Physics-informed neural network methods based on Miura transformations and discovery of new localized wave solutions

TL;DR

This paper introduces two physics-informed neural network schemes utilizing Miura transformations to solve nonlinear PDEs, enabling the discovery of a new localized wave solution and demonstrating effective reproduction of known solutions.

Contribution

The novel integration of Miura transformation constraints into PINNs allows data-driven solutions of related nonlinear equations and the discovery of previously unreported localized wave solutions.

Findings

Successfully reproduced solutions for KdV and mKdV equations

Discovered a new kink-bell type localized wave solution

Compared advantages and disadvantages of the two proposed schemes

Abstract

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.