Understanding the predication mechanism of deep learning through error propagation among parameters in strong lensing case

The error propagation among estimated parameters reflects the correlationamong the parameters. We study the capability of machine learning of "learning"the correlation of estimated parameters. We show that machine learning canrecover the relation between the uncertainties of different parameters,especially, as predicted by the error propagation formula. Gravitationallensing can be used to probe both astrophysics and cosmology. As a practicalapplication, we show that the machine learning is able to intelligently findthe error propagation among the gravitational lens parameters (effective lensmass $M_{L}$ and Einstein radius $\theta_{E}$) in accordance with thetheoretical formula for the singular isothermal ellipse (SIE) lens model. Therelation of errors of lens mass and Einstein radius, (e.g. the ratio ofstandard deviations $\mathcal{F}=\sigma_{\hat{ M_{L}}}/\sigma_{\hat{\theta_{E}}}$) predicted by the deep convolution neural networkare consistent with the error propagation formula of SIE lens model. As aproof-of-principle test, a toy model of linear relation with Gaussian noise ispresented. We found that the predictions obtained by machine learning indeedindicate the information about the law of error propagation and thedistribution of noise. Error propagation plays a crucial role in identifyingthe physical relation among parameters, rather than a coincidence relation,therefore we anticipate our case study on the error propagation of machinelearning predictions could extend to other physical systems on searching thecorrelation among parameters.