2026-08-19
Invariant GNNs on 201k metal dielectric functions show broken data scaling that steepens past ~30k samples; parameter scaling saturates near 5M weights.
Materials data are scarce and expensive. Neural scaling laws are supposed to say how much loss drops if you compute another batch of structures or widen the net. Vision and language models mostly follow a smooth power law. For materials GNNs the measurement has barely started. If the curve bends, a lab that extrapolates from a small set will spend the next million CPU hours on the wrong side of the bend.
Metal optics is a task that can actually be scaled. Screening is strong and excitons are weak, so the independent-particle approximation plus a Drude term already tracks experiment. Existing semiconductor optical sets sit under 30,000 spectra. This work raises that by about an order of magnitude, on purpose, to watch the scaling.
Intermetallics with only metallic elements and at most ten atoms per cell were filtered from Alexandria and run through a Quantum ESPRESSO plus SIMPLE DFT workflow. 201,361 converged dielectric functions and Drude frequencies came back, at about 2.3 million CPU hours. Splits are 80/10/10 by chemical formula. The models are stripped-down invariant GNNs, OptiMetal2B and 3B, differing only in two- versus three-body message passing. They jointly regress the full spectrum and the scalar Drude frequency; the loss is the sum of the two MAEs.
Data scaling subsamples the train set into seven sizes from 2,500 to 160,000, with hidden width fixed at 256 (about 10M parameters). Parameter scaling sweeps width 16 to 1,024 (about 100k–100M parameters) at 20,000 samples. Four candidate curves are scored with small-sample AIC: a plain power law, a power law with a floor, and two smoothly broken power laws.
Data scaling consistently picks the smoothly broken power law. The low-data exponent is αD,1≈0.15–0.18. Past a breakpoint Dc≈10^4.4–10^4.7 (roughly 20k–50k samples) it steepens to αD,2=0.38–0.42. Two- and three-body nets share similar exponents; the three-body curve sits lower. Parameter scaling is a power law with a floor, αN=0.41–0.58, and saturates near 5M weights.
The 2D map L(D,N) says the bend is not an artifact of an oversized model. When data and capacity match, around 1M parameters, a single power law is a decent description. Overparameterized models bring the break back, and the three-body net splits more sharply (0.16 versus 0.46). Swapping Crystal Graph Convolution for Transformer Convolution barely moves the exponents.
| Model | αD,1 → αD,2 | Break Dc | αN |
| OptiMetal2B (CGC) | 0.15 → 0.42 | 10^4.72 | 0.53 |
| OptiMetal2B (TC) | 0.18 → 0.39 | 10^4.62 | 0.58 |
| OptiMetal3B (TC) | 0.17 → 0.38 | 10^4.43 | 0.41 |
Stacking parameters stops paying quickly in this regime. Data is the bottleneck. The data curve itself steepens around 30k samples, so the next expensive spectrum is worth more after the kink than before it. Dataset plans need two budgets. Adding three-body terms shifts the loss; it buys data efficiency, not parameter efficiency.
The PDF in hand is the arXiv version and covers invariant GNNs only. The PRX Intelligence abstract adds an equivariant net; those numbers are not in this file. The task is metals under the independent-particle plus Drude approximation. Insulators and excitonic systems may not bend the same way. Seven dataset sizes feed the AIC choice, so Dc can move if the grid changes. All runs reuse hyperparameters tuned at width 256, then scale the learning rate with N, so optimizer coupling is not fully ruled out.