  We put forth two physics-informed neural network (PINN) schemes based on
Miura transformations and the novelty of this research is the incorporation of
Miura transformation constraints into neural networks to solve nonlinear PDEs.
The most noteworthy advantage of our method is that we can simply exploit the
initial-boundary data of a solution of a certain nonlinear equation to obtain
the data-driven solution of another evolution equation with the aid of PINNs
and during the process, the Miura transformation plays an indispensable role of
a bridge between solutions of two separate equations. It is tailored to the
inverse process of the Miura transformation and can overcome the difficulties
in solving solutions based on the implicit expression. Moreover, two schemes
are applied to perform abundant computational experiments to effectively
reproduce dynamic behaviors of solutions for the well-known KdV equation and
mKdV equation. Significantly, new data-driven solutions are successfully
simulated and one of the most important results is the discovery of a new
localized wave solution: kink-bell type solution of the defocusing mKdV
equation and it has not been previously observed and reported to our knowledge.
It provides a possibility for new types of numerical solutions by fully
leveraging the many-to-one relationship between solutions before and after
Miura transformations. Performance comparisons in different cases as well as
advantages and disadvantages analysis of two schemes are also discussed. On the
basis of the performance of two schemes and no free lunch theorem, they both
have their own merits and thus more appropriate one should be chosen according
to specific cases.
