  Supermodeling is a modern, model-ensembling paradigm that integrates several
self-synchronized imperfect sub-models by controlling a few meta-parameters to
generate more accurate predictions of complex systems' dynamics. Continual
synchronization between sub-models allows for trajectory predictions with
superior accuracy compared to a single model or a classical ensemble of
independent models whose decision fusion is based on the majority voting or
averaging the outcomes. However, it comes out from numerous observations that
the supermodeling procedure's convergence depends on a few principal factors
such as (1) the number of sub-models, (2) their proper selection, and (3) the
choice of the convergent optimization procedure, which assimilates the
supermodel meta-parameters to data. Herein, we focus on modeling the evolution
of the system described by a set of PDEs. We prove that supermodeling is
conditionally convergent to a fixed-point attractor regarding only the
supermodel meta-parameters. We investigate the formal conditions of the
convergence of the supermodeling scheme theoretically. We employ the Banach
fixed point theorem for the supermodeling correction operator, updating the
synchronization constants' values iteratively. The "nudging" of the supermodel
to the ground truth should be well balanced because both too small and too
large attraction to data cause the supermodel desynchronization. The time-step
size can control the convergence of the training procedure, by balancing the
Lipshitz continuity constant of the PDE operator. All the sub-models have to be
close to the ground-truth along the training trajectory but still sufficiently
diverse to explore the phase space better. As an example, we discuss the
three-dimensional supermodel of tumor evolution to demonstrate the supermodel's
perfect fit to artificial data generated based on real medical images.
