Deep symbolic regression method based on Transformer

doi:10.11772/j.issn.1001-9081.2024050609

Journal of Computer Applications ›› 2025, Vol. 45 ›› Issue (5): 1455-1463.DOI: 10.11772/j.issn.1001-9081.2024050609

• Artificial intelligence • Previous Articles

Deep symbolic regression method based on Transformer

Pengcheng XU¹^,², Lei HE²(), Chuan LI¹, Weiqi QIAN², Tun ZHAO²

^1.College of Computer Science，Sichuan University，Chengdu Sichuan 610065，China
^2.State Key Laboratory of Aerodynamics （China Aerodynamics Research and Development Center），Mianyang Sichuan 621000，China

Received:2024-05-15 Revised:2024-09-21 Accepted:2024-09-29 Online:2024-10-09 Published:2025-05-10
Contact: Lei HE
About author:XU Pengcheng， born in 1999， M. S. candidate. His research interests include deep reinforcement learning， deep symbolic regression.
HE Lei， born in 1988， Ph. D.， associate research fellow. His research interests include machine learning， pneumatic modeling.
LI Chuan， born in 1977， Ph. D.， associate professor. His research interests include deep reinforcement learning， data mining， knowledge engineering.
QIAN Weiqi， born in 1973， Ph. D.， research fellow. His research interests include aerodynamics， aerodynamic parameter identification.
ZHAO Tun， born in 1989， Ph. D.， associate research fellow. His research interests include flight control， artificial intelligence.
Supported by:
Intellectual Strength Foundation （ISF） Project for Talent Excellence

基于Transformer的深度符号回归方法

许鹏程¹^,², 何磊²(), 李川¹, 钱炜祺², 赵暾²

^1.四川大学计算机学院，成都 610065
^2.空天飞行空气动力科学与技术全国重点实验室（中国空气动力研究与发展中心），四川绵阳 621000

通讯作者: 何磊
作者简介:许鹏程（1999—），男，重庆人，硕士研究生，主要研究方向：深度强化学习、深度符号回归
何磊（1988—），男，四川绵竹人，副研究员，博士，主要研究方向：机器学习、气动建模
李川（1977—），男，河南郑州人，副教授，博士，主要研究方向：深度强化学习、数据挖掘、知识工程
钱炜祺（1973—），男，江苏无锡人，研究员，博士生导师，博士，主要研究方向：空气动力学、气动参数辨识
赵暾（1989—），男，甘肃西和人，副研究员，博士，主要研究方向：飞行控制、人工智能。
基金资助:
智强基金卓越人才项目

Abstract

Abstract:

To address the challenges of reduced population diversity and sensitivity to hyperparameters in solving Symbolic Regression （SR） problems by using genetic evolutionary algorithms， a Deep Symbolic Regression Technique （DSRT） method based on Transformer was proposed. This method employed autoregressive capability of Transformer to generate expression symbol sequence. Subsequently， the transformation of the fitness value between the data and the expression symbol sequence was served as a reward value， and the model parameters were updated through deep reinforcement learning， so that the model was able to output expression sequence that fitted the data better， and with the model’s continuous converging， the optimal expression was identified. The effectiveness of the DSRT method was validated on the SR benchmark dataset Nguyen， and it was compared with DSR （Deep Symbolic Regression） and GP （Genetic Programming） algorithms within 200 iterations. Experimental results confirm the validity of DSRT method. Additionally， the influence of various parameters on DSRT method was discussed， and an experiment to predict the formula for surface pressure coefficient of an aircraft airfoil using NACA4421 dataset was performed. The obtained formula was compared with the Kármán-Tsien formula， yielding a mathematical formula with a lower Root Mean Square Error （RMSE）.

Key words: Symbolic Regression (SR), Transformer, deep reinforcement learning, NACA4412, Kármán-Tsien formula

摘要：

针对利用遗传进化算法解决符号回归（SR）问题时存在的种群多样性降低以及对超参数敏感等问题，提出基于Transformer的深度符号回归（DSRT）方法。该方法在利用Transformer自回归的方式生成表达式符号序列后，将数据和表达式符号序列的拟合度值的变换值当作奖励值，再利用深度强化学习的方法更新模型参数，使模型输出的表达式序列更加拟合数据，并随着模型的不断收敛找出最优的表达式。在SR基准数据集Nguyen上对DSRT方法进行有效性测试，并在200次迭代内将它与DSR（Deep Symbolic Regression）和GP（Genetic Programming）算法进行对比，实验结果验证了DSRT方法的有效性。另外，讨论了各参数对DSRT方法的影响，并在NACA4421数据上进行飞机翼型表面压力系数公式预测实验，将所得到的公式与卡门-钱学森公式作对比，找到了均方根误差（RMSE）较小的数学公式。

