A model lattice ab initio parameterized Heisenberg–Landau magnetic cluster expansion Hamiltonian spanning a broad range of alloy compositions and a large variety of chemical and magnetic configurations has been developed for face-centered cubic Fe–Ni alloys. The thermodynamic and magnetic properties of the alloys are explored using configuration and magnetic Monte Carlo simulations over a temperature range extending well over 1000 K. The predicted face-centered cubic–body-centered cubic coexistence curve, the phase stability of ordered Fe3Ni, FeNi, and FeNi3 intermetallic compounds, and the predicted temperatures of magnetic transitions simulated as functions of alloy composition agree well with experimental observations. Simulations show that magnetic interactions stabilize the face-centered cubic phase of Fe–Ni alloys. Both the model Hamiltonian simulations and ab initio data exhibit a particularly large number of magnetic configurations in a relatively narrow range of alloy compositions corresponding to the occurrence of the Invar effect.