Graph neural networks stop at predictions: GSD turns message passing into readable formulas and recovers expert descriptors in three systems
Synopsis
Graph neural networks learn structure–property relationships from graphs of atoms and bonds, but those relationships stay in a latent space and are hard to read as scientific rules. Graph symbolic distillation (GSD) first uses a constrained message-passing architecture search to fix how information is transformed, propagated and aggregated, then fits each layer's learned transformation with a sparse symbolic expression while keeping graph connectivity and aggregation. Across halide-perovskite band gaps, molecular hydration free energies and multidentate adsorption on transition-metal surfaces, it rebuilds expert descriptors from elementary node and edge features and yields testable local-geometry, polarity–size and electronic–connectivity relationships.
Interpretation
Interpretable modeling on graph data can now expose the whole path from graph to descriptor to formula, not only attribute individual predictions. Earlier graph symbolization either relied on a manually specified message-passing skeleton or pruned an overcomplete flow before symbolizing, so the symbolic part did not reconstruct the full message-passing computation. On three graph-regression tasks the searched architectures compare with reported graph baselines at MAD/MAE 17.27 versus 9.01 for -CsPbI3, MAE 0.62 kcal mol-1 for FreeSolv, and RMSE 0.19 eV versus 0.32 eV for ComplexAdsorbate; data splits and training protocols may differ.
The distilled formula writes the prediction as a sum of graph motifs, each built from original input features and explicit aggregation operations. Direct symbolic regression maps fixed-length tabular descriptors to the target, flattening topology-dependent information during pooling. Layer-wise distillation fits each scalar output channel of the teacher with SISSO compressed-sensing regression through TorchSISSO, then hierarchical token pruning drops negligible terms; on -CsPbI3 the targeted-residual variant cuts formula complexity from 397 to 304 while test RMSE rises from 0.00269 to 0.00490 eV.
The formulas offer testable local structural relationships, not only fits. Conventional molecular descriptors summarize chemical features at the molecular level, making individual atomic environments hard to separate. For -CsPbI3, a linear combination of two extremal neighborhood descriptors predicts the band gap with an RMSE of 0.0163 eV over the sampled volume-preserving strains; for FreeSolv, the compact formula reaches an overall RMSE of 1.81 kcal mol-1, comparable to 1.87 kcal mol-1 for direct symbolic regression.
Perspective
The demonstrated scope is three graph-level regression tasks: band gaps of strained -CsPbI3, hydration free energies of small neutral molecules in FreeSolv, and adsorption enthalpies of complex adsorbates on transition-metal surfaces; the formulas and intermediate symbolic kernels can serve as task-adaptive graph descriptors for human inspection, while the longer distilled formulas remain accessible to computer-assisted analysis.
GSD inherits the representational ceiling of its deep teacher and can expose only relationships the teacher can express; the current search space is largely built from conventional local message-passing transformations and invariant aggregations, so relationships requiring explicit angular correlations, rotational equivariance, adaptive attention or long-range coupling may need geometry-aware graph networks or graph transformers together with symbolic operators able to represent their computations.