Scalar Residual Thermodynamic Head

Integrate this at the scalar prediction stage of a crystal graph neural network or atomistic transformer. First run each relaxed structure through the pretrained encoder to obtain a structure embedding \(h=\operatorname{Encoder}(\bm{x}^{\circ})\) and obtain its frozen 0 K energy \(U^{\circ}\) from the base potential. Concatenate \(h\), normalized temperature \(\tilde T\), and normalized pressure \(\tilde P\) in a two- or three-layer MLP residual head, producing \(g_\delta=\operatorname{MLP}_\delta(h,\tilde T,\tilde P)\); return \(\hat G=U^{\circ}+g_\delta\). During each batch, mark \(\tilde T\) and \(\tilde P\) as differentiable, compute \(\partial_T\hat G\) and \(\partial_P\hat G\) with reverse-mode autodiff, and optimize the loss above. Use per-dataset normalization for energy, entropy, and volume so derivative terms have comparable scale. Start with the encoder frozen, then unfreeze only the final encoder block if residual fine-tuning underfits. Estimate no mathematical quantities: \(U^{\circ}\) is directly supplied by the pretrained model, while all derivatives are exact autodiff derivatives of the learned scalar. The first cheap experiment should use a small pretrained crystal GNN, a few hundred structures sampled over several \((T,P)\) points, and compare against (a) direct \(G\) prediction with an identical MLP and (b) independent \(G,S,V\) heads. Evaluate held-out free-energy MAE, derivative MAE, Maxwell-consistency error from mixed derivatives, and phase-crossing temperature error for pairs of structures. Success is lower error at the same number of expensive labels, smoother phase diagrams, and substantially smaller violation of derivative identities.