2.2.2. Proposed GCNN Architecture

Our GCNN has a classical architecture for graph classification/regression. It contains three main parts: Graph convolution part, i.e., the sequence of graph convolutional layers (GCL); Feature aggregation, i.e. the pool (aggregation) function that maps a set of hidden vectors to output vector; Multilayer perceptron (MLP) part, i.e., MLP at the top of GCNN, for the final prediction.

We use the modified Gated graph convolution⁹³ with gated recurrent unit⁹⁴ as a basic operation for our GCNNs. We account the edge features using the learnable message function, which uses the edge feature vector as an input and applies the k × k matrix A implemented by MLP to transform it. This MLP is trained simultaneously with the GCL layer.

Since a typical molecular graph of a polyimide repeating unit may contain up to about several tens of vertexes, a lot of iterations are required to pass information from one vertex to another. For this reason, along with the graph neural network (GNN) part we use an additional 2-GNN part in our GCNNs, which distinguishes our network from that used in refs (54) and (63). Previously, Grohe et al. have shown that adding a 2-GNN part increases the expressive ability of ordinary graph neural networks, which is usually limited.⁹⁵ Our preliminary tests have also demonstrated that adding a 2-GNN part in parallel to the original graph neural networks improved its performance. A schematic illustration of the proposed GCNNs architecture is presented in Figure ​Figure55.

Schematic illustration of the proposed GCNN architecture. Colored bars indicate feature vectors of vertexes. G denotes input molecular graph of the polyimide repeating unit, G′ is the input graph processed with the first GNN block (first subnet), G₂ is the “2-graph” formed from the G′, G₂′ is the “2-graph” processed with the second GNN block (second subnet), and MLP is the multilayer perceptron. For the convenience of drawing the feature vectors of the edges are not shown.

Each of our GCNNs has five GCLs. There are three GCLs before the “2-graph” conversion procedure and two graph convolutional layers after it. After each GCL, we use Rectified Linear Unit (ReLU) activation.

For each GCL, we apply three message passing sessions. In addition, we use the two-layer MLP with a linear layer consisting of 256 neurons followed by a linear layer consisting of 4096 neurons (the dimension of A matrix is 64 × 64) with ReLU nonlinearity in between in order to compute A matrix. For readout, we use ordinary sum operation. After readout, stage, and concatenation, the resulting vector is processed by a linear layer with 256 neurons with ReLU to get the final feature vector. Then we use a one-neuron layer to get the final prediction. The dimensions of the input graph features are reduced to a fixed size with 64 channels using two linear layers (one for the node features and one for the edge features).

For a more comprehensive review about our GCNN architecture, we refer the reader to Section S4 in the Supporting Information. Testing code and pretrained models are available on https://github.com/polycomplab/GCNN_PI_glass_transition.git.