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Supermodeling is a modern, model-ensembling paradigm that integrates severalself-synchronized imperfect sub-models by controlling a few meta-parameters togenerate more accurate predictions of complex systems' dynamics. Continualsynchronization between sub-models allows for trajectory predictions withsuperior accuracy compared to a single model or a classical ensemble ofindependent models whose decision fusion is based on the majority voting oraveraging the outcomes. However, it comes out from numerous observations thatthe supermodeling procedure's convergence depends on a few principal factorssuch as (1) the number of sub-models, (2) their proper selection, and (3) thechoice of the convergent optimization procedure, which assimilates thesupermodel meta-parameters to data. Herein, we focus on modeling the evolutionof the system described by a set of PDEs. We prove that supermodeling isconditionally convergent to a fixed-point attractor regarding only thesupermodel meta-parameters. We investigate the formal conditions of theconvergence of the supermodeling scheme theoretically. We employ the Banachfixed point theorem for the supermodeling correction operator, updating thesynchronization constants' values iteratively. The "nudging" of the supermodelto the ground truth should be well balanced because both too small and toolarge attraction to data cause the supermodel desynchronization. The time-stepsize can control the convergence of the training procedure, by balancing theLipshitz continuity constant of the PDE operator. All the sub-models have to beclose to the ground-truth along the training trajectory but still sufficientlydiverse to explore the phase space better. As an example, we discuss thethree-dimensional supermodel of tumor evolution to demonstrate the supermodel'sperfect fit to artificial data generated based on real medical images.

Supermodeling of tumor dynamics with parallel isogeometric analysis solver