper Austria（上奥地利应用科技大学）；Johannes Kepler University（约翰内斯·开普勒大学）
Symbolic regression is a powerful system identification technique in industrial scenarios where no prior knowledge on model structure is available. Such scenarios often require specific model properties such as interpretability, robustness, trustworthiness and plausibility, that are not easily achievable using standard approaches like genetic programming for symbolic regression. In this chapter we introduce a deterministic symbolic regression algorithm specifically designed to address these issues. The algorithm uses a context-free grammar to produce models that are parameterized by a non-line…
Scalable Neural Symbolic Regression using Control Variables
School of Computer Science（计算机科学学院）；GaTech（佐治亚理工学院）；School of Computing and Data Science（计算与数据科学学院）；UIUC（伊利诺伊大学厄巴纳-香槟分校）；William & Mary（威廉玛丽学院）；Department of EECS（电子工程与计算机科学系）；University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校）；Wake Forest University（维克森林大学）
University of Applied Sciences Upper Austria（上奥地利应用科技大学）；Johannes Kepler University（约翰内斯·开普勒大学）
In this chapter we take a closer look at the distribution of symbolic regression models generated by genetic programming in the search space. The motivation for this work is to improve the search for well-fitting symbolic regression models by using information about the similarity of models that can be precomputed independently from the target function. For our analysis, we use a restricted grammar for uni-variate symbolic regression models and generate all possible models up to a fixed length limit. We identify unique models and cluster them based on phenotypic as well as genotypic similarity…
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…
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）
SymbolicArena: A Unified Infrastructure for Benchmark Distillation and Dynamic Evaluation in Symbolic Regression
School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）；CDL, Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所CDL）；Nanjing University of Science and Technology（南京理工大学）；School of Future Technology, University of Chinese Academy of Sciences（中国科学院大学未来技术学院）
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…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Adobe Systems, Inc.（奥多比系统公司）
In this paper, we propose a new Recurrent Neural Network (RNN) architecture. The novelty is simple: We use diagonal recurrent matrices instead of full. This results in better test likelihood and faster convergence compared to regular full RNNs in most of our experiments. We show the benefits of using diagonal recurrent matrices with popularly used LSTM and GRU architectures as well as with the vanilla RNN architecture, on four standard symbolic music datasets.
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…
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…
Data-centric prognostics is beneficial to improve the reliability and safety of proton exchange membrane fuel cell (PEMFC). For the prognostics of PEMFC operating under dynamic load, the challenges come from extracting degradation features, improving prediction accuracy, expanding the prognostics horizon, and reducing computational cost. To address these issues, this work proposes a data-driven PEMFC prognostics approach, in which Hilbert-Huang transform is used to extract health indicator in dynamic operating conditions and symbolic-based gated recurrent unit model is used to enhance the accu…
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.
Symbolic regression corresponds to an ensemble of techniques that allow to uncover an analytical equation from data. Through a closed form formula, these techniques provide great advantages such as potential scientific discovery of new laws, as well as explainability, feature engineering as well as fast inference. Similarly, deep learning based techniques has shown an extraordinary ability of modeling complex patterns. The present paper aims at applying a recent end-to-end symbolic regression technique, i.e. the equation learner (EQL), to get an analytical equation for wind speed forecasting. …
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…
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…
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 …
Recurrent neural networks are powerful models for sequential data, able to represent complex dependencies in the sequence that simpler models such as hidden Markov models cannot handle. Yet they are notoriously hard to train. Here we introduce a training procedure using a gradient ascent in a Riemannian metric: this produces an algorithm independent from design choices such as the encoding of parameters and unit activities. This metric gradient ascent is designed to have an algorithmic cost close to backpropagation through time for sparsely connected networks. We use this procedure on gated le…
Technische Universität Darmstadt（达姆施塔特工业大学）；Concordia University（康考迪亚大学）
A regression problem with dependent data is considered. Regularity assumptions on the dependency of the data are introduced, and it is shown that under suitable structural assumptions on the regression function a deep recurrent neural network estimate is able to circumvent the curse of dimensionality.
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）
Symbolic regression for defect interactions in 2D materials
HSE University（高等经济大学）；Institute for Functional Intelligent Materials, National University of Singapore（新加坡国立大学功能智能材料研究所）；National University of Singapore（新加坡国立大学）；Constructor University（康斯特大学）
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）
PruneSymNet: A Symbolic Neural Network and Pruning Algorithm for Symbolic Regression
Institute of Semiconductors, Chinese Academy of Sciences（中国科学院半导体研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Beijing Key Laboratory of Semiconductor Neural Network Intelligent Sensing and Computing Technology（北京半导体神经网络智能感知与计算技术重点实验室）
