Double decision mechanism-based deep symbolic regression algorithm

doi:10.11772/j.issn.1001-9081.2025020174

Journal of Computer Applications ›› 2026, Vol. 46 ›› Issue (2): 406-415.DOI: 10.11772/j.issn.1001-9081.2025020174

• Artificial intelligence • Previous Articles

Double decision mechanism-based deep symbolic regression algorithm

Zeyi GUO¹, Fenglian LI¹(), Lichun XU²

^1.College of Electronic Information Engineering，Taiyuan University of Technology，Taiyuan Shanxi 030600，China
^2.College of Physics and Optoelectronic Engineering，Taiyuan University of Technology，Taiyuan Shanxi 030600，China

Received:2025-02-25 Revised:2025-04-07 Accepted:2025-04-11 Online:2025-04-24 Published:2026-02-10
Contact: Fenglian LI
About author:GUO Zeyi， born in 1997， M. S. candidate. His research interests include symbolic regression， deep learning.
LI Fenglian， born in 1972， Ph. D.， professor. Her research interests include intelligent information processing theory and its application in industrial data analysis Email:lifenglian@tyut.edu.cn
XU Lichun， born in 1985， Ph. D.， associate professor. His research interests include computational physics， artificial intelligence.
Supported by:
Special Project for Scientific and Technological Cooperation and Exchange in Shanxi Province(202304041101035)

基于双重决策机制的深度符号回归算法

郭泽一¹, 李凤莲¹(), 徐利春²

^1.太原理工大学电子信息工程学院，太原 030600
^2.太原理工大学物理与光电工程学院，太原 030600

通讯作者: 李凤莲
作者简介:郭泽一（1997—），男，山西临汾人，硕士研究生，主要研究方向：符号回归、深度学习
李凤莲（1972—），女，山西运城人，教授，博士，CCF会员，主要研究方向：智能信息处理理论及其在工业数据分析中的应用 Email:lifenglian@tyut.edu.cn
徐利春（1985—），男，四川内江人，副教授，博士，主要研究方向：计算物理、人工智能。
基金资助:
山西省科技合作交流专项(202304041101035)

Abstract

Abstract:

Concerning the problem that the Deep Symbolic Regression （DSR） algorithm， which generates expression trees through Recurrent Neural Network （RNN） automatically， cannot ensure both accuracy and structural simplicity simultaneously， a Double decision mechanism-based DSR （DDSR） algorithm was proposed. Firstly， a dual scoring mechanism was employed to evaluate the accuracy and simplicity of the expression trees comprehensively on the basis of initial RNN decision. Then， reinforcement learning was used to train the expression trees， and Risk Proximal Policy Optimization （RPPO） algorithm was utilized to perform reward feedback， so as to update model parameters of the next batch. Experimental results on public datasets show that compared with DSR algorithm， DDSR algorithm achieves a maximum improvement of 0.396 and a minimum improvement of 0.001 in the coefficient related to fitness， with an average gain of 0.116. The above proves the effectiveness of DDSR algorithm.

Key words: symbolic regression, deep learning, scoring mechanism, proximal policy optimization algorithm, risk-seeking policy gradient

摘要：

深度符号回归（DSR）算法由循环神经网络（RNN）自动化生成表达式树，进而获得较高的模型性能，然而，它无法兼顾表达式树的准确性和结构的简洁性。因此，提出一种基于双重决策机制的深度符号回归（DDSR）算法。首先，在RNN初步决策的基础上，利用双评分机制综合评估表达式树的结构简洁性和准确性。其次，采用强化学习对表达式树生成进行训练，将表达式树生成视为序列决策过程，并利用风险近端策略优化（RPPO）算法进行奖励反馈以更新下一批次的模型参数。在公共数据集上的实验结果表明，相较于DSR算法，DDSR算法在拟合度相关系数上最多提高了0.396，最少提高了0.001，而整体性能提升了0.116。以上证明了DDSR算法的有效性。

关键词: 符号回归, 深度学习, 评分机制, 近端策略优化算法, 风险寻优策略梯度

CLC Number:

TP181

Zeyi GUO, Fenglian LI, Lichun XU. Double decision mechanism-based deep symbolic regression algorithm[J]. Journal of Computer Applications, 2026, 46(2): 406-415.

