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Abstract: We build ab initio three-body potential energy surfaces for clusters of the type CO–(pH2) n . The configuration space of the trimers (H2)3 and CO–(H2)2 is sampled with coupled-cluster calculations and the three-body contributions to the nine-dimensional potential energy surface in the van der Waals well are fitted with a neural-network based method of Manzhos and Carrington using dimensionality reduction. The resulting three-body corrections can be used with high-quality pairwise-additive potentials for (H2)2 and CO–H2 available in the literature to build a potential energy surface for CO molecules trapped in solid para-hydrogen. [Copyright &y& Elsevier]