Machine learning property prediction for organic photovoltaic devices

Abstract

Organic photovoltaic (OPV) materials are promising candidates for cheap, printable solar cells. However, there are a very large number of potential donors and acceptors, making selection of the best materials difficult. Here, we show that machine-learning approaches can leverage computationally expensive DFT calculations to estimate important OPV materials properties quickly and accurately. We generate quantitative relationships between simple and interpretable chemical signature and one-hot descriptors and OPV power conversion efficiency (PCE), open circuit potential (V_oc), short circuit density (J_sc), highest occupied molecular orbital (HOMO) energy, lowest unoccupied molecular orbital (LUMO) energy, and the HOMO–LUMO gap. The most robust and predictive models could predict PCE (computed by DFT) with a standard error of ±0.5 for percentage PCE for both the training and test set. This model is useful for pre-screening potential donor and acceptor materials for OPV applications, accelerating design of these devices for green energy applications.

Introduction

Worrying increases in anthropomorphic greenhouse gas emissions have driven a strong expansion of research into discovery, design, and optimization of materials for energy applications. Organic photovoltaic (OPV) materials are of great interest because of their potential to generate cheap, printable semiconductor devices that convert light into electrical energy. They promise sustainable sources of clean energy if their efficiencies and stabilities can be improved. Organic solar cell manufacture is intrinsically a simple and low-cost process, and devices can be lightweight and flexible^1,2. However, the relationships between materials properties, device configuration, and performance are complex and often poorly understood. Given the potentially vast number of materials and device configurations possible exhaustive experimentation, even using high throughput methods, cannot guarantee finding the highest performing materials.

The power conversion efficiency (PCE, % incident light energy converted to electricity) is one of the most crucial properties for OPV solar cells. Density functional theory (DFT) can calculate several important properties of photovoltaic materials that affect PCE:³ V_oc (open circuit potential), J_sc (short circuit density), energy of the donor highest occupied molecular orbital (HOMO), energy of the acceptor lowest unoccupied molecular orbital (LUMO), and the HOMO–LUMO gap, but this requires extensive computational resources and time. Many physiochemical phenomena relating to light absorption in solar cells, such as the exciton formation⁴ and migration⁵ process, charge transport⁶ and recombination, need to be considered^7,8.

Machine learning (ML) can potentially model the complex relationships between materials, device properties, and OPV performance, given sufficient data, allowing efficient leveraging of expensive and time-consuming experiments and quantum chemical calculations. Carefully chosen, a relatively small number of DFT calculations, validated by experiments, can train ML models that predict relevant OPV properties for materials not yet synthesized. Apart from the availability of sufficient and reliable training data, the most important element of ML models is the choice of descriptors, mathematical representations of the structural and physicochemical properties of the donors and acceptors used in the OPV devices. Clearly, device construction parameters are also relevant and can be included in the models if they are available. Different ML algorithms often give similar quality models for a given set of descriptors, whereas a given ML algorithm trained on different types of descriptors can generate models of highly variable quality⁹. Many types of molecular descriptors are available, including topological, electronic, geometrical, molecular fragment, and quantum chemical, among others.