郭泽一, 李凤莲, 徐利春. 基于双重决策机制的深度符号回归算法[J]. 《计算机应用》唯一官方网站, 2026, 46(2): 406-415.

Figures/Tables 14

Fig. 1 Preorder traversal of expression tree

Tab. 1 Symbols used in the paper and their related definitions

符号	定义
$P (t \| x t, h t)$	RNN生成的基础概率分布
$P i (t \| x t, h t)$	双重决策机制得到的新的概率分布
$α$ ， $β$ ， $γ$	自反馈机制所用的权重因子
$C o m p l e x i t y V a r$	复杂度评分运算公式
$C o m p l e x i t y_m a t r i x$	通过 $C o m p l e x i t y V a r$ 进行求解后得到的复杂度评分矩阵
$R 2_m a t r i x$	通过式（4）求解后得到的 $R 2$ 评分矩阵
$r t θ$	衡量新旧策略差异的比值
$L C L I P θ$	RPPO算法中的裁剪损失
$L e n t r o p y θ$	RPPO算法中的熵损失
$τ i$	表达式树中的第i个节点
T	表达式树的遍历长度
$p t$	当前节点的父节点
$s t$	当前节点的兄节点
L	令牌库，包含所使用运算符
n	样本数
$l r$	学习率

Tab. 1 Symbols used in the paper and their related definitions

符号	定义
$P (t \| x t, h t)$	RNN生成的基础概率分布
$P i (t \| x t, h t)$	双重决策机制得到的新的概率分布
$α$ ， $β$ ， $γ$	自反馈机制所用的权重因子
$C o m p l e x i t y V a r$	复杂度评分运算公式
$C o m p l e x i t y_m a t r i x$	通过 $C o m p l e x i t y V a r$ 进行求解后得到的复杂度评分矩阵
$R 2_m a t r i x$	通过式（4）求解后得到的 $R 2$ 评分矩阵
$r t θ$	衡量新旧策略差异的比值
$L C L I P θ$	RPPO算法中的裁剪损失
$L e n t r o p y θ$	RPPO算法中的熵损失
$τ i$	表达式树中的第i个节点
T	表达式树的遍历长度
$p t$	当前节点的父节点
$s t$	当前节点的兄节点
L	令牌库，包含所使用运算符
n	样本数
$l r$	学习率

Fig. 2 Overall flow of proposed algorithm

Fig. 3 Expression tree generation module

Tab. 2 DDSR algorithm’s parameter setting

参数	参数值
批次大小	100
迭代次数	300
令牌库L	［+，-，×，÷，sin，cos，exp，log，**， sqrt］
学习率	0.05
裁剪阈值	0.1
优化器	Adam
风险阈值	0.8

Tab. 3 Introduction of datasets

数据集	样本数	变量数	是否平衡
nikuradse_2	362	1	否
1027_ESL	488	4	否
210_cloud	108	5	是
feynman_Ⅰ_6_2a	100 000	1	是
strogatz_glider1	400	2	是
523_analcatdata_neavote	100	2	否
feynman_Ⅲ_12_43	100 000	2	是
519_vinnie	380	2	否
厚度数据集	999	1	否

Tab. 4 R2 comparison of DDSR algorithm and benchmark algorithms on experimental datasets

