Abstract: Addressing the issues of reduced population diversity and sensitivity to hyperparameters in solving symbolic regression problems with genetic evolution algorithms, a Deep Symbolic Regression Technique (DSRT) based on Transformer was introduced. This approach leveraged the autoregressive capability of Transformer to generate expression symbol sequences. Subsequently, the transformed value of the fitness score between the data and the expression symbol sequence serves as a reward, which was employed to update the model parameters through deep reinforcement learning. This process enabled the model to output expression sequences that better align with the data, and as the model progressively converges, the optimal expression was identified. An evaluation of the DSRT method's effectiveness was conducted on the Nguyen symbolic regression benchmark dataset, where it was compared with DSR (Deep Symbolic Regression) and GP (Genetic Programming) algorithms within 200 iterations. Experimental results confirmed the validity of the DSRT approach. Additionally, the impact of various parameters on the DSRT method was discussed, and an experiment to predict the formula for the surface pressure coefficient of an aircraft airfoil using the NACA4421 dataset was performed. The obtained formula was compared with the Kármán-Tsien formula, yielding a mathematical formula with a lower RMSE.

Key words: symbolic regression, Transformer, deep reinforcement learning, NACA4412, Karman-Tsien formula

摘要： 针对利用遗传进化算法解决符号回归问题时存在的种群多样性减少、对超参数敏感等问题，提出基于Transformer的深度符号回归（DSRT）方法，该方法通过Transformer自回归的方式生成表达式符号序列，然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，通过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Nguyen符号回归基准数据集上进行了有效性测试，在200次迭代数内与DSR（Deep symbolic regression）,和GP(Genetic programming)算法做了对比，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，且NACA4421数据上进行了飞机翼型表面压力系数公式预测实验，将得到的公式与卡门-钱学森公式作对比，找到了RMSE较小的数学公式。

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