关键词: 符号回归, Transformer, 深度强化学习, NACA4412, 卡门-钱学森公式

CLC Number:

TP181

Pengcheng XU, Lei HE, Chuan LI, Weiqi QIAN, Tun ZHAO. Deep symbolic regression method based on Transformer[J]. Journal of Computer Applications, 2025, 45(5): 1455-1463.

许鹏程, 何磊, 李川, 钱炜祺, 赵暾. 基于Transformer的深度符号回归方法[J]. 《计算机应用》唯一官方网站, 2025, 45(5): 1455-1463.

Figures/Tables 22

Fig. 1 Basic framework of RL

Fig. 2 Expression tree and its preorder traversal sequence

Fig. 3 Overall framework of DSRT

Fig. 4 Schematic diagram of expression generator at a moment

Fig. 5 Conceptual diagram of state feature acquisition at moment t5

Fig. 6 One-hot coding for mathematical symbol

Fig. 7 Coding representation for state feature

Fig. 8 Schematic diagram of mathematical expression generation

Fig. 9 Reinforcement learning update sequence

Fig. 10 Schematic diagram of training process of one-round update of entire model

Tab. 1 Parameter setting of DSRT algorithm

参数	参数值
候选符号库	［+，-，*，/，sin，exp，log，1，f］
前馈神经网络维度	128
编码器解码器层数	3
学习率	0.002 5

Tab. 2 Parameter setting of GP algorithm

参数名	参数值	参数名	参数值
初始种群	5 000	子树突变率	0.1
迭代数	200	提升突变率	0.05
交叉率	0.7	点突变率	0.1

Tab. 3 Comparison test’s RMSE results of DSRT vs DSR，GP

Benchmark	表达式	取值范围	RMSE/%
Benchmark	表达式	取值范围	DSRT	DSR	GP
Nguyen-1	$x 3 + x 2 + x$	U（-1，1， 20）	0.00	0.00	0.00
Nguyen-2	$x 4 + x 3 + x 2 + x$		0.00	0.00	0.00
Nguyen-3	$x 5 + x 4 + x 3 + x 2 + x$		0.00	0.00	0.00
Nguyen-4	$x 6 + x 5 + x 4 + x 3 + x 2 + x$		0.00	0.00	9.21
Nguyen-5	$s i n (x 2) c o s (x) - 1$		1.17	1.77	7.16
Nguyen-6	$s i n (x) + s i n (x + x 2)$		0.00	0.00	5.70
Nguyen-7	$l o g (x + 1) + l o g (x 2 + 1)$	U（0，2，20）	3.09	2.38	13.92
Nguyen-8	$x$	U（0，4，20）	0.00	0.00	0.00
Nguyen-9	$s i n (x) + s i n (y 2)$	U（-1，1，100）	1.24	0.00	0.00
Nguyen-10	$2 s i n (x) c o s (y)$	U（-1，1，100）	0.00	0.00	0.00
Nguyen-11	$x y$	U（0，1， 100）	0.00	1.29	6.22
Nguyen-12	$x 4 - x 3 + 12 y 2 - y$	U（0，1， 100）	2.53	1.41	5.36

Tab. 3 Comparison test’s RMSE results of DSRT vs DSR，GP

Benchmark	表达式	取值范围	RMSE/%
Benchmark	表达式	取值范围	DSRT	DSR	GP
Nguyen-1	$x 3 + x 2 + x$	U（-1，1， 20）	0.00	0.00	0.00
Nguyen-2	$x 4 + x 3 + x 2 + x$		0.00	0.00	0.00
Nguyen-3	$x 5 + x 4 + x 3 + x 2 + x$		0.00	0.00	0.00
Nguyen-4	$x 6 + x 5 + x 4 + x 3 + x 2 + x$		0.00	0.00	9.21
Nguyen-5	$s i n (x 2) c o s (x) - 1$		1.17	1.77	7.16
Nguyen-6	$s i n (x) + s i n (x + x 2)$		0.00	0.00	5.70
Nguyen-7	$l o g (x + 1) + l o g (x 2 + 1)$	U（0，2，20）	3.09	2.38	13.92
Nguyen-8	$x$	U（0，4，20）	0.00	0.00	0.00
Nguyen-9	$s i n (x) + s i n (y 2)$	U（-1，1，100）	1.24	0.00	0.00
Nguyen-10	$2 s i n (x) c o s (y)$	U（-1，1，100）	0.00	0.00	0.00
Nguyen-11	$x y$	U（0，1， 100）	0.00	1.29	6.22
Nguyen-12	$x 4 - x 3 + 12 y 2 - y$	U（0，1， 100）	2.53	1.41	5.36

