Symbolic regression (SR) is the process of finding an unknown mathematical expression given the input and output and has important applications in interpretable machine learning and knowledge discovery. The major difficulty of SR is that finding the expression structure is an NP-hard problem, which makes the entire process time-consuming. In this study, the solution of expression structures was regarded as a classification problem and solved by supervised learning such that SR can be solved quickly by using the solving experience. Techniques for classification tasks, such as equivalent label merging and sample balance, were used to enhance the robustness of the algorithm. We proposed a symbolic network called DeepSymNet to represent symbolic expressions to improve the performance of the algorithm. DeepSymNet has been proven to have a strong representation ability with a shorter label compared to the current popular representation methods, reducing the search space when predicting. Moreover, DeepSymNet conveniently decomposes SR into two smaller subproblems, which makes solving the problem easier. The proposed algorithm was tested on artificially generated expressions and public datasets and compared with other algorithms. The results demonstrate the effectiveness of the proposed algorithm.

Mathematical formulas are the language of communication between humans and nature. Discovering latent formulas from observed data is an important challenge in artificial intelligence, commonly known as symbolic regression(SR). The current mainstream SR algorithms regard SR as a combinatorial optimization problem and use Genetic Programming (GP) or Reinforcement Learning (RL) to solve the SR problem. These methods perform well on simple problems, but poorly on slightly more complex tasks. In addition, this class of algorithms ignores an important aspect: in SR tasks, symbols have explicit numerical meaning. So can we take full advantage of this important property and try to solve the SR problem with more efficient numerical optimization methods? Extrapolation and Learning Equation (EQL) replaces activation functions in neural networks with basic symbols and sparsifies connections to derive a simplified expression from a large network. However, EQL's fixed network structure can't adapt to the complexity of different tasks, often resulting in redundancy or insufficient, limiting its effectiveness. Based on the above analysis, we propose MetaSymNet, a tree-like network that employs the PANGU meta-function as its activation function. PANGU meta-function can evolve into various candidate functions during training. The network structure can also be adaptively adjusted according to different tasks. Then the symbol network evolves into a concise, interpretable mathematical expression. To evaluate the performance of MetaSymNet and five baseline algorithms, we conducted experiments across more than ten datasets, including SRBench. The experimental results show that MetaSymNet has achieved relatively excellent results on various evaluation metrics.

Two series of laser diodes with different AlGaN cladding layers were investigated, and it was found that the internal absorption loss was reduced by using a u-AlGaN and p-AlGaN composite upper cladding layer, and the performance of LD was improved. The optimized thickness of u-AlGaN is related to the waveguide layer structure of LDs. For LDs with a thick waveguide layer, the u-AlGaN cladding layer should be thinner to achieve high hole concentration and high hole injection efficiency. For LDs with a thin waveguide layer, the u-AlGaN cladding layer can be thick to suppress the internal absorption loss. Thus, a composite upper waveguide structure with a thinner waveguide layer, a thicker u-AlGaN cladding layer, and a p-AlGaN cladding layer is an optimized structure for short waveguide UV LDs due to lower carrier loss and absorption loss.