As mentioned above, nowadays, data are becoming a sort of extremely valuable and scarce resource. Due to the lack of complete ultraviolet (UV)-visible absorption spectra and spectral data for mixtures, we cannot fully verify the performance of the entire 2-stage framework. In order to prove the superiority of our model, we will use as few data as possible to verify the 2 models logically.

First, we collected 432 groups of UV absorption spectra data from the literature as the training set (sample features are shown in Fig. 2; for more details, see the Materials and Methods section) and 40 groups of data as the test set to verify the performance of CGC-BNN model; the molecules in the dataset contain sufficient chemical groups (the name and chemical formula of the molecule in the dataset are provided in the Supplementary Materials). The mean absolute error (MAE), mean square error (MSE), and the coefficient of determination (R²) of the prediction results are presented in Table 1. According to our design, we only need to know the chemical formula of the molecule and convert it into a vector of features at the electronic, atomic, and group levels with the revised GC method and molecular descriptor (see the Materials and Methods section for more details); the vector form is used next as input to obtain the complete absorption spectrum (see Fig. 3 for the prediction process). Since we only use small samples, it is important to verify whether the models are overfitted in training. Thus, we randomly select 10% of the samples for internal prediction each time during the internal validation and repeat 50 times of prediction to afford an average value as the final result (see Table 1). With this method, each molecule in the dataset gets its turn for prediction. Next, we used the 432 sets of data for training to perform external validation using 40 molecules. As shown in Table 1, the error of external test is only slightly larger than that of internal verification. Moreover, due to the lack of vis-spectra data, the maximum absorption wavelengths in our training data may not be accurate. In fact, many of these molecules also have absorption peaks in the visible light band, which is supposed to reduce the accuracy of the model considerably. To further support our conclusion, we used another 73 groups of visible spectra data (this training set contains different dyes dissolved in different solvents; see the Materials and Methods section for details) for internal 10% validation, and the results are shown in Table 2 (the average values of 50 time repeats are presented here). As (somewhat) expected, due to the lack of full-spectrum data, although we used 432 sets of data for training, the prediction of UV spectra (which may not include the maximum absorption wavelengths of the molecules) is still unsatisfactory. By contrast, the small-sample training affords higher accuracy in the prediction of vis-spectra as compared to the training using 432 groups of UV-spectra data. Here, for comparison, Shao et al. [22] used ~1,200 samples in training to eventually obtain 1.52% mean relative error (MRE) in the prediction of maximum absorption wavelength with a self-designed ML system, SMFluo1-DP.

Although the lack of data will dramatically reduce the accuracy of the ML model, the final prediction results are still excellent in our case. As shown in Fig. 4, there is no serious deviation between the predicted spectral waveform and the experimental value. Note that Urbina et al. [23] used more than 2,000 samples as the dataset but still could not converge successfully with the feedforward neural network. Zhang et al. [31] used quantitative calculation data as input and also used more than 2,000 sets of data to accurately predict the UV absorption spectrum of protein. To our delight, the CGC-Bayesian neural network (BNN) model in this work reaches comparable accuracy with the above 2 literatures but requires a much smaller dataset and much shorter time.

Compared with a data-driven model, the CGC model is more like a data-fitted chemical equation. Thus, a series of factors influential to the molecular properties have been taken into account to afford the accurate spectrum. This enables our model to adapt to small data samples. To demonstrate the crucial roles played by different descriptors in the model, we tried the revGC method and the molecular descriptor method (by describing the characteristics of atoms and charges) separately. As shown in Table 3, their performance is not as good as that of the CGC method. In addition, guided by chemical principles, the algorithm itself is not expected to dominate the accuracy of the prediction. To justify this, we also modeled the CGC method using other ML models. As a result, all predictions afford similar results (see Table S3 and Fig. S1).

After obtaining the absorbance vector of the complete absorption spectrum of a single molecule, it remains challenging to use these data to afford the spectra of mixture. To this end, we developed the MC method. Different from the GC method, the MC method determines the characteristics of the mixture based on the size of the characteristic value contributed by each molecule. It is constructed based on 2 assumptions: (a) the interactions between different molecules can be expressed as the mixing functions of group vectors (especially the difference between vectors); (b) the mixture properties are contributed by linear or nonlinear combination of each species upon considering both the mixing functions and the relative concentrations of different molecules. For the traditional GC method, the interactions of different groups in the same or different molecules are considered by introducing complex reciprocal parameters. In this work, after dealing with intramolecular group interactions in the first stage, the second stage treats intermolecular group interactions as functions of the group vector difference. According to the Beer–Lambert–Bouguer law, the absorption spectrum of the mixture is additive combination of the spectra of individual components. However, for a multicomponent system, the intermolecular interaction force will make the mixed spectrum show a nonlinear addition (see Fig. S2). Here, if one considers the intermolecular interaction by counting the contribution of each type of force, e.g., by employing energy decomposition analysis (EDA) [32–34], the entire model can be much too complex and may encounter convergence problems. Thus, we can count the interactions of different molecules by introducing a mixing function, as, for example, the following interaction factor k_ij (Eq. 1):where {\overrightarrow{a}}_i and {\overrightarrow{a}}_j are the group vectors for the components i and j, and the spectrum for mixture can be obtained using Eq. 2:in which L_i is the absorption spectrum of pure component i, and X is the transformation vector. The logic represented by the above formula is the fundamental framework for the MC method, but it would be rather difficult for artificial modeling to provide relatively an accurate solution. Thanks to the data-driven tools, both Eqs. 1 and 2 can be modeled with the neural network during training with the 2-component data; we assume that the multicomponent interaction can be interpreted using the 2-component data in a functional manner afforded by selected neural network logics. Thus, the task left is to migrate the 2-component models to multicomponent systems without additional training of experimental data; such a transfer learning process is illustrated in Fig. 5. Notably, the contribution of solvents to the spectra via various solvent effects has been taken into account within the CGC model in the first stage. Therefore, one does not need to consider the influence of solvent in the MC-BNN model of the second stage.

