Peanut DualCutoff LearnedBoundary Architecture

This model extends the PEANUT architecture by introducing learned, element-pair–dependent cutoff distances and soft distance-based gating to interpolate between close-range and non-close-range interactions.

Graph construction

For a molecular structure, construct a single directed neighbor graph with atoms as nodes and edges connecting all atom pairs within a global cutoff radius \(r_{\mathrm{cut}}\).

The neighbor graph is fixed per forward pass. Interaction regimes are distinguished at the edge message level.

Learned atom-pairwise cutoff distances

Let \(Z_i, Z_j \in \{0, \dots, N_{\mathrm{el}}-1\}\) denote atomic numbers.

Pair index mapping

Map the unordered element pair \((Z_i, Z_j)\) to a unique index \(p(i,j)\):

\[p(i,j) = \min(Z_i,Z_j)\,N_{\mathrm{el}} - \frac{\min(Z_i,Z_j)\bigl(\min(Z_i,Z_j)+1\bigr)}{2} + \max(Z_i,Z_j)\]

Learned pairwise cutoff

Each element pair is assigned a learnable scalar parameter \(\theta_p \in \mathbb{R}\). The effective cutoff distance is

\[r^{\ast}_{ij} = r_{\mathrm{cut}} \,\sigma\!\left(\theta_{p(i,j)}\right), \qquad \sigma(x) = \frac{1}{1 + e^{-x}}\]

which ensures

\[r^{\ast}_{ij} \in (0, r_{\mathrm{cut}})\]

Soft distance-based gating

Define a smooth distance-dependent gating function

\[\alpha_{ij} = \sigma\!\left( \frac{r^{\ast}_{ij} - r_{ij}}{T} \right), \qquad T > 0\]

where \(r_{ij}\) is the interatomic distance. At the learned cutoff,

\[r_{ij} = r^{\ast}_{ij} \quad \Rightarrow \quad \alpha_{ij} = 0.5\]

Dual edge message functions

Edge features are given by

\[\mathbf{x}_{ij} = (F_{ij} \mid h_j^{(t-1)})\]

Two separate message functions are applied:

\[\mathbf{m}_{ij}^{\mathrm{close}} = \mathrm{MLP}_{\mathrm{close}}(\mathbf{x}_{ij})\]

\[\mathbf{m}_{ij}^{\mathrm{nonclose}} = \mathrm{MLP}_{\mathrm{nonclose}}(\mathbf{x}_{ij})\]

with

\[\mathbf{m}_{ij}^{\mathrm{close}}, \mathbf{m}_{ij}^{\mathrm{nonclose}} \in \mathbb{R}^{d_m}\]

Distance-weighted message combination

The final edge message is computed as

\[\mathbf{m}_{ij} = \alpha_{ij}\,\mathbf{m}_{ij}^{\mathrm{close}} + \bigl(1-\alpha_{ij}\bigr)\,\mathbf{m}_{ij}^{\mathrm{nonclose}}\]

This yields a smooth, fully differentiable interpolation between short-range and long-range interaction regimes.