ML approaches have been a popular choice for predicting photovoltaic properties¹⁰. For example, Padula and co-workers modeled the photovoltaic properties of 249 organic donor–acceptors pairs to using k-NN (k-nearest neighbor)¹¹ regression and kernel ridge regression^12,13 methods trained on a combination of electronic and structural parameters¹⁴. Sahu et al. used random forest (RF)^15,16, gradient boosting (GB)¹⁷, and deep neural networks (DNN) to model PCE for 280 small OPV molecules using 13 microscopic properties of as descriptors to train the models. The models predicted PCE for 30 molecules in a test set with modest R² values of 0.44, 0.50, 0.46, 0.61, and 0.62 for linear regression, k-NN, neural network, RF, GB models, respectively¹⁸. Root-mean-square error (RMSE) values, a robust estimate of model quality¹⁹, ranged from 1.07% PCE for GB to 1.34% PCE for linear regression. Note: as PCE is the percentage conversion of light energy, in this paper when we refer to standard error or RMSE values we mean the uncertainty in this property (e.g. 10.6 ± 0.5%), not the percentage error in this property (e.g. 100 × 0.5/10.6). Pereira and co-workers²⁰ also generated ML models for the energy of the HOMO and LUMO of OPV materials. They used a dataset of 111,725 molecules, fingerprint and modified distance descriptors, and RF^15,16, support vector machine²¹, and a standard feedforward neural network to perform feature selection and property modeling. They found that the RF algorithm trained on modified distance descriptors generated the best predictions for HOMO and LUMO for an external test set of 9989 compounds, with R² values of 0.89 and 0.93 and RMSE of 0.21 and 0.23 eV for the HOMO and LUMO energies, respectively. The HOMO–LUMO gap could be predicted with an R² of 0.91 and RMSE of 0.30 eV.

Although ML methods can model OPV properties well, one of the main problems is that the models are opaque, and the descriptors used to train them arcane. It is hard to extract information from the models that is useful for designing improved OPV materials. Here we show how efficient and chemically interpretable descriptors that can be computed quickly and do not require additional experimental measurements or resource-intensive DFT calculations can predict important OPV properties with good accuracy.

Results and discussion

Model development

Here, we used the Harvard Photovoltaic Dataset (HOPV15) dataset²² that includes data from quantum chemical calculations and that calculated by the Scharber model plus experimental properties collected from literature²³. The Scharber model uses a single parameter, the computed HOMO–LUMO gap, in which V_oc is assumed to be the HOMO–LUMO gap minus a band offset, J_sc is assumed to be 0.65 of the current resulting from absorbing all incident photons above the HOMO–LUMO gap, and FF is set 0.65 (ref. ²⁴).

Our aim is to demonstrate that simple, interpretable molecular descriptors and ML methods can model and predict important OPV properties. While it is clearly ideal to model experimentally measured properties directly, there are many variables that can affect the OPV performance metrics, for example, the device design; processing conditions; dopants, dyes, solvents, and other additives; and others. Thus, measured OPV properties can vary from experiment to experiment and between labs. Data points from different sources can be inconsistent and affect reproducibility, constituting a relatively large source of error. Our goal in this work was to show that ML methods in general, and signatures specifically, are well suited to modeling and predicting a wide range of OPV properties. Therefore, we used the reproducible large dataset of photovoltaic properties calculated by a range of DFT methods in HOPV15. Clearly, these methods can be usefully applied to experimental data collected under conditions where all relevant information and device characteristics are carefully controlled, once the results are available in sufficient quantity to train ML models. Generally, there is not a good correlation between experimental and raw DFT calculated values²⁵. DFT calculations occasionally calculate physically unrealistic negative values for PCE, and these uncorrected computed values PCE do not correlate well with the experimental values. However, Lopez et al.²⁶ showed that calibration of PCE values in the HOPV15 dataset can significantly improve the correlation between experimental and Scharber PCEs (Fig. 1).

Fig. 1: Effects of calibration on HOMO–LUMO gap and PCE.

The distributions for a HOMO–LUMO gap and b PCE are presented as a histogram and a kernel density estimate from the difference between a baseline and a calibration. The band gap also shows the general statistics of the distribution. Reprinted with permission from ref. ²⁶. Copyright 2017, with permission from Elsevier.

As our goal was to generate models with good predictive performance that are chemically interpretable, we employed molecular signature descriptors. These represent chemical fragments in the donor and acceptor molecules. The “Methods” section describes the signature descriptors fully. We generated models for PCE, V_oc, J_sc, HOMO energy, LUMO energy, and the HOMO–LUMO gap for the 344 compounds in the dataset. We initially generated multiple linear regression models using an expectation maximization algorithm with a Laplacian prior²⁷ to select a sparse subset of descriptors. These linear models generally exhibited low test set predictivities, with R² values ≤0.2. Consequently, we modeled these properties using the well-proven nonlinear BRANNLP (Bayesian Regularized Artificial Neural Network with Laplacian prior)^28,29 method.