Up until here, we have been able to output the absorbance at each wavelength through the molecular formula. If the vector composed of this absorbance is put into the MC-BNN model, the spectrum of the multicomponent system can be obtained. There is no doubt that if a certain amount of complete UV-vis absorption spectra and the associated mixture spectra data were available, the 2-stage models could have predicted the mixture spectra with high accuracy. Unfortunately, such experimental data are not abundantly available in the literature. However, we can still verify the performance of the MC-BNN model separately. Here, we use only 73 groups of 2-component data as the training set. The MAE of internal test is 0.1576; R² is 0.9496 (again, 10% of the samples were selected for test each time, and 50 repeats of prediction gave the average values as the final result). Then, we selected 3 dyes covering 3 primary colors and used the above 73 groups of data as the training set for prediction purposes. As shown in Fig. 6, the MC method has been proven from theory to practice—the MC-BNN model can learn the intermolecular interactions and output a relatively accurate spectrum for mixture. Considering the satisfactory performances of the first and second stages in modeling the intra- and intermolecular interactions, in principle, it is able to afford spectra of any organic systems upon training with sufficient data. Such sufficient data are expected to be much smaller than that required for traditional data-driven ML protocols.

Finally, we extended the prediction of mixture spectra from a binary system to multicomponent systems. In order to compare the impact of different virtual mixing paths in prediction, we used 3 mixing paths to predict. Surprisingly and as expected, although multicomponent experimental data were not involved in training, the migration of the model to the ternary system did not enlarge the prediction error. As shown in Fig. 7, the prediction of multicomponent systems is satisfactory, and the error only comes from the relative intensity of absorbance. This not only proves the reliability of the MC-BNN model but also reflects its good migration ability and obvious modularity.

For the multicomponent mixture, the spectrum can be obtained accurately by using only 2-component data for training. Moreover, note that the MC method is logically available as well for other mixture properties: as long as the intermolecular interaction matters for the mixture properties, the 2-stage protocol may be operative.

A total of 432 training sets and 40 test sets for full UV spectral prediction were collected from literature [23]. These compounds were obtained as 10 mM solutions in 100% DMSO. Each compound was diluted 50-fold (to 200 mM and 2% DMSO) with water and transferred to black, clear-bottom Greiner UV-STAR microplates. The UV absorption of each compound was read in a SpectraMax iD5 Multi-Mode Microplate spectrophotometer from 230 to 400 nm in 1-nm increments. The resulting spectra were scaled by setting the minimum absorbance to zero and normalizing to a maximum absorbance of 1.0. In order to speed up the training of the model, we increased the wavelength interval of these data to 3 nm. The processing of 3 spectral characteristics is as follows: (a) The maximum absorption wavelength is set as the wavelength at the peak of the spectrum: λ (as previously described, this is obviously not necessarily the maximum absorption wavelength). (b) The spectral area is the sum of absorbance at each wavelength: S (not the actual area, which is more conducive to model learning). (c) For some molecules without wave peak in the UV region: H; we set the wave peak height as the absorbance at 250 nm.

The maximum absorption wavelength data between visible light regions for comparison are obtained from literature (including 38 azo dyes dissolved in different organic solvents and 35 water-soluble rhodamine dyes). Its distribution in different solvents is shown in Table 4.

BNN: Deep neural network often uses gradient descent algorithm with fast convergence in realizing nonlinear regression. For small sample learning, one can sacrifice the convergence speed to obtain high accuracy. On the other hand, due to the strong learning ability of the neural network, overfitting is commonly encountered when there are few data. Therefore, we use Bayesian regularization to reduce overfitting and improve the generalization ability of the neural networks. In the Bayesian framework, the weights of the network are considered as random variables. After the data are taken, the density function for the weights can be updated according to Bayes' rule [38,39]:in which D represents the dataset; M is the particular neural network model used; w is the vector of network weights; P(wIα,M) is the prior density, i.e., the known weights before any data are collected; P(DIw,β,M) is the likelihood function, i.e., the probability of the data occurring under the weights w; and P(Dlα,β,M) is a normalization factor, which guarantees the total probability of 1 [39]. The Hessian matrix of Gauss Newton approximation can be implemented within the framework of the Levenberg–Marquardt algorithm to reduce the computing cost.

According to the established chemical characteristics, we used MATLAB to establish 3 BNN models and obtained the results by inputting the characteristics (the code and hyperparameter settings are available in the Supplementary Materials). In addition, we only divided the training set and test set in the prediction of the complete UV spectrum, the prediction of the maximum absorption wavelength of visible light was only used as a comparison for internal testing, and only 4 dyes were used in the prediction of the mixed spectrum. As mentioned above, we adopt the same form for the internal verification of all training sets: 10% of the samples are randomly selected each time as the prediction, and the average value is taken as the evaluation index after 50 repetitions. In addition, compared with 10-fold cross-validation, this method is more suitable for such a small sample. Each datum in the training set can be predicted as a prediction set for many times, and each prediction is a completely random combination. For 10-fold cross-validation, each datum has only one chance to be predicted, which may not objectively reflect the performance of the model.

We use the following indicators to evaluate the prediction performance of the model: MRE, MAE, MSE, and R². In the prediction of complete spectrum, we will calculate the error of absorbance at each wavelength and then take the average of the errors at all wavelengths as the final error. Of course, for multiple samples, the error is the average of the errors of each sample.