A universal numerical approach for computing loop amplitudes would permit to achieve high precision in theoretical predictions for a wide range of phenomenologically relevant processes. A major obstacle in developing such methods is the treatment of local infrared and ultraviolet singularities. These need to be removed at the integrand level before numerical integration becomes feasible. In these proceedings we discuss a framework under development for the construction of locally finite two-loop amplitudes for arbitrary processes. It is based on the universality of infrared singularities, realizing infrared factorization manifestly in momentum space. As a specific example, the construction of the locally finite two-loop amplitude for single Higgs production in gluon fusion is discussed in detail. It generalizes to the production of an arbitrary number of electroweak final states in gluon fusion.