Group Contribution-Based Graph Convolution Network: Pure Property Estimation Model

Abstract

Properties data for chemical compounds are essential information for the design and operation of chemical processes. Experimental values are reported in the literature, but that are too scarce compared with exploding demand for data. When the data are not available, various estimation methods are employed. The group contribution method is one of the standards and simple techniques used today. However, these methods have inherent inaccuracy due to the simplified representation of the molecular structure. More advanced methods are emerging, including improved molecular representations and handling experimental data. However, such processes also suffer from a lack of valid data for adjusting many parameters. We suggest a compromise between a complex machine learning algorithm and a linear group contribution method in this contribution. Instead of representing a molecule using a graph of atoms, we employed bulkier blocks—a graph of functional groups. The new approach dramatically reduced the number of adjustable parameters for machine learning. The result shows higher accuracy than the conventional methods. The whole process was also examined in various aspects—incorporating uncertainties in the data, the robustness of the fitting process, and detecting outlier data.

Graphical Abstract

Appendix A

Description of the Molecular Graph Convolution Network (M-gcn) used in this work.

Similar to the proposed model GC-gcn, the compound is represented in molecular graph ${\varvec{G}} {^{\prime}}(d{^{\prime}},V{^{\prime}}, E{^{\prime}})$. ${\varvec{d}} {^{\prime}}$ represents the dimension of the graph, ${\varvec{V}} {^{\prime}}$ represents the vertex, and ${\varvec{E}} {^{\prime}}$ represents the edge. A vertex should not need to be connected to another vertex but must be connected once in the entire dimension ${\varvec{d}} {^{\prime}}$. The atom feature (or called as node feature) is represented at ${\varvec{G}} {^{\prime}}(V{^{\prime}}),$ while the edge feature is represented at ${\varvec{G}} {^{\prime}}(d{^{\prime}}, E{^{\prime}})$

The rows of the ${\varvec{V}} {^{\prime}}$ indicate the vertex, and the columns indicate the atom types of the vertex. For consistency of model inputs, the shape of the atom feature matrix is fixed to $({N}_{max}\times {N}_{A})$, where ${N}_{max}$ represents the maximum number of atoms and ${N}_{A}$ represents the number of atom descriptors. Dimension ${\varvec{d}} {^{\prime}}$ represents the type of connection between vertices. The description of atoms and bonds is summarized in Table 15. To consider the atom itself, the identity matrix is generally added to ${\varvec{G}} {^{\prime}}(d{^{\prime}}, E{^{\prime}})$. The functions of multi-dimensional graph convolution networks are also designed to have values between -1 and 1. So matrix must be normalized with symmetrical Laplacian.

$${\varvec{B}}{^{\prime}}={{\varvec{D}} {^{\prime}}}^{-1/2}({\varvec{G}} {^{\prime}}(d{^{\prime}}, E{^{\prime}})+{\varvec{I}}){{\varvec{D}} {^{\prime}}}^{-1/2}$$

(11)

where ${\varvec{B}}{^{\prime}}$ is called the bond feature matrix, ${\varvec{D}} {^{\prime}}$ is the degree matrix of ${\varvec{G}} {^{\prime}}(d{^{\prime}}, E{^{\prime}})+{\varvec{I}},$ and I is the identity matrix.

Table 15 Descriptors of the molecule used in M-gcn

1.1 An Example of a Molecular Graph Interpretation of a Molecule—Glycolonitrile

The maximum number of groups ${N}_{max}$ was set to 4, and the descriptor we used in this example is shown in Table 16.

Table 16 Simplified description used in M-gcn example

First, atoms are numbered. The process is shown in Fig. 6.

