Mathematical formulas serve as the means of communication between humans and nature, encapsulating the operational laws governing natural phenomena. The concise formulation of these laws is a crucial objective in scientific research and an important challenge for artificial intelligence (AI). While traditional artificial neural networks (MLP) excel at data fitting, they often yield uninterpretable black box results that hinder our understanding of the relationship between variables x and predicted values y. Moreover, the fixed network architecture in MLP often gives rise to redundancy in both network structure and parameters. To address these issues, we propose MetaSymNet, a novel neural network that dynamically adjusts its structure in real-time, allowing for both expansion and contraction. This adaptive network employs the PANGU meta function as its activation function, which is a unique type capable of evolving into various basic functions during training to compose mathematical formulas tailored to specific needs. We then evolve the neural network into a concise, interpretable mathematical expression. To evaluate MetaSymNet's performance, we compare it with four state-of-the-art symbolic regression algorithms across more than 10 public datasets comprising 222 formulas. Our experimental results demonstrate that our algorithm outperforms others consistently regardless of noise presence or absence. Furthermore, we assess MetaSymNet against MLP and SVM regarding their fitting ability and extrapolation capability, these are two essential aspects of machine learning algorithms. The findings reveal that our algorithm excels in both areas. Finally, we compared MetaSymNet with MLP using iterative pruning in network structure complexity. The results show that MetaSymNet's network structure complexity is obviously less than MLP under the same goodness of fit.

翻译：数学公式是人与自然沟通的媒介，承载着自然现象的运行规律。这些规律的简洁表述是科学研究的关键目标，也是人工智能（AI）面临的重要挑战。传统人工神经网络（MLP）虽擅长数据拟合，但常产生不可解释的黑箱结果，阻碍了我们对变量x与预测值y之间关系的理解。此外，MLP中固定的网络架构往往导致网络结构和参数的冗余。为解决这些问题，我们提出MetaSymNet——一种新型神经网络，可实时动态调整自身结构，实现网络扩展与收缩。该自适应网络采用PANGU元函数作为激活函数，这种特殊函数能在训练过程中演变为多种基本函数，从而组合出满足特定需求的数学公式。我们进而将神经网络演化为简洁、可解释的数学表达式。为评估MetaSymNet的性能，我们在包含222个公式的10余个公开数据集上，将其与四种最先进的符号回归算法进行对比。实验结果表明，无论是否存在噪声，我们的算法均一致优于其他算法。此外，我们评估了MetaSymNet与MLP、SVM在拟合能力和外推能力这两个机器学习算法关键方面的表现，结果显示我们的算法在这两个领域均具优势。最后，我们通过迭代剪枝在网络结构复杂度方面对比了MetaSymNet与MLP。结果表明，在相同拟合优度下，MetaSymNet的网络结构复杂度明显低于MLP。