数据集	DSR	GP	FFX	GGGP-STGP	qlattic
nikuradse_2	0.583±0.003	0.707±0.003	0.971±0.002	0.268±0.003	0.978±0.001
1027_ESL	0.679±0.004	0.578±0.004	0.812±0.002	0.802±0.003	0.754±0.002
210_cloud	0.946±0.002	0.946±0.002	0.119±0.005	0.948±0.002	0.887±0.003
feynman_Ⅰ_6_2a	0.966±0.001	0.975±0.001	0.998±0.001	0.943±0.002	0.999±0.001
strogatz_glider1	0.766±0.003	0.976±0.001	0.897±0.002	0.124±0.005	0.886±0.003
523_analcatdata_neavote	0.955±0.002	0.950±0.003	0.943±0.003	0.944±0.002	0.946±0.002
feynman_Ⅲ_12_43	0.945±0.001	0.952±0.001	0.998±0.001	0.982±0.002	0.998±0.001
519_vinnie	0.720±0.003	0.721±0.003	0.643±0.003	0.727±0.003	0.711±0.003
厚度数据集	0.643±0.004	0.708±0.002	0.814±0.002	0.514±0.004	0.892±0.002
数据集	NSGA-DCGP	BSR	PySR	RILS-ROLS	DDSR
nikuradse_2	0.332±0.003	0.409±0.003	0.941±0.001	0.978±0.001	0.979±0.001
1027_ESL	0.679±0.003	0.799±0.002	0.810±0.002	0.831±0.002	0.828±0.002
210_cloud	0.946±0.002	0.836±0.003	0.946±0.002	0.891±0.002	0.950±0.002
feynman_Ⅰ_6_2a	0.964±0.001	0.961±0.001	0.999±0.001	0.999±0.001	0.999±0.001
strogatz_glider1	0.762±0.003	0.129±0.005	0.999±0.001	0.999±0.001	0.956±0.002
523_analcatdata_neavote	0.919±0.003	0.948±0.002	0.936±0.003	0.946±0.002	0.956±0.002
feynman_Ⅲ_12_43	0.998±0.001	0.976±0.001	0.999±0.001	0.999±0.001	0.999±0.001
519_vinnie	0.706±0.003	—	0.706±0.003	0.709±0.003	0.734±0.003
厚度数据集	0.712±0.003	—	0.864±0.002	0.896±0.001	0.845±0.002

Tab. 5 RMSE comparison of DDSR algorithm and benchmark algorithms on experimental datasets

数据集	DSR	GP	FFX	GGGP-STGP	qlattic
nikuradse_2	0.161±0.005	0.135±0.004	0.042±0.002	0.213±0.005	0.036±0.002
1027_ESL	0.709±0.005	0.814±0.005	0.542±0.003	0.557±0.003	0.620±0.004
210_cloud	0.287±0.003	0.287±0.003	1.170±0.005	0.283±0.003	0.417±0.004
feynman_Ⅰ_6_2a	0.012±0.002	0.011±0.001	0.002±0.001	0.016±0.002	0.001±0.001
strogatz_glider1	0.385±0.004	0.124±0.002	0.255±0.003	0.746±0.005	0.268±0.004
523_analcatdata_neavote	0.749±0.005	0.799±0.005	0.844±0.005	0.834±0.005	0.824±0.005
feynman_Ⅲ_12_43	0.190±0.003	0.178±0.003	0.036±0.002	0.109±0.003	0.002±0.001
519_vinnie	1.599±0.005	1.598±0.005	1.808±0.005	1.579±0.005	1.626±0.005
厚度数据集	0.592±0.004	0.536±0.003	0.451±0.002	0.691±0.004	0.325±0.002
数据集	NSGA-DCGP	BSR	PySR	RILS-ROLS	DDSR
nikuradse_2	0.204±0.004	0.191±0.004	0.060±0.003	0.036±0.002	0.035±0.002
1027_ESL	0.709±0.005	0.560±0.003	0.546±0.003	0.514±0.002	0.518±0.002
210_cloud	0.287±0.003	0.504±0.004	0.287±0.003	0.411±0.003	0.277±0.002
feynman_Ⅰ_6_2a	0.013±0.001	0.013±0.001	0.001±0.001	0.001±0.001	0.001±0.001
strogatz_glider1	0.388±0.004	0.743±0.005	0.001±0.001	0.001±0.001	0.162±0.003
523_analcatdata_neavote	1.010±0.005	0.807±0.005	0.898±0.005	0.826±0.005	0.737±0.004
feynman_Ⅲ_12_43	0.032±0.002	0.125±0.002	0.001±0.001	0.001±0.001	0.001±0.001
519_vinnie	1.641±0.005	—	1.641±0.005	1.633±0.005	1.546±0.005
厚度数据集	0.531±0.003	—	0.364±0.002	0.319±0.002	0.391±0.003