Tab. 4 Influence of encoder and decoder layers on results

编码器解码器层数	最小RMSE出现时间/s	出现轮数	最小MSE/%	数学公式
1	1 096.13	15	2.52	$- y - c o s (y) + e - x (0.633 - c o s (1))$
2	2 415.91	29	3.86	$c o s (y + c o s (0.90) + c o s (y))$
3	803.15	11	3.96	$0.96 - y - c o s (y)$
4	696.76	9	4.07	$- 2.26 + e c o s (c o s (- 2.26 - y))$
7	539.50	7	3.96	$1.04 - y - c o s (y)$

Tab. 4 Influence of encoder and decoder layers on results

编码器解码器层数	最小RMSE出现时间/s	出现轮数	最小MSE/%	数学公式
1	1 096.13	15	2.52	$- y - c o s (y) + e - x (0.633 - c o s (1))$
2	2 415.91	29	3.86	$c o s (y + c o s (0.90) + c o s (y))$
3	803.15	11	3.96	$0.96 - y - c o s (y)$
4	696.76	9	4.07	$- 2.26 + e c o s (c o s (- 2.26 - y))$
7	539.50	7	3.96	$1.04 - y - c o s (y)$

Tab. 5 Influence of feedforward neural network dimensionality on results

前馈神经网络维度	最小RMSE出现时间/s	出现轮数	最小RMSE/%	数学公式
16	2 037.09	26	3.91	$l o g 0.53 c o s (1 - y) c o s 0.53$
32	1 654.32	22	3.92	$2.052 - y - c o s (y)$
64	459.43	6	3.75	$- s i n y e x 1.046 + y e x$
128	2 085.66	21	3.00	$- - 3.16 y - 4.16 y + x + y - e$
256	1 593.14	21	3.06	$s i n c o s l o g - 0.22 x - y - 1 0.22$
512	2 098.43	27	2.58	$- 0.08 x - y + c o s (- 0.08) - c o s (y)$
1 024	1 993.78	25	3.12	$y l o g (- 0.31 (3 x + y + 1) + x)$
2 048	1 937.79	26	3.85	$- 1.94 c o s (1) c o s (y) - y + 1$

Tab. 5 Influence of feedforward neural network dimensionality on results

前馈神经网络维度	最小RMSE出现时间/s	出现轮数	最小RMSE/%	数学公式
16	2 037.09	26	3.91	$l o g 0.53 c o s (1 - y) c o s 0.53$
32	1 654.32	22	3.92	$2.052 - y - c o s (y)$
64	459.43	6	3.75	$- s i n y e x 1.046 + y e x$
128	2 085.66	21	3.00	$- - 3.16 y - 4.16 y + x + y - e$
256	1 593.14	21	3.06	$s i n c o s l o g - 0.22 x - y - 1 0.22$
512	2 098.43	27	2.58	$- 0.08 x - y + c o s (- 0.08) - c o s (y)$
1 024	1 993.78	25	3.12	$y l o g (- 0.31 (3 x + y + 1) + x)$
2 048	1 937.79	26	3.85	$- 1.94 c o s (1) c o s (y) - y + 1$

Tab. 6 Influence of learning rate on results

学习率	最小RMSE出现时间/s	出现轮数	最小RMSE/%	数学公式
0.000 1	638.73	9	3.84	$s i n (0.28 - e 0.28) (y + c o s (y)) + 1$
0.000 5	439.74	5	3.87	$c o s y + c o s 0.90 + c o s (y)$
0.001 0	1 914.95	26	3.86	$c o s l o g - 0.772 0.77 - e 0.77 + c o s (y - 1)$
0.002 0	115.03	8	3.80	$c o s 0.63 + y + c o s (y)$
0.002 5	1 936.37	17	3.35	$- 0.63 l o g - x + y 0.40 + e s i n (x)$
0.005 0	2 041.99	28	3.86	$s i n - 1.28 + 1.28 e + y + c o s (y)$
0.010 0	827.10	10	3.87	$c o s 0.63 + y + c o s (y)$