Clearly, OPV devices comprise donor and acceptor materials, and ML models must encode the properties of both. We employed three modeling strategies (Supplementary Methods) with increasing complexity in how the acceptor material was encoded. The first and simplest strategy generated separate OPV properties models for each type of acceptor (Supplementary Table 2). The second strategy accounted for different acceptors using a simple “1-hot” indicator variable. Here the different acceptors were encoded in the model as 1 if present and 0 if absent (Appendix B, Supplementary Information). This captures essential differences between the acceptors, the most relevant being the acceptor LUMO energies. Thirdly, we generated models for the six OPV properties in which donor and acceptors were encoded using signature descriptors (Supplementary Table 3). We aimed to make the best predictions for materials and, if possible, to interpret the models in terms of molecular functionality in the donor and acceptor materials structures.

All three modeling strategies predicted PCE, V_oc, J_sc, HOMO, LUMO, and HOMO–LUMO gap with moderate to good efficacy. The PCE models were always robust and predictive, with R² > 0.64 for training set and >0.58 for test set prediction. Exceptions were the HOMO energy and V_oc prediction for the PC61BM acceptor subset, the J_sc prediction for the TiO₂ subset, and the V_oc prediction for the total dataset with acceptors encoded by signature descriptors (Supplementary Tables 2 and 3). In the latter case, it is likely that the V_oc model is overfitted given that that it employs 90 effective parameters in the neural network.

The best models were generated by the second strategy, trained on 344 donor–acceptors pairs, with donors encoded by signature descriptors and acceptors captured by 1-hot binary vectors. We summarize the results of this study below, and the results of modeling OPV properties using strategies 1 and 3 in the Supplementary Material. For strategy 2 models, the dataset was divided into a training set of 276 donor–acceptor pairs and a test set of 68 donor–acceptor pairs by k-means clustering. The BRANNLP nonlinear modeling and variable selection method was used to generate the QSPR models. Table 1 summarizes the performance of these models.

Table 1 Performance of BRANNLP models in predicting OPV properties.

Figure 2 illustrates the performance of the BRANNLP models for the six OPV properties. The majority of models predicting V_oc, J_sc, HOMO, LUMO, and HOMO–LUMO gap resulted in R² for the training and test sets greater than 0.5, which indicates that all these models are sufficiently predictable to provide useful estimates for these properties.

Fig. 2: Prediction of six OPV properties for the training set and test set using the BRANNLP method and signature descriptors for the donors and 1-hot descriptors for the acceptors.

a PCE, b V_oc, c J_sc, d HOMO energy, e LUMO energy, f HOMO–LUMO gap. Training set predictions are in black and test set predictions in orange. Data on the performance of these models are shown in Table 1.

Model validation

It is essential to validate models to determine their predictive power, robustness, and reliability. We assessed this in three ways: predicting properties of a test set partitioned from the dataset and never used in training; randomly scrambling the property values and rebuilding the models (y-scrambling); de novo prediction of OPV properties of materials from the literature not used in the modeling study. Model predictivity was assessed by the R² statistic and the standard error of estimation or prediction for training and test set^30,31. The ability of the ML models to recapitulate the properties of materials in the test set partitioned from the dataset is summarized in Table 1.

In y-scrambling, we randomly distributed the property values and generated ML models using this randomized data³¹. Low R² values for the training set and test set compared to the initial model shows that these models are not chance correlations nor overfitted³². We conducted three y-scrambling tests for each model as shown in Table 2. The R² values were near zero for the ML models trained on these data, showing that the primary models, whose predictions are presented in Table 1, are robust, reliable, and predictive.