Tab. 6 Nguyen symbolic regression’s benchmark test suite

基准测试名称	相关公式	数据样本数	所用标识符
Nguyen-1	$x 3 + x 2 + x$	520	$£ 0$
Nguyen-2	$x 4 + x 3 + x 2 + x$	520	$£ 0$
Nguyen-3	$x 5 + x 4 + x 3 + x 2 + x$	520	$£ 0$
Nguyen-4	$x 6 + x 5 + x 4 + x 3 + x 2 + x$	520	$£ 0$
Nguyen-5	$s i n (x 2) c o s (x) - 1$	520	$£ 0$
Nguyen-6	$s i n (x) + s i n (x + x 2)$	520	$£ 0$
Nguyen-7	$l o g (x + 1) + l o g (x 2 + 1)$	520	$£ 0$
Nguyen-8	$x$	520	$£ 0$ ∪｛sqrt｝
Nguyen-9	$s i n (x) + s i n (y 2)$	1 020	$£ 0$ ∪｛y｝
Nguyen-10	$2 s i n (x) c o s (y)$	1 020	$£ 0$ ∪｛y｝
Nguyen-11	$x y$	1 020	$£ 0$ ∪｛y｝
Nguyen-12	$x 4 - x 3 + 12 y 2 - y$	1 020	$£ 0$ ∪｛y｝

Tab. 6 Nguyen symbolic regression’s benchmark test suite

基准测试名称	相关公式	数据样本数	所用标识符
Nguyen-1	$x 3 + x 2 + x$	520	$£ 0$
Nguyen-2	$x 4 + x 3 + x 2 + x$	520	$£ 0$
Nguyen-3	$x 5 + x 4 + x 3 + x 2 + x$	520	$£ 0$
Nguyen-4	$x 6 + x 5 + x 4 + x 3 + x 2 + x$	520	$£ 0$
Nguyen-5	$s i n (x 2) c o s (x) - 1$	520	$£ 0$
Nguyen-6	$s i n (x) + s i n (x + x 2)$	520	$£ 0$
Nguyen-7	$l o g (x + 1) + l o g (x 2 + 1)$	520	$£ 0$
Nguyen-8	$x$	520	$£ 0$ ∪｛sqrt｝
Nguyen-9	$s i n (x) + s i n (y 2)$	1 020	$£ 0$ ∪｛y｝
Nguyen-10	$2 s i n (x) c o s (y)$	1 020	$£ 0$ ∪｛y｝
Nguyen-11	$x y$	1 020	$£ 0$ ∪｛y｝
Nguyen-12	$x 4 - x 3 + 12 y 2 - y$	1 020	$£ 0$ ∪｛y｝

Tab. 7 Benchmark test suite recovery rates

基准测试	DSR	GP	NSGA-DCGP	PySR	DDSR
平均恢复率	61.0	29.7	21.8	67.5	73.0
Nguyen-1	100	100	33	100	100
Nguyen-2	100	100	10	100	100
Nguyen-3	99	0	0	93	100
Nguyen-4	55	0	0	37	86
Nguyen-5	1	0	7	13	1
Nguyen-6	71	0	4	100	100
Nguyen-7	43	0	0	0	56
Nguyen-8	95	100	81	100	100
Nguyen-9	43	0	27	100	100
Nguyen-10	29	56	16	80	33
Nguyen-11	96	0	84	87	100
Nguyen-12	0	0	0	0	0

Tab. 8 R2 results of ablation experiments

数据集	DSR	DSR-D	DSR-R	DDSR
nikuradse_2	0.683	0.736	0.903	0.979
1027_ESL	0.679	0.695	0.685	0.828
210_cloud	0.946	0.936	0.950	0.950
feynman_Ⅰ_6_2a	0.966	0.974	0.991	0.999
strogatz_glider1	0.766	0.851	0.842	0.956
523_analcatdata_neavote	0.955	0.940	0.940	0.956
feynman_Ⅲ_12_43	0.945	0.948	0.999	0.999
519_vinnie	0.720	0.736	0.723	0.734
厚度数据集	0.643	0.738	0.751	0.845

