This paper , accepted for publication in Briefings in Bioinformatics, develops a deep neural network that accepts cell descriptors and molecules of multiple administered drugs and predicts the joint dose-response hypersurface of the combinatorial treatment. Since the dose-response hypersurface over several concentration dimensions fully characterizes the interaction dynamics of the administered drugs, the model is a computational tool that guides the discovery of synergistic treatments. The neural network is a biochemistry-informed universal approximator; it can estimate any shape of a dose-response hypersurface and has desirable invariances built into its architecture. The model excels at interpolating and extrapolating dose-response surfaces; its predictions align well with known mechanisms of action. It is the first model that can estimate joint dose-response hypersurfaces of arbitrarily many drugs, including untried combinations, in the presence of arbitrary, potentially nonlinear interactions between drugs. We release the model itself as well as a database of likely synergistic drug triplets. Our code is available at https://github.com/alonsocampana/PanThera/; the database of likely synergistic drug triplets at https://zenodo.org/records/14001717.