Table 1 List of symbols used in this paper.

\(\mathscr {G}\)

A large chemical space

\(\mathscr {D}^\text {train} = \{(G_i,y_i){\}_{i=1}^N}\subset \mathscr {G}\)

Training dataset of N molecules

\(\mathscr {D}^\text {test} = \{G_i{\}}_{i=N+1}^{N+M}\)

Test dataset of M molecules

\(G_i =(\mathscr {V}_i,\mathscr {E}_i, \sigma _i)\in \mathscr {G}\)

Molecular graph

\(\mathscr {V}_i\)

Set of graph nodes of \(G_i\)

\(\mathscr {E}_i \subseteq \mathscr {V}_i \times \mathscr {V}_i \)

Set of edges of \(G_i\)

\(\sigma : \mathscr {V}_i \cup \mathscr {E}_i \rightarrow \mathscr {L}\)

Node and edge label function

\(\mathscr {L}\)

Set of node and edge labels

\(y_i \in \mathbb {R}\)

Target chemical property value       

Symbol in METHODS                 

Description

\(\mathbf {m}_v^t \in \mathbb {R}^D\)

Message of node v in layer t

\(\mathbf {h}_v^t \in \mathbb {R}^D\)

Feature vector of node v in layer t

\(d_i \in \{0,1\}\)

Domain of \(G_i\)

\(m_t: (\mathbf {h}_v^t,\mathbf {h}_u^t,\sigma (u,v))\rightarrow \mathbb {R}^D\)

GNN message function

\(a: \mathbb {R}^D\rightarrow \mathbb {R}^D\)

GNN activation function

\(u_t: (\mathbf {h}_v^t,\mathbf {m}_v^t)\rightarrow \mathbb {R}^D\)

GNN update function

\(r: \{\mathbf {h}_v^T\}\rightarrow \mathbb {R}^D\)

GNN graph-level readout function

\(f: \mathscr {G}\rightarrow \mathbb {R}\)

Property predictor

\(f_{\text{ F }}: \mathscr {G}\rightarrow \mathbb {R}^D\)

Feature extractor

\(f_{\text{ L }}: \mathbb {R}^D\rightarrow \mathbb {R}\)

Label predictor

\(f_{\text {IPM}}:\mathbb {R}^D\times \mathbb {R}^D\rightarrow \mathbb {R}\)

Internal probability metric

\(f_{\text{ W }}: \mathbb {R}^D\rightarrow \mathbb {R}^2\)

Weight estimator

\(\pi : \mathscr {G}\rightarrow [0,1]\)

Propensity score function

\(\ell : \mathbb {R}\times \mathbb {R} \rightarrow \mathbb {R}^{\ge 0}\)

Regression loss function

\(c: \{0,1\}\times [0,1]\rightarrow \mathbb {R}^{\ge 0}\)

Classification loss function