Tab. 9 RMSE results of ablation experiments

数据集	DSR	DSR-D	DSR-R	DDSR
nikuradse_2	0.141	0.128	0.078	0.035
1027_ESL	0.709	0.692	0.704	0.518
210_cloud	0.287	0.315	0.279	0.277
feynman_Ⅰ_6_2a	0.012	0.011	0.007	0.001
strogatz_glider1	0.385	0.308	0.317	0.162
523_analcatdata_neavote	0.749	0.867	0.867	0.737
feynman_Ⅲ_12_43	0.190	0.185	0.001	0.001
519_vinnie	1.599	1.544	1.592	1.546
厚度数据集	0.592	0.508	0.495	0.391

Tab. 10 Analysis results of ablation experiments’ complexity

数据集	DSR	DSR-D	DSR-R	DDSR
nikuradse_2	32	4 096	∞	1 024
1027_ESL	16	4	20	42
210_cloud	16	10	4	54
feynman_Ⅰ_6_2a	∞	4	∞	9
strogatz_glider1	∞	9	32	12
523_analcatdata_neavote	∞	136	4 096	7
feynman_Ⅲ_12_43	65 536	56	∞	4
519_vinnie	65 536	22	∞	34
厚度数据集	∞	5	∞	5

Tab. 11 Wilcoxon signed-rank test

算法	DDSR（p值）	DDSR（h值）
DSR	0.003 906 25	1
GP	0.019 531 25	1
FFX	0.003 906 25	1
GGGP-STGP	0.003 906 25	1
qlattic	0.035 691 90	1
NSGA-DCGP	0.003 906 25	1
BSR	0.003 906 25	1
PySR	0.398 024 72	0
RILS-ROLS	0.735 316 69	0