Table 2 Statistics of three rounds of Y-scrambling for the models presented in Table 1.

We also used another external test set to assess model predictivity³². After generating the OPV property models and validating using the test sets partitioned from the data and by y-scrambling, we returned to the literature to find additional donor and acceptor materials whose properties could be predicted by the ML models. It is preferable that the DFT method that calculate the properties in external validation set is the same as that used for the dataset used to train the models. Our external validation dataset comprised eight donors and one acceptor (PC61BM) whose OPV properties were reported in the literature³³. The HOMO, LUMO, and HOMO–LUMO gap energies were calculated by the same B3LYP/def2-SVP functional and basis set employed in the HOPV15 dataset. Table 3 shows the statistics for the line of best fit (trend) between the reported and predicted frontier orbital properties.

Table 3 Performance of the models trained using HOPV15 data in predicting the HOMO, LUMO, and HOMO–LUMO gap values of the external validation set (trend line fit).

Table 4 shows the predicted absolute values for the HOMO, LUMO, and HOMO–LUMO gap energies compared to the reported values. The RMSE values for these predictions for the external test set were 0.19, 0.43, and 0.41 eV respectively. These results show that the models have useful abilities to predict at least the frontier orbital energies of materials, provided that are within or close to the domains of applicability of the models used to predict these properties. This proof of concept test of the ability of this type of descriptor and machine-learning method suggests that when larger training sets are available, it will be possible to predict important OPV properties of a larger range of materials.

Table 4 Reported, predicted, and error values of HOMO, LUMO, and HOMO–LUMO gap energies of the external test set.

Descriptor analysis

Machine-learning models, including artificial neural networks, can be hard to interpret in terms of the chemistries needed to improve the OPV properties^34,35. Often the problem is due to the use of arcane descriptors rather than the modeling algorithm per se. This was the motivation to assess the ability of chemically interpretable signature descriptors to model OPV properties.

To this end, we performed analysis on the most relevant descriptors used to build the ML models based on strategy 2. Supplementary Table 4 shows the molecular structures of the most relevant descriptors selected by the BRANNLP method for each of the six OPV properties. In general, for the six properties modeled, there needs to be a balance between electron-withdrawing and donating functional groups, hydrophilicity and hydrophobicity, and conjugation length.

The OPV properties are not completely independent, some are significantly correlated, as reflected in the ML models. PCE is related to V_oc, J_sc, fill factor (FF), and the input power (P_in) by Eq. (1).

$${\mathrm {PCE}} = 100\, \times \,\frac{{V_{\mathrm {oc}} \times {\mathrm {FF}} \times J_{\mathrm {sc}}}}{{P_{\mathrm {in}}}}.$$

(1)

The device efficiency PCE in the Scharber and related models is related to V_oc (Eq. 1), which is a function of the HOMO–LUMO gap. Frontier orbital energies are generally raised by electron-donating substituents and lowered by electron-withdrawing substituents^36,37. The gap is influenced by the presence of strong electron-withdrawing substituents, such as nitro, trifluoromethyl, sulfone, nitrile, and methylene malononitrile (one of the strongest electron-withdrawing functional groups) moieties which lower the energy of the HOMO and LUMO and by electron-donating functional groups such as amine that raise the energy of the frontier orbitals. Thus, HOMO–LUMO gap can be raised or lowered by these substituents and is usually less influenced by these substituent effects. The key molecular fragments identified by sparse feature selection for the V_oc, HOMO, LUMO, and gap models are largely consistent with this theory and experimental observations. HOMO energies were also modulated by F and S substitution in the polyene chain. Hydrophilic groups such as carboxylic acid and N–O–N also modulated the PCE, as did fragments with extended conjugated double bonds such as polyenes, especially those with heteroatoms (O, N) embedded functionality. The most important functional groups for J_sc were hydrophilic moieties such as carboxylic acid and amines, and polyenes with sulfur substitution. Figure 3 shows an example of effective fragments on PCE. Additional examples to illustrate the rest of the relevant descriptors for each OPV properties that we described above are shown in Supplementary Fig. 1.