Tab. 6 Influence of learning rate on results

学习率	最小RMSE出现时间/s	出现轮数	最小RMSE/%	数学公式
0.000 1	638.73	9	3.84	$s i n (0.28 - e 0.28) (y + c o s (y)) + 1$
0.000 5	439.74	5	3.87	$c o s y + c o s 0.90 + c o s (y)$
0.001 0	1 914.95	26	3.86	$c o s l o g - 0.772 0.77 - e 0.77 + c o s (y - 1)$
0.002 0	115.03	8	3.80	$c o s 0.63 + y + c o s (y)$
0.002 5	1 936.37	17	3.35	$- 0.63 l o g - x + y 0.40 + e s i n (x)$
0.005 0	2 041.99	28	3.86	$s i n - 1.28 + 1.28 e + y + c o s (y)$
0.010 0	827.10	10	3.87	$c o s 0.63 + y + c o s (y)$

Tab. 7 NACA4412 real data

$M a ∞$	$C p$	$M a ∞$	$C p$
0.069 282	-0.626 16	0.350 32	-0.689 80
0.098 867	-0.601 15	0.484 85	-0.790 64
0.188 260	-0.636 57	0.531 01	-0.793 30
0.220 030	-0.622 63	0.569 55	-0.816 01
0.226 720	-0.664 31	0.603 83	-0.801 06
0.258 940	-0.626 75	0.587 74	-0.893 89
0.292 330	-0.667 03	0.629 72	-0.853 86
0.331 560	-0.647 07

Tab. 7 NACA4412 real data

$M a ∞$	$C p$	$M a ∞$	$C p$
0.069 282	-0.626 16	0.350 32	-0.689 80
0.098 867	-0.601 15	0.484 85	-0.790 64
0.188 260	-0.636 57	0.531 01	-0.793 30
0.220 030	-0.622 63	0.569 55	-0.816 01
0.226 720	-0.664 31	0.603 83	-0.801 06
0.258 940	-0.626 75	0.587 74	-0.893 89
0.292 330	-0.667 03	0.629 72	-0.853 86
0.331 560	-0.647 07

Tab. 8 Parameter setting of DSRT algorithm in NACA4421 experiments

参数	参数值
候选符号库	［+，-，*，/，sin，cos，exp，log，1，f］
前馈神经网络维度	256
编码器解码器层数	1
学习率	0.002 5

Fig. 11 NACA4412 data results fitted by DSRT and Kármán-Tsien formula

Fig. 12 NACA4412 data results fitted by DSRT and Kármán-Tsien formula considering physical priori

Tab. 9 Situations of interpolation

序号	公式	RMSE/10^-2
序号	公式	DSRT	卡门-钱学森公式
1	$- 1.614 + M a ∞ - 0.614 e 1 M a ∞ - c o s (C p I) e 1 M a ∞ - c o s (C p I)$	1.19	4.62
2	$C p I (1.939 - C p I) e C p I - M a ∞ 2 e - x 1.939 - C p I$	1.15	3.74
3	$l o g - 0.815 - C p I c o s (M a ∞) - 2 C p I - e - 0.815$	2.19	3.23

Tab. 9 Situations of interpolation

序号	公式	RMSE/10^-2
序号	公式	DSRT	卡门-钱学森公式
1	$- 1.614 + M a ∞ - 0.614 e 1 M a ∞ - c o s (C p I) e 1 M a ∞ - c o s (C p I)$	1.19	4.62
2	$C p I (1.939 - C p I) e C p I - M a ∞ 2 e - x 1.939 - C p I$	1.15	3.74
3	$l o g - 0.815 - C p I c o s (M a ∞) - 2 C p I - e - 0.815$	2.19	3.23

