The error propagation among estimated parameters reflects the correlation among the parameters.We study the capability of machine learning of"learning"the correlation of estimated parameters.We show that mac...The error propagation among estimated parameters reflects the correlation among the parameters.We study the capability of machine learning of"learning"the correlation of estimated parameters.We show that machine learning can recover the relation between the uncertainties of different parameters,especially,as predicted by the error propagation formula.Gravitational lensing can be used to probe both astrophysics and cosmology.As a practical application,we show that the machine learning is able to intelligently find the error propagation among the gravitational lens parameters(effective lens mass ML and Einstein radiusθ_(E))in accordance with the theoretical formula for the singular isothermal ellipse(SIE)lens model.The relation of errors of lens mass and Einstein radius,(e.g.,the ratio of standard deviations F=σ_(ML)/σ_(θ_(E)))predicted by the deep convolution neural network are consistent with the error propagation formula of the SIE lens model.As a proof-of-principle test,a toy model of linear relation with Gaussian noise is presented.We found that the predictions obtained by machine learning indeed indicate the information about the law of error propagation and the distribution of noise.Error propagation plays a crucial role in identifying the physical relation among parameters,rather than a coincidence relation,therefore we anticipate our case study on the error propagation of machine learning predictions could extend to other physical systems on searching the correlation among parameters.展开更多
In the context of strong gravitational lensing, the magnification of an image is crucially important for constraining various lens models. For several commonly used quadruple lens models, the magnification invariants,...In the context of strong gravitational lensing, the magnification of an image is crucially important for constraining various lens models. For several commonly used quadruple lens models, the magnification invariants, defined as the sum of the signed magnifications of images, have been analytically derived when the image multiplicity is a maximum. In this paper, we further study the magnification of several disk lens models, including （a） exponential disk lens, （b） Gaussian disk lens, （c） modified Hubble profile lens, and another two of the popular three-dimensional symmetrical lens models, （d） NFW lens and （e） Einasto lens. We find that magnification invariant exists for each lens model. Moreover, our results show that magnification invariants can be significantly changed by the characteristic surface mass density kc.展开更多
基金supported by the National Natural Science Foundation of China(grant No.11922303)the Natural Science Foundation of Chongqing(grant No.CSTB2023NSCQ-MSX0103)+1 种基金the Key Research Program of Xingtai 2020ZC005the Fundamental Research Funds for the Central Universities(grant No.2042022kf1182)。
文摘The error propagation among estimated parameters reflects the correlation among the parameters.We study the capability of machine learning of"learning"the correlation of estimated parameters.We show that machine learning can recover the relation between the uncertainties of different parameters,especially,as predicted by the error propagation formula.Gravitational lensing can be used to probe both astrophysics and cosmology.As a practical application,we show that the machine learning is able to intelligently find the error propagation among the gravitational lens parameters(effective lens mass ML and Einstein radiusθ_(E))in accordance with the theoretical formula for the singular isothermal ellipse(SIE)lens model.The relation of errors of lens mass and Einstein radius,(e.g.,the ratio of standard deviations F=σ_(ML)/σ_(θ_(E)))predicted by the deep convolution neural network are consistent with the error propagation formula of the SIE lens model.As a proof-of-principle test,a toy model of linear relation with Gaussian noise is presented.We found that the predictions obtained by machine learning indeed indicate the information about the law of error propagation and the distribution of noise.Error propagation plays a crucial role in identifying the physical relation among parameters,rather than a coincidence relation,therefore we anticipate our case study on the error propagation of machine learning predictions could extend to other physical systems on searching the correlation among parameters.
基金supports of the National Natural Science Foundation of China(Nos.11403103,11603032,11333008 and 11273061)the 973 program(Nos.2015CB857003 and 2013CB834900)+1 种基金China Postdoctoral Science Foundation(2014M551681)the Natural Science Foundation of Jiangsu Province(No.BK20140050).
文摘In the context of strong gravitational lensing, the magnification of an image is crucially important for constraining various lens models. For several commonly used quadruple lens models, the magnification invariants, defined as the sum of the signed magnifications of images, have been analytically derived when the image multiplicity is a maximum. In this paper, we further study the magnification of several disk lens models, including （a） exponential disk lens, （b） Gaussian disk lens, （c） modified Hubble profile lens, and another two of the popular three-dimensional symmetrical lens models, （d） NFW lens and （e） Einasto lens. We find that magnification invariant exists for each lens model. Moreover, our results show that magnification invariants can be significantly changed by the characteristic surface mass density kc.
