On the Steeper Scaling Laws of Equivariant Neural Force Fields

Equivariance to Euclidean symmetries is a critical inductive bias for neural networks that model 3D atomistic systems. Whether to build equivariance into a network or learn it through data augmentation remains an active debate. Prior work has empirically shown that equivariant networks achieve steeper scaling laws on certain tasks, raising the open question of what causes this gap. This paper takes a step toward an answer in the setting of machine learning interatomic potentials (MLIPs), where the targets are equivariant forces. In particular, we show that a theory of scaling laws based on source-capacity conditions explains this gap. The capacity and source exponents of each architecture that reflect the decay rate of the network's kernel spectrum and its alignment with the target forces, respectively, account for the observed exponents in the empirical scaling laws. Our results show that for equivariant targets, equivariant architectures scale better than their unconstrained counterparts, because their kernels align better with the targets. Since these exponents can be estimated from an early training checkpoint, they provide a practical way to forecast the scaling laws of MLIP architectures without full-scale training.