Fig. 13 Three interpolation fittings

References 29

1	KRIZHEVSKY A， SUTSKEVER I， HINTON E G. ImageNet classification with deep convolutional neural networks［J］. Communications of the ACM， 2017， 60（6）：84-90.
2	HE K， ZHANG X， REN S， et al. Spatial pyramid pooling in deep convolutional networks for visual recognition［J］. IEEE Transactions on Pattern Analysis and Machine Intelligence， 2015， 37（9）：1904-1916.
3	THIRUNAVUKARASU A J， TING D S J， ELANGOVAN K， et al. Large language models in medicine［J］. Nature Medicine， 2023， 29（8）： 1930-1940.
4	WU L， LI T， WANG L， et al. Improving hybrid CTC/Attention architecture with time-restricted self-attention CTC for end-to-end speech recognition［J］. Applied Sciences， 2019， 9（21）： No.4639.
5	ZHOU G， ZHU X， SONG C， et al. Deep interest network for click-through rate prediction［C］// Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. New York： ACM， 2018： 1059-1068.
6	WANG H， FU T， DU Y， et al. Scientific discovery in the age of artificial intelligence［J］. Nature， 2023， 620（7972）：47-60.
7	LU Q， REN J， WANG Z. Using genetic programming with prior formula knowledge to solve symbolic regression problem［J］. Computational Intelligence and Neuroscience， 2016， 2016： No.1021378.
8	KOZA J R. Genetic programming： a paradigm for genetically breeding populations of computer programs to solve problems［M］. Cambridge： MIT Press， 1992：12-20.
9	VIRGOLIN M， ALDERLIESTEN T， WITTEVEEN C， et al. Improving model-based genetic programming for symbolic regression of small expressions［J］. Evolutionary Computation， 2020， 29（2）： 211-237.
10	ZHOU J， FENG L， CAI W， et al. Multifactorial genetic programming for symbolic regression problems［J］. IEEE Transactions on Systems， Man， and Cybernetics： Systems， 2020， 50（11）： 4492-4505.
11	AL-HELALI B， CHEN Q， XUE B， et al. Multitree genetic programming with new operators for transfer learning in symbolic regression with incomplete data［J］. IEEE Transactions on Evolutionary Computation， 2021， 25（6）： 1049-1063.
12	VLADISLAVLEVA E J， SMITS G F， DEN HERTOG D. Order of nonlinearity as a complexity measure for models generated by symbolic regression via Pareto genetic programming［J］. IEEE Transactions on Evolutionary Computation， 2009， 13（2）： 333-349.
13	DUBČÁKOVÁ R. Eureqa： software review［J］. Genetic Programming and Evolvable Machines， 2011， 12（2）：173-178.
14	UDRESCU S M， TAN A， FENG J， et al. AI Feynman 2.0： Pareto-optimal symbolic regression exploiting graph modularity［C］// Proceedings of the 34th International Conference on Neural Information Processing Systems. Red Hook： Curran Associates Inc.， 2020： 4860-4871.
15	鲁强，张洋. 基于蒙特卡洛树搜索的符号回归算法［J］. 计算机工程与设计， 2020， 41（8）：2158-2164.
	LU Q， ZHANG Y. Solving symbol regression based on Monte Carlo tree search［J］. Computer Engineering and Design， 2020， 41（8）： 2158-2164.
16	ALAA A M， VAN DER SCHAAR M. Demystifying black-box models with symbolic metamodels［C］// Proceedings of the 33rd International Conference on Neural Information Processing Systems. Red Hook： Curran Associates Inc.， 2019：11304-11314.
17	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.
18	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］. ［2024-04-05］..
19	MUNDHENK T N， LANDAJUELA M， GLATT R， et al. Symbolic regression via neural-guided genetic programming population seeding［C］// Proceedings of the 35th International Conference on Neural Information Processing Systems. Red Hook： Curran Associates Inc.， 2021： 24912-24923.
20	KAMIENNY P A， D'ASCOLI S， LAMPLE G， et al. End-to-end symbolic regression with Transformers［C］// Proceedings of the 36th International Conference on Neural Information Processing Systems. Red Hook： Curran Associates Inc.， 2022：10269-10281.
21	VASTL M， KULHÁNEK J， KUBALÍK J， et al. SymFormer： end-to-end symbolic regression using Transformer-based architecture［J］. IEEE Access， 2024， 12： 37840-37849.
22	LI W， LI W， SUN L， et al. Transformer-based model for symbolic regression via joint supervised learning［EB/OL］. ［2024-08-22］..
23	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.
24	VASWANI A， SHAZEER N， PARMAR N， et al. Attention is all you need［C］// Proceedings of the 31st International Conference on Neural Information Processing Systems. Red Hook： Curran Associates Inc.， 2017： 6000-6010.
25	ELMAN J L. Finding structure in time［J］. Cognitive Science， 1990， 14（2）： 179-211.
26	LeCUN Y， BOTTOU L， BENGIO Y， et al. Gradient-based learning applied to document recognition［J］. Proceedings of the IEEE， 1998， 86（11）： 2278-2324.
27	CHAMBERS L G. Review： Practical methods of optimization （2nd ed.）［J］.The Mathematical Gazette， 2001， 85（504）： 562-563.
28	PETERSEN B K， SANTIAGO C P， LANDAJUELA M. Incorporating domain knowledge into neural-guided search［EB/OL］. ［2024-08-02］..
29	WILLIAMS R J. Simple statistical gradient-following algorithms for connectionist reinforcement learning［J］. Machine Learning， 1992， 8（3/4）： 229-256.

