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Neural networks possess formidable representational power, rendering theminvaluable in solving complex quantum many-body systems. While they excel atanalyzing static solutions, nonequilibrium processes, including criticaldynamics during a quantum phase transition, pose a greater challenge for neuralnetworks. To address this, we utilize neural networks and machine learningalgorithms to investigate the time evolutions, universal statistics, andcorrelations of topological defects in a one-dimensional transverse-fieldquantum Ising model. Specifically, our analysis involves computing the energyof the system during a quantum phase transition following a linear quench ofthe transverse magnetic field strength. The excitation energies satisfy apower-law relation to the quench rate, indicating a proportional relationshipbetween the excitation energy and the kink numbers. Moreover, we establish auniversal power-law relationship between the first three cumulants of the kinknumbers and the quench rate, indicating a binomial distribution of the kinks.Finally, the normalized kink-kink correlations are also investigated and it isfound that the numerical values are consistent with the analytic formula.

Learning topological defects formation with neural networks in a quantum phase transition