Fig. 3: Examples of most relevant descriptors for PCE.

a Examples of most relevant descriptors for PCE. b An example donor molecule with the descriptors circled. Additional examples of effective descriptors for PCE and other OPV properties are provided in Supplementary Fig. 1.

In summary, we have shown that chemically interpretable chemical fragment-based descriptors can be used to train ML models that predict six key properties of OPV devices. Our approach leverages resource-intensive DFT calculations into larger regions of materials space, allowing fast and accurate estimates of these important photovoltaic properties for a relatively large number of donor and acceptor materials that may not yet be synthesized. Our study used a synergistic combination of efficient and chemically interpretable descriptors, sparse feature selection, and self-optimizing Bayesian regularized neural networks. The most relevant descriptors for each model provide guidance for materials chemists as to how to synthesize materials with improved OPV properties, or to mine them from larger databases of real or virtual materials. The ML models predicted PCE, V_oc, J_sc, HOMO, LUMO, and HOMO–LUMO gap using simple signature descriptors that encode the molecular properties of the molecules with good efficacy. Although some individual models had relatively low prediction accuracies for the test set, a consensus of all models for each property could identify a statistically robust model for each of the six OPV properties. This work demonstrated the importance of using nonlinear ML methods to map molecular descriptors to important OPV properties, and the value of signature descriptors in building robust, chemically interpretable, and predictive models of these properties. When using these models to screen large libraries of candidate OPV materials prior to synthesis, care must be taken to ensure such libraries lie near the domain of applicability of the models. Clearly, the quality of predictions in this study is dependent of the size, diversity, and accuracy of the underlying dataset. The ML algorithms we have developed in this study can be applied to any dataset of OPV structures and in future work we intend to extend the scope of this work to include large and more accurate computed and experimental datasets.

Methods

Dataset

Being data-driven methods, machine-learning methods are critically dependent on the amount and quality of training data. As QSPR models are only valid within their domain of applicability, a large, diverse dataset with a wide range of properties can generate models with a better generalization ability³⁸. Clearly, all objects in the dataset must be measured under same condition and be reproducible and accurate. The closer the distribution of training data is to a normal distribution, the more accurate the models generated from it are likely to be³⁰. In this study, we employed the Harvard Photovoltaic Dataset (HOPV15, https://www.nature.com/articles/sdata201686#Tab1)²², one of the largest and most diverse datasets available in literature for OPV properties, to train QSPR models. This dataset was compiled from 350 small molecule and polymer electron donors and acceptors, and includes experimental properties collected from literature plus data from quantum chemical calculations and the Scharber model²³. The calculated properties include the values of open circuit potential (V_oc), short circuit density (J_sc), and PCE, which were derived from the model given by Scharber, whereas HOMO energy, LUMO energy of donor, and the HOMO–LUMO gap where calculating using ab initio (Hybrid DFT) methods. Lopez et al.²² generated all possible conformers of each donor materials in the HOPV15 dataset and used various DFT functionals combined with def2-SVP, and the Scharber model to calculate the OPV properties. In this dataset, four different DFT functionals (B3LYP, BP86, M06-2X, and PBE0) were used in the quantum chemical calculations. The property values were averaged over all conformers as there was negligible dependence of properties on conformation. We compared the properties calculated by these methods and found that B3LYP, BP86, and PBE0 generated very similar values, while M06-2X dramatically overestimated the HOMO–LUMO gap and consequently predicted PCEs of ~0. Our QSPR models used the values of PCE, V_oc, and J_sc calculated using the Scharber model and HOMO, LUMO, and HOMO–LUMO gap using the B3LYP^39,40 DFT functional combined with the def2-SVP⁴¹ basis set. While it is well known that the B3LYP functional overestimates electron delocalization, reasonable reproduction of experimental HOMO–LUMO gaps for conjugated systems is still possible⁴². Moreover, the errors tend to be systematic, and trends based on relative values can still be meaningful. We chose B3LYP in order to be consistent with previous studies, but calibration to experimental results has been shown to remove the dependence on the specific functional chosen for DFT calculations^25,26. The advantage of using calculated properties over the experimental ones is that we are confident that these properties are measured with the same method, while the experimental data could have been measured under different conditions. The chemical names of electron acceptors, their SMILES (simplified molecular input line entry system) strings for donors, and values of OPV properties for each pair of donor and acceptors are listed in Appendix A of the Supplementary Information. We removed three donors due to the lack of information about the acceptors used in the PV device (compounds number 18, 82, and 273 in Appendix A) and another three donors because of duplications (compounds number 73, 204, and 334 in Appendix A). Table 5 presents the range of reported properties. A k-means clustering algorithm was used to divide the datasets for each property into a training set (80% of the dataset) used to train the model and a test set (20% of the dataset) used to evaluate the prediction accuracies of the model. This was done to ensure the test set lay within the domain of applicability of the model, and to allow others to reproduce the results we report here.

