Data-driven discoveries of Bäcklund transformations and soliton evolution equations via deep neural network learning schemes

Abstract

We introduce a deep neural network learning scheme to discover the Bäcklund transforms (BTs) of soliton evolution equations and an enhanced deep learning scheme for data-driven soliton equation discovery based on the known BTs, respectively. The first deep learning scheme takes advantage of some solution (or soliton equation) informations to train the data-driven BT discovery, and is valid in the study of the BT of the sine-Gordon equation, and complex and real Miura transforms between the defocusing (focusing) mKdV equation and KdV equation, as well as the data-driven mKdV equation discovery via the Miura transforms. The second deep learning scheme uses the higher-order solitons generated by the explicit/implicit BTs to study the data-driven discoveries of mKdV and sine-Gordon equations, in which the high-order soliton informations are more powerful for the enhanced leaning soliton equations with higher accuracies.