Molecular Dynamics Simulations. The high efficiency of the trained MTPs allows for extensive sampling via large-scale classical molecular dynamics (MD) simulations using the atomic simulation environment (ASE) and the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package . Bulk liquid Cu simulations are conducted in a supercell with 864 atoms in the NPT ensemble at 1356 K to provide the bulk liquid density. Surface liquid Cu NVT simulations involve a 1589 Cu atom slab and yield the surface tension. Adding a 364 C atom Gr sheet, as shown in Figure , gives access to the "gap" on liquid Cu (Gr-Cu(l)). For Gr on solid Cu(111) (Gr-Cu(111)), a large simulation cell composed of a slab with 11664 Cu atoms (corresponding to a centered rectangular c(27 x 27√3) Cu(111) surface unit-cell and eight atomic layers) covered with a Gr sheet containing 3136 C atoms is used to compute the "gap" on solid Cu. Here, the lattice constant of Gr is adjusted according to the relaxed lattice of Cu at 0K. For MTP@PBE+MBD, the lattice mismatch in this large simulation cell leads to a residual compression of only 0.1% of Gr at 0K. We further compare two smaller models of Gr-Cu(111) in rectangular c(13 x 7√3) and c(24 x 24√3) Cu(111) surfaces with strains of +3.6% and -0.5% at 0K, respectively (see details in the SI).