Gaussian Nonlinear Activator

In modern deep learning architectures, activation functions play a critical role in introducing nonlinearity, enabling neural networks to learn complex patterns and representations. Among the many activation functions available, those based on Gaussian probability theory have gained significant attention for their unique properties.

The Gaussian Error-based Nonlinear Activator is a smooth, continuously differentiable activation function that combines the input value with the probability of it being greater than other values under a standard normal distribution. Unlike simpler activators like ReLU which have hard cutoffs, this function creates a soft gating mechanism where inputs are weighted by their own probability under the Gaussian CDF.

Mathematical Formulation: The activation is computed as:

$$\text{Activator}(x) = x \cdot \Phi(x)$$

Where Φ(x) is the cumulative distribution function (CDF) of the standard normal distribution:

$$\Phi(x) = \frac{1}{2} \left[1 + \text{erf}\left(\frac{x}{\sqrt{2}}\right)\right]$$

This can also be approximated using the tanh function for computational efficiency:

$$\text{Activator}(x) \approx 0.5x \left[1 + \tanh\left(\sqrt{\frac{2}{\pi}}\left(x + 0.044715x^3\right)\right)\right]$$

Key Properties:

• Smooth and differentiable everywhere — no discontinuities like ReLU at x=0
• Non-monotonic — the function has a slight dip in the negative region before approaching zero
• Probabilistic interpretation — outputs can be understood as expectations under the normal distribution
• Self-gating — the input gates itself based on its percentile in the Gaussian distribution

Your Task: Implement a function that applies this Gaussian-based nonlinear activation to a NumPy array of input values. The function should:

• Compute the activation for each element in the input array using vectorized operations
• Round each output value to 4 decimal places
• Return a NumPy array of the same shape as the input