A non-transitory machine readable storage medium having a machine readable program stored therein, wherein the machine readable program, when executed on a processing system, causes the processing system to perform a method of modeling a many particle system, wherein the method includes determining a compensation function, wherein the compensation function compensates errors introduced by an approximation of at least one of a plurality of interaction equations applied on the plurality of interaction equations, wherein the plurality of interaction equations includes the approximation. The method additionally includes solving at least a system of the many particle system without the plurality of interaction equations to extract an uncompensated solution of the many particle system. Additionally, the method includes solving the many particle system with the plurality of interaction equations by a first iteration to extract a plurality of observables in the many particle system, wherein the first iteration includes