problem setting in that approach is also significantly different from the one studied here, as their main focus is to extend Bayesian optimisation for materials design from a single-task sample-efficient setup to multi-fidelity models capable of leveraging information from different sources. Previously, Patra et al. have also employed multi-fidelity Gaussian processes to predict polymer band gaps on a small dataset of 382 polymers. Perhaps the closest to our work is the custom message passing algorithm by Chen et al. , which is the first notable work on multi-fidelity graph neural networks. The algorithm, which we refer to as multi-fidelity state embedding or MFSE for short, proposes a small modification to the standard message passing workflow, where in addition to node (atom) and edge (bond) messages and update functions, a state message is constructed and updated alongside them. The state embedding is a global attribute of the graph (molecule), and it is updated according to a fidelity encoding that is present for each data point. In other words, during training each molecule is associated with its structure, regression label, and a fidelity indicator (e.g. low or high) that is used to propagate fidelity-specific messages. The approach has two shortcomings: i) as in Fare et al., jointly training on all fidelities can be problematic when the low-fidelity data outnumbers the high-fidelity samples by more than two orders of magnitude, as the high-fidelity information can effectively be lost, and it remains to be explored if the state embedding mechanism can account for this (see Results, Section 2.5), and ii) coupling low-and high-fidelity training means that to enable high-fidelity inference, the model must be trained on the entirety of the low-fidelity data, dramatically increasing the training times and resource utilisation. In a general approach has been proposed for recursively modelling more than two fidelity levels by means of feature augmentation where a high-fidelity data representation includes a predicted low-fidelity label associated with the previous screening step. This approach is similar to our baseline for the inductive setup where a low-fidelity proxy is obtained using a surrogate model.