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<p>Activation functions (AFs) in artificial neural networks (ANNs) significantly affect system performance. In this study, a new complex AF obtained using Schwarz lemma is proposed. As a result, three theorems for analytical functions have been proposed by evaluating distinct variants of the boundary Schwarz lemma, and AFs have been generated using these theorems. Obtained AF is an intuitive result of the considered problem. The performance of the proposed AF is extensively evaluated through classification and regression problems on both real and complex-valued datasets and compared with commonly used functions such as sigmoid, tanh, arcsinh, zReLU, swish and gelu. Experimental results show that the proposed function is highly competitive, especially for complex-valued problems. As the network complexity increases, it maintains a stable performance profile and produces stable and reliable results in both training and testing phases. With these features, the proposed function stands out as a powerful and practical alternative, especially for complex neural network applications, based on a theoretical foundation rather than an arbitrary choice. Results from experiments indicate that the proposed function can be used in ANNs successfully.</p>

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Activation function design based on Schwarz lemma for neural networks

  • 期刊论文

  • 发表日期：

    2025年7月8日

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  • Bülent Nafi Örnek,
  • Canan Oral,
  • Erhan Bergil,
  • Süleyman Dirik

摘要

Activation functions (AFs) in artificial neural networks (ANNs) significantly affect system performance. In this study, a new complex AF obtained using Schwarz lemma is proposed. As a result, three theorems for analytical functions have been proposed by evaluating distinct variants of the boundary Schwarz lemma, and AFs have been generated using these theorems. Obtained AF is an intuitive result of the considered problem. The performance of the proposed AF is extensively evaluated through classification and regression problems on both real and complex-valued datasets and compared with commonly used functions such as sigmoid, tanh, arcsinh, zReLU, swish and gelu. Experimental results show that the proposed function is highly competitive, especially for complex-valued problems. As the network complexity increases, it maintains a stable performance profile and produces stable and reliable results in both training and testing phases. With these features, the proposed function stands out as a powerful and practical alternative, especially for complex neural network applications, based on a theoretical foundation rather than an arbitrary choice. Results from experiments indicate that the proposed function can be used in ANNs successfully.