The proposed Elementary Universal Activation Function (EUAF) network demonstrates significant advantages over traditional neural networks, particularly those utilizing ReLU or other commonly used activation functions. One of the primary distinctions lies in the trade-offs between width, depth, and approximation accuracy.
In the context of the EUAF network, it achieves the super approximation property with a fixed number of neurons, specifically 366d + 365 intrinsic neurons, allowing for arbitrary accuracy in approximating functions in C([a, b]^d). This is in stark contrast to ReLU networks, which often require a width that scales polynomially or even exponentially with the input dimension d to achieve similar levels of accuracy. For instance, a ReLU network may necessitate a width of O(d^3) or more, depending on the desired approximation accuracy, leading to a substantial increase in the number of neurons as the dimensionality of the input increases.
Moreover, the depth of the EUAF network is fixed, which simplifies the architecture and reduces the computational complexity associated with training deeper networks. In contrast, networks using ReLU often require deeper architectures to enhance their approximation capabilities, which can lead to issues such as vanishing gradients and increased training times. The EUAF network's ability to maintain a fixed architecture while achieving high approximation accuracy represents a significant advancement in neural network design, particularly for high-dimensional continuous functions.