utilizes only the relaxed host structure as input. In the depicted unit cell with only one inequivalent red crystal site node (ν 2 ), the supercell contains multiple inequivalent paths (p ̸ = p ′ ̸ = p ′′ ) between this symmetrically equivalent endpoint. A given path is decorated with nodes ν p1..n=3 that break the supercell symmetry. The ordering of the energy sequence (0 . . . n + 1) is with respect to the vacancy motion (dashed red circle), which is a reverse of the index that represents the occupation of the diffusing species (filled red circle) at a given image coordinate (n + 1 . . . 0). (b-c) The d 2 GNN architecture consists of six general steps: graph encoding of the crystal and image nodes, an embedding function for the node features, a nearest neighbor filter that enforces directed message passing in the convolution function, a sequence extraction, a sequence-to-sequence translation via Transformer Encoder (TE), and a decoder step. Purple indicates steps involving learnable parameters, while yellow denotes parameterless operations. Black arrow represent the flow of information, while red arrows represent a discard of features. The improvement of (c) over (b) involves physically constraining the model to preserve Equation (5) via independent decoders for the end states and the images. is used to decode the endpoint feature vectors, but using only the pre-transformed sequence of feature vectors. Meanwhile, v (t ′′ ) p0 and v (t ′′ ) pn+1 from the second MLP are discarded. A third and final MLP decoder predicts the final energies. This second architecture, which preserves Equations ( ) to (7), yields a large improvement in the model performance (see Supplementary Section 4).