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We introduce a deep neural network learning scheme to learn the B\"acklundtransforms (BTs) of soliton evolution equations and an enhanced deep learningscheme for data-driven soliton equation discovery based on the known BTs,respectively. The first scheme takes advantage of some solution (or solitonequation) information to study the data-driven BT of sine-Gordon equation, andcomplex and real Miura transforms between the defocusing (focusing) mKdVequation and KdV equation, as well as the data-driven mKdV equation discoveryvia the Miura transforms. The second deep learning scheme uses theexplicit/implicit BTs generating the higher-order solitons to train thedata-driven discovery of mKdV and sine-Gordon equations, in which thehigh-order solution informations are more powerful for the enhanced leaningsoliton equations with higher accurates.

Data-driven discoveries of B\"acklund transforms and soliton evolution equations via deep neural network learning schemes