Table 1 Results of phDOS prediction on the test set.

ML model	Setting		phDOS prediction				Calculated C_V (300 K)		Calculated \(\bar{\omega }\)
	Scaling	Loss	R²	MAE	MSE	WD	MAE	MSE	MAE	MSE
E3NN	MaxNorm	MSE	0.56	0.094	0.034	39	3.58	56	30.9	3161
GATGNN	MaxNorm	MSE	0.45	0.105	0.042	44	4.66	80	35.1	3392
Mat2Spec	MaxNorm	MSE	0.63	0.086	0.029	33	3.30	49	26.2	2284
E3NN	SumNorm	WD	−0.48	0.339	1.884	132	11.7	393	90.0	17343
GATGNN	SumNorm	WD	−2.78	0.185	0.065	194	19.6	591	183	42753
Mat2Spec	SumNorm	WD	0.57	0.085	0.026	21	1.32	10	10.6	348
E3NN	SumNorm	KL	0.48	0.105	0.036	50	4.88	77	41.1	3718
GATGNN	SumNorm	KL	−1.05	0.177	0.057	215	22.4	756	205	51609
Mat2Spec	SumNorm	KL	0.62	0.078	0.023	24	1.96	11	17.1	625

For each combination of 3 ML models and 3 settings, the performance metrics include 4 measures of the prediction of the 51-D phDOS and 2 measures each for the properties CV (heat capacity at 300 K, J/K ⋅ mol) at 300K and \(\bar{\omega }\) (average phonon frequeny, cm⁻¹) calculated from each material’s phDOS. Each loss metric is aggregated over all materials in the test set. Note that for phDOS predictions with MaxNorm scaling, each prediction is re-scaled to a maximum value of 1 prior to evaluating prediction loss, and analogously the predictions with SumNorm scaling are re-scaled to a sum of 1 prior to evaluating prediction loss. MAE and MSE metrics for phDOS prediction inherit the arbitrary units from the respective scaling and should not be directly compared across scaling. R² (unitless) and WD (units of cm⁻¹ from the energy axis) are insensitive to this difference in scaling. The best value in each column is noted in bold. In each setting and for each loss metric, Mat2Spec provides the best predictions.

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