Many integrable hierarchies of differential equations allow a variational description, called Lagrangian multiform or pluri-Lagrangian structure. The fundamental object in this theory is not Lagrange function but $d$-form that integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for field to solution the problem. In paper we present procedure obta...

and Applied Analysis 3 2. Existence and Uniqueness In this section, the existence and uniqueness of the minimizer for the model 1.4 are proved. Some basic notations and properties concerning with the BV 2 space can be seen in 8, 13 . Motivated by 5 , we have the following existence and uniqueness result for the optimization problem 1.4 . Theorem 2.1. Assume Ω is an open, bounded Lipschitz domai...

• We present a novel hybrid framework that enables discovery of unknown fields in inverse partial differential problems. implement trainable finite discretization solver layers are composable with pre-existing neural layers. The network can be pre-trained self-supervised fashion and used on unseen data without further training. This consideration domain specific knowledge about the fields. In c...

Physics-informed neural networks (PINNs) have been widely adopted to solve partial differential equations (PDEs), which could be used simulate physical systems. However, the accuracy of PINNs does not meet needs industry, and severely degrades, especially when PDE solution has sharp transitions. In this paper, we propose a ResNet block-enhanced network architecture better capture transition. Me...

Pre-Hamiltonian matrix operators in total derivatives are considered; they are defined by the property that their images are subalgebras of the Lie algebra of evolutionary vector fields. This construction is close to the Lie algebroids over infinite jet spaces. We assign a class of these operators and the brackets induced in their pre-images to integrable KdV-type hierarchies of symmetry flows ...

This thesis investigates techniques for the solution of partial differential equations (PDE) on advanced parallel architectures comprising central processing units (CPU) and graphics processing units (GPU). Many physical phenomena studied by scientists and engineers aremodelled with PDEs, and these are often computationally expensive to solve. This is one of the main drivers of large-scale comp...

This paper contains three results about generating functions for Lie-theoretic integration of Poisson brackets and their relation to quantization. In the first, we show how construct a function associated germ any local symplectic groupoid provide an explicit (smooth, non-formal) universal formula $$S_\pi $$ integrating structure $$\pi on coordinate space. The second result involves semiclassic...

This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling Laser propagation through turbulent media. In some typical physical contexts, problem reduces to Schrödinger equation with space–time white noise Gaussian or Poisson Lévy type. We identify their resolution via quantization. Nonlinear phenomena such as Kerr effect can be modeled by a n...

Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of R. This family of operators includes the case of the linearized Poisson-Boltzmann equation used to compute the electrostatic free energy...

This paper gives a new integrable hierarchy of nonlinear evolution equations. The DHH equation: mt + umx + 3mux = 0, m = u − uxx, proposed very recently by Desgaperis, Holm and Hone, is the first one in the negative hierarchy while the first one in the positive hierarchy is: mt = 4(m 2 3 )x − 5(m − 2 3 )xxx + (m − 2 3 )xxxxx. The whole hierarchy is shown Laxintegrable through solving a key matr...