Although this approach is physically motivated, it is limited to modestly-sized molecular systems, as the computational cost of evaluating the energy scales exponentially with system size, making it impractical to consider interactions beyond the three-body order. ACE overcomes this limitation by introducing the concept of atomic neighbour density, where the energy of each atom depends on the many-body interactions with its N neighbours within a defined cut-off radius. The validity of this concept is based on the assumption that the energy of each atom only depends on its local environment. Second, it projects these densities onto physically invariant basis functions. This procedure ensures that the evaluation cost of many-body terms scales linearly, rather than exponentially, with the number of neighbours, regardless of the body order. A detailed description of the method is provided in Ref. 85. The capability of the ACE descriptor to accurately map the PES enables the use of simple linear regression for fitting, resulting in an accurate and data-efficient approach to train MLPs. MACE combines the ACE descriptor with an equivariant message-passing neural network architecture, incorporating body-order contributions as node features. Using graph neural networks (GNNs), the body-order term and cluster region implicitly expand with the number of message-passing layers, resulting in a more accurate representation of atomic environments. Both ACE and MACE have been shown to reliably predict the energies and forces of molecules and condensed phase systems. Linear ACE provides high accuracy in lowdata regimes, making it particularly suitable for use in the early stages of AL, where typically small data sets are used. In this work, we use linear ACE to build the training data sets using the AL loop, while MACE is used to expand the data sets and fine-tune the final potentials used for production MD simulations.