Data-driven discoveries of Bäcklund transforms and soliton evolution equations via deep neural network learning schemes
This work addresses the challenge of automating the discovery of soliton equations and transforms in mathematical physics, representing an incremental advancement by applying deep learning to known bottlenecks in the field.
The authors tackled the problem of discovering Bäcklund transforms and soliton evolution equations by introducing deep neural network learning schemes, achieving data-driven discoveries for equations like sine-Gordon and mKdV with enhanced accuracy using higher-order soliton information.
We introduce a deep neural network learning scheme to learn 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 scheme takes advantage of some solution (or soliton equation) information to study the data-driven BT of 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 explicit/implicit BTs generating the higher-order solitons to train the data-driven discovery of mKdV and sine-Gordon equations, in which the high-order solution informations are more powerful for the enhanced leaning soliton equations with higher accurates.