PDE-constrained optimization refers to the optimization of systems governed by partial differential equations (PDEs). The simulation problem is to solve the PDEs for the state variables (e.g. displacement, velocity, temperature, electric field, magnetic field, species concentration), given appropriate data (e.g. geometry, coefficients, boundary conditions, initial conditions, source functions)....

This paper is concerned with the Sobolev type weak solutions of one class second order quasilinear parabolic partial differential equations (PDEs, for short). First all, similar to Feng, Wang and Zhao [9] Wu Yu [29], we use a family coupled forward-backward stochastic (FBSDEs, short) which satisfy monotonous assumption represent classical PDEs. Then, based on PDEs approximating PDEs, prove exis...

A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The method avoids advanced differential-geometric tools. Instead, it is solely based on calculus, variational calculus, and linear algebra. Densities are constructed as linear combinations of scaling homogeneous terms with...

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...

In many of machine learning problems, it is essential to use not only the training data, but also a priori knowledge about how the world is constrained. In many cases, such knowledge is given in the forms of constraints on differential data or more specifically partial differential equations (PDEs). Neural networks with capabilities to learn differential data can take advantage of such knowledg...

A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank 2, such as the operator associated wi...

The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only which provably overcomes curse of dimensionality general time horizons if coefficient functions and nonlinearity are globally Lipschitz continuous gradient-independent. In this article we extend result to locally monotone functions. Our results cover a ran...

Einstein vacuum equations, that is a system of nonlinear partial differential equations (PDEs) are derived from Weyl metric by using relation between Einstein tensor and metric tensor. The symmetries of Einstein vacuum equations for static axisymmetric gravitational fields are obtained using the Lie classical method. We have examined the optimal system of vector fields which is further used to ...

Travelling wave solutions are important in nonlinear science. These solutions describe phenomena such as vibrations, solitons and propagation with finite speed. In recent years, direct search for exact solutions of nonlinear partial differential equations (PDEs) has become more and more attractive, partly due to the availability of computer systems like Maple or Mathematica, which allow to perf...

A novel mesh-free approach for solving differential equations based on Evolution Strategies (ESs) is presented. Any structure is assumed in the equations making the process general and suitable for linear and nonlinear ordinary and partial differential equations (ODEs, PDEs), as well as systems of ordinary differential equations (SODEs). Candidate solutions are expressed as partial sums of Four...