The data that support the findings of this study are openly available in Github, at https://doi.org/10.5281/zenodo.7528885. Physical models can help improve solar cell efficiency during the design phase as well as quality control after the fabrication process. We present a data-driven approach to inverse modeling that can predict the underlying parameters of a finite element method (FEM) solar cell model based on an electroluminescence (EL) image of the solar cell. For training the inverse model, 75,000 synthetic EL images were generated with randomized parameters of the physical cell model. We combine 17 deep convolutional neural networks (CNNs) based on a modified VGG19 architecture into a deep ensemble to add uncertainty estimates. Using the silicon solar cell model, we show that such a novel approach to data-driven statistical inverse modeling can help apply recent developments in deep learning to new engineering applications that require real-time parameterizations of physical models augmented by confidence intervals. The trained network was tested on four different physical solar cell samples and the estimated parameters were used to create the corresponding model representations. Resimulations of the measurements yielded relative deviations of the calculated from the measured junction voltage values of 0.2\% on average with a maximum of 10\%, demonstrating the validity of the approach.