Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks

Assessing basic research : Some partial indicators of scientific progress in radio astronomy

Aiming at the heat conduction equation, the heat conduction model is a very typical example of the partial differential equation. This paper focuses on the development and preliminary knowledge of the partial differential equation, the establishment of the heat conduction model and the solution of the heat conduction model. The partial differential equation with fractional Laplace operator is a kind of typical fractional partial differential equation, which has important applications in the fields of science and engineering. Fractional Laplace operator is a kind of nonlocal quasi-differential operator, which is the infinitesimal generator of L é vy steady-state process. It is essentially different from the classical Laplace operator, which leads to the disappearance of some classical properties, which generally brings difficulties to the study of this kind of problems. It is very difficult to solve the explicit solution of partial differential equation with fractional Laplace operator. In order to improve the accuracy and speed of solving partial differential equation, a high precision method for solving partial differential equation based on deep learning network is proposed. A multilayer radial basis function neural network is constructed and its structural layers are determined. By introducing subnetworks into each layer of the multilayer radial basis function neural network, a composite multilayer radial basis function neural network is constructed to improve the real function approximation performance and operation accuracy of the multilayer radial basis function neural network. The high-precision compound multi-layer radial basis function neural network is used to solve the partial differential equation. By giving a specific example of solving the partial differential equation, the solution effect of the method is tested. The results show that the method has very high solution accuracy under the four-layer network structure, which can improve about 1.5 orders of magnitude compared with the first layer. Under different number of training samples, it has higher solution accuracy and speed, and the comprehensive performance is superior. Therefore, it is a practical but challenging work to study the existence of solutions of partial differential equations with fractional Laplace operators. This paper mainly describes the research progress and trends of the existence of solutions of several classes of partial differential equations with fractional Laplace operators, including some of the work done by the author in this field in recent years.

ABSTRACTThe problem of feedback control of bandwidth allocation for Internet links is analysed in this article. It is shown that the procedure for numerical solution of a fluid flow partial differential equation (PDE) which describes the bandwidth allocation on a link results into a set of ordinary differential equations (ODEs) and into an associated state-space equations model. For the local subsystems, into which the flow PDE is decomposed, it becomes possible to apply boundary-based feedback control. The solution of the control problem proceeds by showing that the state-space model of the flow PDE stands for a differentially flat system. Next, for each subsystem which is related to an ODE, a virtual control input is computed, which can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the data-flow PDE is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the data flow PDE of the communication network so as to assure that all its state variables will converge to the desirable set points.

Moving boundary and boundary value problems occur in many physical and engineering processes involving heat transfer and phase changes, they range from icing problems, for example the Stefan problem, to the melting of metals and plastics using lasers. This book reviews and compares some well known and some recently developed methods to solve these problems. Recent advances in the finite difference solution of linear and non-linear partial differential equations are presented and algorithmic manipulations that enhance the computational efficiency are incorporated into the overall schemes. In this new approach, the the finite difference substitutions are made into the solutions of the partial differential equation rather than classically into the partial differential equation itself. This approach can be exploited for many types of partial differential equation and they may therefore be of interest to researchers studying a wide range of applications in physical problems such as diffusion and heat transfer equations, optoelectronic paraxial wave equations, the quantum mechanical wave equations, time dependent pulse propagation, and similar WKB-derived problemsM

Deep neural networks based temporal-difference methods for high-dimensional parabolic partial differential equations

To model biochemical systems in which both noise in the chemical reaction process and spatial movement of molecules is important, both the reaction-diffusion master equation (RDME) and Smoluchowski diffusion-limited reaction (SDLR) partial differential equation (PDE) models have been used. In previous work we showed that the solution to the RDME may be interpreted as an asymptotic approximation in the reaction radius to the solution of the SDLR PDE [S. A. Isaacson, SIAM J. Appl. Math. 70, 77 (2009)]. The approximation was shown to be divergent in the limit that the lattice spacing in the RDME approached zero. In this work we expand upon these results for the special case of the two-molecule annihilation reaction, A+B-->Ø. We first introduce a third stochastic reaction-diffusion PDE model that incorporates a pseudopotential based bimolecular reaction mechanism. The solution to the pseudopotential model is then shown to be an asymptotic approximation to the solution of the SDLR PDE for small reaction radii. We next illustrate how the RDME may be obtained by a formal discretization of the pseudopotential model, motivating why the RDME is itself an asymptotic approximation of the SDLR PDE. Finally, we give a more detailed numerical analysis of the difference between solutions to the RDME and SDLR PDE models as a function of both the reaction-radius and the lattice spacing (in the RDME).

We investigate the existence and Borel summability of formal power series in one variable with holomorphic coefficients, solutions of nonlinear partial differential equations with shrinkings having non-Kowalewski type in the complex domain. Sufficient conditions are given in terms of the shape of the functional partial differential equations and initial conditions. The existence and asymptotic behavior of local sectorial holomorphic solutions of these nonlinear functional partial differential equations are obtained as a by-product. We apply the results to the study of asymptotic the behavior of solutions of some advanced-argument partial differential equations in a neighborhood of infinity.

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional partial differential equations (PDEs), as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerical PDEs in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs’ and PINNs’ residual into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton–Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in less than one hour for 1000 dimensions and in 12 h for 100,000 dimensions on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions.

SONets: Sub-operator learning enhanced neural networks for solving parametric partial differential equations

The fractional partial differential equation is an equation with non-integer partial derivative which is now widely used in solving various problems so that in recent years, many researchers are interested to study fractional partial differential equation. The equation which is used in this paper is the non-homogeneous fractional partial differential equation. The aim of this study is to determine general solution of the partial differential equation through three different order equations (α; 1), (β; 2) and (α; β), where 0 < α ≤ 2 and 0 < β ≤ 1 by using Homotopy Analysis Method (HAM). Furthermore, by taking a convergent sequence, it will be analyzed that the convergence of the sequence of differential equations results in the sequence of solutions of partial fractional non-homogeneous partial differential equations converging to the solution function of the fractional partial differential equation.

Several research papers experimentally validated the derived theoretical solution of the partial differential equation (PDE) of Transient Land Drainage (TLD) using laboratory sand tanks, Hele-Shaw models, resistance networks, and field data. In this research, the main objective is to use an Arduino microcontroller linked with the LabVIEW software to derive experimentally-by using an RC analog circuit connected to the Arduino microcontroller - a solution to Kirchhoff’s partial differential equation describing the unsteady electrical flow through the RC circuit. The RC circuit electrically simulates a non-dimensional virtual aquifer for which an experimental Fourier sine series solution in terms of the analog components is to be derived and used for comparison with the theoretical Fourier sine series solution of the transient partial differential equation of land drainage. The derived Fourier sine series solution of the partial differential equation governing the unsteady electrical flow through the RC analog is written in a non-dimensional form to enable its general application to the case study of this research. Isochrones of the water table drawdown based on those derived from the analog solution are similar in trend and behavior to those of the theoretical solution, indicating that both Fourier solutions are physically and mathematically identical.

A comparison of differential quadrature methods for the solution of partial differential equations

Deep-time neural networks: An efficient approach for solving high-dimensional PDEs

Interface solutions of partial differential equations with point singularity