We propose a novel type of nonlinear solver acceleration for systems of nonlinear partial differential equations (PDEs) that is based on online/adaptive learning. It is applied in the context of multiphase porous media flow. The presented method is built on four pillars: compaction of the training space using dimensionless numbers, offline training in a representative simplistic (two-dimensional) numerical model, control of the numerical relaxation (or other tuning parameter) of a classical nonlinear solver, and online learning to improve the machine learning model in run time (online training). The approach is capable of reducing the number of nonlinear iterations by dynamically adjusting one single global parameter (the relaxation factor) and by learning on-the-job the characteristics of each numerical model. Its implementation is simple and general. In this work, we have also identified the key dimensionless parameters required, compared the performance of different machine learning models, showed the reduction in the number of nonlinear iterations obtained by using the proposed approach in complex realistic (three-dimensional) models, and for the first time properly coupled a machine learning model into an open-source multiphase flow simulator achieving up to 85\% reduction in computational time.