We studied the shear deformation of a laminar Al-Cu composite with (100) and (110) interfaces with a shear perpendicular to the lamellae in comparison with pure single crystal Al and Cu at strain rates of 109 s−1 and 108 s−1 and different initial pressures in the range from −3 GPa to +50 GPa. The results of molecular dynamics (MD) for the plasticity initiation are generalized by means of an artificial neural network (ANN) trained by MD data for the (100) interface, and a rate sensitivity parameter identified using MD data for different strain rates. The ANN-based approach allows us to extrapolate MD data to much lower strain rates, which are more relevant for typical dynamic loadings. The considered problem is of interest as an example of the application of the developed ANN-based approach to bimetallic systems, whereas previously it was tested only for pure metals; in addition, Al-Cu composites are of practical interest for technology. The interface between metals reduces the shear strength of the composite in comparison with both pure metals. At an initial pressure below 10 GPa, the plasticity begins in the aluminum part of the composite, while at higher pressures, the plasticity of the copper part starts first. At a pressure above 40 GPa, a phase transition in the aluminum part governs the plasticity development. All this leads to a nonmonotonic pressure dependence of the critical shear stress of the Al-Cu composite in the case of (100) and (110) interfaces without misorientation. Misorientation decreases the critical stress of the nucleation of lattice dislocation and makes the pressure dependence of this stress monotonic. Deformation modes, with a defect-free copper part and a strain-accommodating aluminum part are observed in the MD and can be useful for technological applications related to deformable conducting materials.