Table 1 Performance of baseline, D-MPNN, combined ML + QM and TD-DFT approaches to calculate transition energies (in eV)

ID	Method		R²	MAE	R² (sys. corr.)	MAE (sys. cor.)	R² (sys. corr.)^c	MAE (sys. cor.)^c
1	Baseline HOMO-LUMO gap (GFN2-xTB)		−6.27	1.271	0.277	0.308	–	–
2	Application of a published D-MPNN trained only on artificial colourants		0.181	0.321	0.324	0.267	–	–
6	D-MPNN trained using the collected natural colourants^a		–	–	–	–	0.581	0.184
3	Linear regression + QM descriptors^b	⍵B97X-D4	–	–	–	–	0.443	0.191
4	Linear regression + QM descriptors^b	⍵B97X-D4 + CPCM	–	–	–	–	0.441	0.191
5	TD/TDA-DFT calculation	PBE0	0.430	0.278	0.504	0.244	0.650	0.201
6		PBE0 + CPCM	0.450	0.263	0.509	0.247	0.653	0.205
7		⍵B97X-D4	−0.798	0.615	0.619	0.209	0.766	0.166
8		⍵B97X-D4 + CPCM	−0.302	0.514	0.630	0.208	0.776	0.164
9		⍵B97X-D4 + TDA	−1.79	0.779	0.600	0.214	0.754	0.169
10		⍵B97X-D4 + TDA + CPCM	−1.01	0.659	0.623	0.210	0.772	0.164
11		BMK	0.037	0.423	0.569	0.227	0.727	0.181
12		BMK + CPCM	0.290	0.364	0.577	0.229	0.724	0.184
13		CAM-B3LYP	0.247	0.347	0.566	0.226	0.727	0.180
14		CAM-B3LYP + CPCM	0.434	0.299	0.579	0.227	0.728	0.182
15		M06-2X	−0.11	0.46	0.596	0.217	0.757	0.172
16		M06-2X + CPCM	0.204	0.382	0.607	0.217	0.759	0.171
17		B2PLYP	0.461	0.262	0.553	0.222	0.717	0.175
18		B2PLYP + CPCM	0.551	0.220	0.593	0.212	0.757	0.163
19		SCS-PBE-QIDH	0.269	0.346	0.597	0.207	0.761	0.159
20		SCS-PBE-QIDH + CPCM	0.518	0.255	0.657^d	0.183	0.822	0.133
21		SCS-⍵PBEPP86	0.404	0.304	0.603	0.206	0.764	0.158
22		SCS-⍵PBEPP86 + CPCM	0.605	0.220	0.657	0.183	0.816	0.135

Determination coefficient (R²) and mean absolute error (MAE) are used to rank the performance of the approaches before and after the removal of systematic error components. The last two columns on the right are used to compare the ML model trained on the cleaned dataset obtained based on the five-fold cross-validation and TD-DFT calculation results.
^aCalculated in cross-validation for a cleaned natural colourants dataset containing 595 compounds versus 647 molecules in the original set.
^b10-fold CV.
^cDataset with the outliers removed which was used to train the ML models.
^dBold font marks the best performing techniques.

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