Table 5 The ranges of the OPV properties calculated at the B3LYP/def2-SVP level of theory.

Molecular descriptors

An important aim of this project was to use molecular descriptors to describe the donors and acceptors that are both efficient and chemically interpretable. In models used for virtual screening of potentially unsynthesized materials, the use of experimentally determined electronic properties or those derived from computationally expensive DFT calculations as descriptors would be costly and time-consuming, as the descriptors for each new molecule of interest would have to be determined individually. On the other hand, signature descriptors can be generated for thousands of candidate materials in a matter of minutes, using freely available software. We used signature descriptors in this study to generate interpretable and predictive models of OPV properties because they are better able to guide synthesis towards improved materials than the arcane descriptors commonly used. As well as generating good models, they are also easier to visualize and provide better guidance as to what functionality in the molecules contributes to, or degrades, properties. Signature descriptors, shown to be effective in other areas of property prediction, are based on the connection path of atoms in the molecule. They provide a systematic calculation system that can describe the “neighborhood” of atoms in a molecule. To understand the concept of signatures, it is important to define the molecular graphs which represent a molecule based on atoms and bonds. Atoms are defined by a set of atom types, which could be provided either from the periodic table or a molecular force field. In molecular graphs, every atom will be assigned an atom type by a function, where atom type considers the possible covalent bonds of each atom. The signature of an atom is a subgraph of molecular graph in a shape of a tree that contains all atoms and all bonds within a specified distance. That is, the signature descriptor of a given atom is the connected path of atoms of a specified length, l. This effectively dissects molecules into an ensemble of fragments, creating a fingerprint whose elements indicate the number of each type of fragment exists in each molecule that is characteristic of the material^43,44,45. Figure 4 demonstrates how signature descriptors can be computed from chemical structures. In this project, we applied the MolSig program⁴⁶ to compute the signature descriptors. We used the Open Babel package⁴⁷ to convert the SMILES strings, which encode the 2D structure of molecules as a string of characters, to Cartesian coordinates of atomic positions in.mol file format. The signature descriptors were generated for path lengths 0–4. These descriptors were then collected, sorted based on size, and any with less than two examples were removed. A total pool of 695 acceptors and donor materials descriptors was generated. We then used the variable selection method to choose the most relevant molecular features for each OPV properties. By mapping back the most relevant signature descriptors onto prototype molecules in the dataset, we can provide important guidance for material scientists to synthesize new materials or improve existing ones. The most relevant signature descriptors for each property are provided in Supplementary Table 4.

Fig. 4: Illustration of the process of generating signature descriptors.

C1 and C2 define a complete ring with the length of 0 to 4. Reprinted with permission from ref. ⁴⁵. Copyright 2003 American Chemical Society.

