Supermodeling, a convergent data assimilation meta-procedure used in simulation of tumor progression

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 is an attractive alternative allowing for accurate trajectory predictions 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. In our proof, we assume constant parametrization of the sub-models. 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. From the theoretical estimate, we make the following conclusions. The nudging of the supermodel to the ground truth (real data assimilated) 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.