References 28

[1]	MAKKE N， CHAWLA S. Interpretable scientific discovery with symbolic regression： a review［J］. Artificial Intelligence Review， 2024， 57： No.2.
[2]	WITTENBERG D， ROTHLAUF F， GAGNÉ C. Denoising autoencoder genetic programming： strategies to control exploration and exploitation in search［J］. Genetic Programming and Evolvable Machines， 2023， 24： No.17.
[3]	许鹏程，何磊，李川，等. 基于Transformer的深度符号回归方法［J］. 计算机应用， 2025， 45（4）： 1455-1463.
	XU P C， HE L， LI C， et al. Deep symbolic regression method based on Transformer［J］. Journal of Computer Applications， 2025， 45（4）： 1455-1463.
[4]	LA CAVA W， ORZECHOWSKI P， BURLACU B， et al. Contemporary symbolic regression methods and their relative performance［EB/OL］. ［2025-04-03］..
[5]	刘博弈，王海渝，龚严，等. 基于天气预报和符号回归算法的参考作物腾发量预测研究［J］. 中国农村水利水电， 2018（8）： 22-26.
	LIU B Y， WANG H Y， GONG Y， et al. Predicting daily reference evapotranspiration based on weather forecast data using symbolic regression method［J］. China Rural Water and Hydropower， 2018（8）： 22-26.
[6]	ANGELIS D， SOFOS F， KARAKASIDIS T E. Artificial intelligence in physical sciences： symbolic regression trends and perspectives［J］. Archives of Computational Methods in Engineering， 2023， 30（6）： 3845-3865.
[7]	AGHBASHLO M， PENG W， TABATABAEI M， et al. Machine learning technology in biodiesel research： a review［J］. Progress in Energy and Combustion Science， 2021， 85： No.100904.
[8]	PETERSEN B K， LANDAJUELA M， MUNDHENK T N， et al. Deep symbolic regression： recovering mathematical expressions from data via risk-seeking policy gradients［EB/OL］. ［2025-04-03］..
[9]	谢树钦，陈梓天，徐超，等. 针对不可微多阶段算法的环境升级式强化学习方法［J］. 重庆邮电大学学报（自然科学版）， 2020， 32（5）： 857-866.
	XIE S Q， CHEN Z T， XU C， et al. Environment upgrade reinforcement learning for non-differentiable multi-stage pipelines［J］. Journal of Chongqing University of Posts and Telecommunications （Natural Science Edition）， 2020， 32（5）： 857-866.
[10]	KIM S， LU P Y， MUKHERJEE S， et al. Integration of neural network-based symbolic regression in deep learning for scientific discovery［J］. IEEE Transactions on Neural Networks and Learning Systems， 2021， 32（9）： 4166-4177.
[11]	BIGGIO L， BENDINELLI T， NEITZ A， et al. Neural symbolic regression that scales［C］// Proceedings of the 38th International Conference on Machine Learning. New York： JMLR.org， 2021： 936-945.
[12]	CRANMER M， SANCHEZ-GONZALEZ A， BATTAGLIA P， et al. Discovering symbolic models from deep learning with inductive biases［C］// Proceedings of the 34th International Conference on Neural Information Processing Systems. Red Hook： Curran Associates Inc.， 2020： 17429-17442.
[13]	ABOLAFIA D A， NOROUZI M， SHEN J， et al. Neural program synthesis with priority queue training［EB/OL］. ［2025-04-03］..
[14]	BASTIANI Z， KIRBY R M， HOCHHALTER J， et al. Complexity-aware deep symbolic regression with robust risk-seeking policy gradients［EB/OL］. ［2025-04-03］..
[15]	LANDAJUELA M， PETERSEN B K， KIM S， et al. Discovering symbolic policies with deep reinforcement learning［C］// Proceedings of the 38th International Conference on Machine Learning. New York： JMLR.org， 2021： 5979-5989.
[16]	SCHULMAN J， WOLSKI F， DHARIWAL P， et al. Proximal policy optimization algorithms［EB/OL］. ［2025-04-03］..
[17]	QU Z， WEI X M， CHEN S Q. An algorithm of image mosaic based on binary tree and eliminating distortion error［J］. PLoS ONE， 2019， 14（1）： No.e0210354.
[18]	KOMMENDA M， BEHAM A， AFFENZELLER M， et al. Complexity measures for multi-objective symbolic regression［C］// Proceedings of the 2015 International Conference on Computer Aided Systems Theory， LNCS 9520. Cham： Springer， 2015： 409-416.
[19]	ROMANO J D， LE T T， LA CAVA W， et al. PMLB v1.0： an open-source dataset collection for benchmarking machine learning methods［J］. Bioinformatics， 2022， 38（3）： 878-880.
[20]	ZENG P， SONG X， LENSEN A， et al. Differentiable genetic programming for high-dimensional symbolic regression［EB/OL］. ［2025-04-03］..
[21]	VADDIREDDY H， SAN O. Equation discovery using fast function extraction： a deterministic symbolic regression approach［J］. Fluids， 2019， 4（2）： No.111.
[22]	ESPADA G， INGELSE L， CANELAS P， et al. Data types as a more ergonomic frontend for grammar-guided genetic programming［C］// Proceedings of the 21st ACM SIGPLAN International Conference on Generative Programming： Concepts and Experiences. New York： ACM， 2022： 86-94.
[23]	WILSTRUP C， KASAK J. Symbolic regression outperforms other models for small data sets［EB/OL］. ［2025-04-03］..
[24]	JIN Y， FU W， KANG J， et al. Bayesian symbolic regression［EB/OL］. ［2025-04-03］..
[25]	CRANMER M. Interpretable machine learning for science with PySR and SymbolicRegression.jl［EB/OL］. ［2025-04-03］..
[26]	KARTELJ A， DJUKANOVIĆ M. RILS-ROLS： robust symbolic regression via iterated local search and ordinary least squares［J］. Journal of Big Data， 2023， 10（1）： No.71.
[27]	ELYASAF A， SIPPER M. Software review： the HeuristicLab framework［J］. Genetic Programming and Evolvable Machines， 2014， 15（2）： 215-218.
[28]	MEURER A， SMITH C P， PAPROCKI M， et al. SymPy： symbolic computing in Python［J］. PeerJ Computer Science， 2017， 3： No.e103.

Double decision mechanism-based deep symbolic regression algorithm

基于双重决策机制的深度符号回归算法

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