MACE#
1. Recall MPNN Interatomic Potentials#
When applied to parameterize properties of atomistic structures (materials or molecules), the graph is embedded in 3-dimensional (3D) Euclidean space, where each node represents an atom, and edges connect nodes if the corresponding atoms are within a given distance of each other. We represent the state of each node \(i\) in layer \(t\) of the MPNN by a tuple.
\[
\sigma_i^{(t)}=(r_i,z_i,h_i^{(t)})
\]
Where \(r\in \mathbb{R}^3\) is the position of the atom \(i\), \(z_i\) is the chemical element, and \(h^{(t)}\) are its learnable features.
A forward pass of the network consists of multiple message construction, update, and readout steps.
During message construction, a message \(m^{(t)}_i\) is created for each node by pooling over its neighbors:
\[
m_i^{(t)}=\bigoplus_{j\in\mathcal{N}(i)}M_t(\sigma^{(t)},\sigma^{(j)})
\]
where \(M_t\) is a learnable message function and \(\bigoplus_{j\in\mathcal{N}(i)}\) is a learnable, permutation invariant pooling operation over the neighbors of atom \(i\) (e.g., a sum).
In the update step, the message \(m_i^{(t)}\) is transformed into new features
\[
h^{(t+1)}_i= U_t(\sigma^{(t)}_i,m_i^{(t)})
\]
where \(U_t\) is a learnable update function, After \(T\) message construction and update steps, the learnable readout functions \(R_t\) map the node states \(σ^{(t)}_i\) to the target, in this case the site energy of atom \(i\).
\[
E_i = \sum^T_{t=1}\mathcal{R}_t(\sigma_i^{(t)})
\]