PEANUT mini model architecture
==============================

To explore how little information is sufficient for meaningful energy predictions, a simplified PEANUT model was implemented that ignores the species of neighboring atoms. 
It still uses radial and spherical features, but instead of combining them via a tensor product, they are concatenated. 
The smoothing function is applied to both radial and spherical features to ensure smoothness. 
Messages are summed over neighbors, concatenated with the initial node embedding :math:`h_i^{(0)}`, and passed through an MLP to update the node vector, which is then used to predict atom-wise energy contributions. 
The full energy prediction is given by

.. math::
    E_{\mathit{full}} = \sum_i MLP_{\mathit{energy}}\Bigg(MLP_{\mathit{node}}\Big(h_i^{(0)} \,\big\|\, \sum_{j \in \mathcal{N}(i)}
    \begin{bmatrix}
        R_{ij} \\
        Y_{ij}^{\mathit{smooth}}
    \end{bmatrix}
    \Big)\Bigg),

where

.. math::
    \hat{h}_i = \Big(h_i^{(0)} \,\big\|\, \sum_{j \in \mathcal{N}(i)}
    \begin{bmatrix}
        R_{ij} \\
        Y_{ij}^{\mathit{smooth}}
    \end{bmatrix}\Big) \in \mathbb{R}^{D + B + (L+1)^2}.