mTanh: A Low-Cost Inkjet-Printed Vanishing Gradient Tolerant Activation Function

Inkjet-printed circuits on flexible substrates are rapidly emerging as a key technology in flexible electronics, driven by their minimal fabrication process, cost-effectiveness, and environmental sustainability. Recent advancements in inkjet-printed devices and circuits have broadened their applications in both sensing and computing. Building on this progress, this work has developed a nonlinear computational element coined as mTanh to serve as an activation function in neural networks. Activation functions are essential in neural networks as they introduce nonlinearity, enabling machine learning models to capture complex patterns. However, widely used functions such as Tanh and sigmoid often suffer from the vanishing gradient problem, limiting the depth of neural networks. To address this, alternative functions like ReLU and Leaky ReLU have been explored, yet these also introduce challenges such as the dying ReLU issue, bias shifting, and noise sensitivity. The proposed mTanh activation function effectively mitigates the vanishing gradient problem, allowing for the development of deeper neural network architectures without compromising training efficiency. This study demonstrates the feasibility of mTanh as an activation function by integrating it into an Echo State Network to predict the Mackey–Glass time series signal. The results show that mTanh performs comparably to Tanh, ReLU, and Leaky ReLU in this task. Additionally, the vanishing gradient resistance of the mTanh function was evaluated by implementing it in a deep multi-layer perceptron model for Fashion MNIST image classification. The study indicates that mTanh enables the addition of 3–5 extra layers compared to Tanh and sigmoid, while exhibiting vanishing gradient resistance similar to ReLU. These results highlight the potential of mTanh as a promising activation function for deep learning models, particularly in flexible electronics applications.