Fig. 6

The atom labeling result of glycolonitrile

From Fig. 6, we can obtain the atom feature matrix as:

$${\varvec{V}} {^{\prime}}={\varvec{V}} {^{\prime}}\quad \begin{array}{cccc}& C& O& N\\ \{1\}& 0& 0& 1\\ \{2\}& 1& 0& 0\\ \{3\}& 1& 0& 0\\ \{4\}& 0& 1& 0\end{array}$$

${\varvec{V}} {^{\prime}}$ is also can be expressed as ${{\varvec{V}} {^{\prime}}}^{0}$, where 0 indicates that any processing has been held. The bond feature matrix ${\varvec{B}} {^{\prime}}$_{(d', j, k)}, where d' is the dimension, j and k are the indexes of the atom, can be obtained by Eq. 11 as:

$${{\varvec{G}}}^{ {{\prime}}}\left(1,{E}^{{\prime}}\right)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 0\end{array}\right), \quad{\varvec{B}} {^{\prime}}=\left(\begin{array}{cccc}0.50& 0& 0& 0\\ 0& 0.50& 0.41& 0\\ 0& 0.41& 0.33& 0.41\\ 0& 0& 0.41& 0.50\end{array}\right)$$

$${{\varvec{G}}}^{ {{\prime}}}\left(3,{E}^{{\prime}}\right)=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right), \quad{\varvec{B}} {^{\prime}}=\left(\begin{array}{cccc}0.5& 0.5& 0& 0\\ 0.5& 0.5& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

Because there is no double bonds and rings in glycolonitrile, the dimensions 2,4 and 5 of ${{\varvec{G}}}^{ {{\prime}}}\left({\varvec{d}},{E}^{{\prime}}\right)$ are zero matrix.

$${{\varvec{G}}}^{ {{\prime}}}\left(2,{E}^{{\prime}}\right)\quad {{\varvec{G}}}^{ {{\prime}}}\left(4,{E}^{{\prime}}\right)= {{\varvec{G}}}^{ {{\prime}}}\left(5,{E}^{{\prime}}\right)= \left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$$

$${\varvec{B}} {^{\prime}}={{\varvec{B}} {^{\prime}}}_{(4)}= {{\varvec{B}} {^{\prime}}}_{(5)}= \left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

1.2 Multi-Dimensional Graph Convolution

In the multi-dimensional graph convolution process, or message passing phase, the contributions of atoms are updated according to their neighbor atoms and bond types. Unlike a graph convolution network, this procedure consists of two steps. First, the atoms are updated to each dimension. The result of this process is called the message passing function ${{\varvec{H}} {^{\prime}}}^{k+1}$, which can be expressed as follows:

$${{{\varvec{H}} {^{\prime}}}^{k+1}}_{(i)}=ReLU({{{\varvec{B}}}^{ {{\prime}}}}_{\left({\varvec{i}}\right)}\times {{{\varvec{V}}}^{ {{\prime}}}}^{{\varvec{k}}}\times {{{\varvec{W}}}^{ {{\prime}}}}^{k+1}+{{{\varvec{b}}}^{ {{\prime}}}}^{k+1})$$

(12)

where ${{\varvec{W}} {^{\prime}}}^{k}$ is a weight and ${{\varvec{b}} {^{\prime}}}^{k}$ is a bias for the layer k. ReLU is the Rectified Linear Unit function. In this work, we assume that there is no interaction across the graph dimensions. Then, the contribution of atoms ${{\varvec{V}} {^{\prime}}}^{{\varvec{k}}}$ is updated as:

$${{\varvec{V}} {^{\prime}}}^{{\varvec{k}}+1}=ReLU({{{\varvec{W}}}^{ {{\prime}} {^{\prime}}}}^{k+1}\times {{{{Concat}_{i=1}^{{\varvec{D}} {^{\prime}}}{\varvec{H}}}^{ {{\prime}}}}^{k+1}}_{(i)})$$

(13)

where $Concat$ is concatenating the all the matrix, ${{\varvec{W}} {^{\prime}} {^{\prime}}}^{k}$ is a weight, ReLU is the Rectified Linear Unit function.