Feature selection

The signature descriptor method can generate fingerprints containing a very large number of fragment elements. While large pools of descriptors for materials are very useful, great care must be taken to choose a subset of the most relevant descriptors to avoid overfitting models. Overfitted models predict the training set very well but have little predictive power for new data. To aid interpreting models and to avoid overfitting them, it is essential to reduce the dimensionality of the descriptors. Careless selection of subsets of descriptors from a larger pool can also lead to chance correlations⁴⁸. We employed very sparse feature selection methods based on L1 regression and Bayesian regularized neural networks with sparse (Laplacian) prior to achieve very efficient selection of the most relevant descriptors for each OPV property modeled. These methods have been shown in many studies to yield parsimonious subsets of description in a context-dependent way that provide models derived from them with excellent predictive power.

Nonlinear property modeling

Neural networks are universal approximators⁴⁹ that can model any linear or nonlinear and continuous relationships given sufficient training data. However, they have some drawbacks such as overfitting (too many adjustable parameters relative to the number of training data) and overtraining (memorizing training data better but generalizing worse) that generate models with low predictability. ANN (Artificial Neural Network) models are also said to be difficult to interpret, although this is as much to do with interpretable descriptors as the ML method used⁵⁰. Burden and Winkler⁵¹ showed that Bayesian regularization of standard backpropagation neural networks can overcome many of the disadvantages of ANN used to model molecules or materials. The BRANNGP method^50,52 (Bayesian Regularized Artificial Neural Network with Gaussian Prior) was shown to generate robust models of diverse ranges of molecules and properties. BRANNGP can effectively prune less relevant weights from networks (making the models effectively invariant to the number of hidden layer nodes), providing instead an estimation of number of effective parameters²⁸. When the Gaussian prior is replaced by a sparsity-inducing Laplacian prior Bayesian the resulting neural network (BRANNLP) can also prune less relevant descriptors and well as less relevant weights. Thus, a Bayesian regularized neural network with a Laplacian prior is a feedforward, fully connected neural network that uses Bayesian regularization to optimize the sparsity of models, that is, to find the right balance between model complexity (variance) and simplicity (bias)²⁸. This sparse feature selection method is based on L1 regression, similar to the LASSO method^53,54. This neural network method has been shown to generate robust and optimally sparse models of diverse materials properties^55,56,57,58. Here we have used BRANNLP implemented in the CSIRO-Biomodeller package^59,60,61 to predict the properties photovoltaic and electronic properties of compounds. These networks generate predictions of the training and test set properties that are relatively insensitive to the number of nodes in the hidden layer, with the effective parameters in the models being relatively constant as the number of hidden layer nodes increase above a minimum. These networks rarely need more than 2–3 nodes in the hidden layer to model most materials properties. Two hidden layer nodes were used in this study. Our shallow neural networks consist of input (descriptors), hidden (computation), and output layers and are fully connected⁹. The input and output nodes use linear transfer functions, and the hidden layer node sigmoidal functions. The data are mean centered and normalized prior to modeling. A Levenberg–Marquardt algorithm is used for backpropagation. In contrast, deep learning methods use a very large number of hidden layer nodes and non-differentiable transfer functions (e.g. ReLU) and weight dropout to avoid overfitting^62,63. The universal approximation theorem states that shallow neural networks (like the BRANNLP network here) generate models of similar predictive accuracies to DNN given the same training data. Comparative modeling of large dataset of drugs by shallow and DNN algorithms showed that the predictability of models is indeed similar⁹.

To evaluate the performance of the models, the R² statistic and the standard error of estimation (SEE) and standard error of prediction (SEP) for training set and test set, respectively, were calculated. R² is the square correlation coefficient between the predicted and measured values of data points in training set and test set. SEE and SEP represent the root-mean-square error between the predicted and measured values of data points, adjusted for degrees of freedom, in the training and test set, respectively^27,28. SEE and SEP are more robust measures of model quality than R² values¹⁹.