Physics Informed Neural Networks (PINNs) For Approximating Nonlinear Dispersive PDEs

Stabilised Nonlinear Inverse Diffusion for Approximating Hyperbolic PDEs

Stabilised backward diffusion processes have shown their use for a number of image enhancement tasks. The goal of this paper is to show that they are also highly useful for designing shock capturing numerical schemes for hyperbolic conservation laws. We propose and investigate a novel flux corrected transport (FCT) type algorithm. It is composed of an advection step capturing the flow dynamics,...

Diophantine equations of nonlinear physics . Part 1 : Nonlinear evolution PDEs ∗

Approximating Rough Stochastic Pdes

We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in [Hairer, Weber, Probab. Theory Related Fields, 2013]. The key idea was to use the theory of controlled rough paths to give definitions of weak / mild solutions and to set up a Picard iteratio...

Exact Traveling Wave Solutions of Nonlinear PDEs in Mathematical Physics

In the present article, we construct the exact traveling wave solutions of nonlinear PDEs in mathematical physics via the variant Boussinesq equations and the coupled KdV equations by using the extended mapping method and auxiliary equation method. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differe...

HIDDEN CONVEXITY IN SOME NONLINEAR PDEs FROM GEOMETRY AND PHYSICS

There is a prejudice among some specialists of non linear partial differential equations and differential geometry: convex analysis is an elegant theory but too rigid to address some of the most interesting and challenging problems in their field. Convex analysis is mostly attached to elliptic and parabolic equations of variational origin, for which a suitable convex potential can be exhibited ...