[1]	Hui LI, Bingzhi JIA, Chenxi WANG, Ziyu DONG, Jilong LI, Zhaoman ZHONG, Yanyan CHEN. Generative adversarial network underwater image enhancement model based on Swin Transformer [J]. Journal of Computer Applications, 2025, 45(5): 1439-1446.
[2]	Pengyu CHEN, Xiushan NIE, Nanjun LI, Tuo LI. Semi-supervised video object segmentation method based on spatio-temporal decoupling and regional robustness enhancement [J]. Journal of Computer Applications, 2025, 45(5): 1379-1386.
[3]	Dingmu YANG, Longqiang NI, Jing LIANG, Zhaoyuan QIU, Yongzhen ZHANG, Zhiqiang QI. Protocol conversion method based on semantic similarity [J]. Journal of Computer Applications, 2025, 45(4): 1263-1270.
[4]	Baohua YUAN, Jialu CHEN, Huan WANG. Medical image segmentation network integrating multi-scale semantics and parallel double-branch [J]. Journal of Computer Applications, 2025, 45(3): 988-995.
[5]	Jing WANG, Xuming FANG. Intelligent joint power and channel allocation algorithm for Wi-Fi7 multi-link integrated communication and sensing [J]. Journal of Computer Applications, 2025, 45(2): 563-570.
[6]	Yalun WANG, Yangsen ZHANG, Siwen ZHU. Headline generation model with position embedding for knowledge reasoning [J]. Journal of Computer Applications, 2025, 45(2): 345-353.
[7]	Huahua WANG, Liang HUANG, Jiajie CHEN, Jiening FANG. Dynamic allocation algorithm for multi-beam subcarriers of low orbit satellites based on deep reinforcement learning [J]. Journal of Computer Applications, 2025, 45(2): 571-577.
[8]	Jietao LIANG, Bing LUO, Lanhui FU, Qingling CHANG, Nannan LI, Ningbo YI, Qi FENG, Xin HE, Fuqin DENG. Point cloud registration method based on coordinate geometric sampling [J]. Journal of Computer Applications, 2025, 45(1): 214-222.
[9]	Zijun MIAO, Fei LUO, Weichao DING, Wenbo DONG. Traffic signal control algorithm based on overall state prediction and fair experience replay [J]. Journal of Computer Applications, 2025, 45(1): 337-344.
[10]	Yunchuan HUANG, Yongquan JIANG, Juntao HUANG, Yan YANG. Molecular toxicity prediction based on meta graph isomorphism network [J]. Journal of Computer Applications, 2024, 44(9): 2964-2969.
[11]	Xin YANG, Xueni CHEN, Chunjiang WU, Shijie ZHOU. Short-term traffic flow prediction of urban highway based on variant residual model and Transformer [J]. Journal of Computer Applications, 2024, 44(9): 2947-2951.
[12]	Jinjin LI, Guoming SANG, Yijia ZHANG. Multi-domain fake news detection model enhanced by APK-CNN and Transformer [J]. Journal of Computer Applications, 2024, 44(9): 2674-2682.
[13]	Jieru JIA, Jianchao YANG, Shuorui ZHANG, Tao YAN, Bin CHEN. Unsupervised person re-identification based on self-distilled vision Transformer [J]. Journal of Computer Applications, 2024, 44(9): 2893-2902.
[14]	Liehong REN, Lyuwen HUANG, Xu TIAN, Fei DUAN. Multivariate long-term series forecasting method with DFT-based frequency-sensitive dual-branch Transformer [J]. Journal of Computer Applications, 2024, 44(9): 2739-2746.
[15]	Jiepo FANG, Chongben TAO. Hybrid internet of vehicles intrusion detection system for zero-day attacks [J]. Journal of Computer Applications, 2024, 44(9): 2763-2769.

Deep symbolic regression method based on Transformer

基于Transformer的深度符号回归方法

RichHTML

PDF

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 22

References 29

Related Articles 15

Recommended Articles

Metrics