相关论文: From Tensor Equations to Numerical Code -- Compute…
In this paper, we propose a novel machine learning method based on adaptive tensor neural network subspace to solve linear time-fractional diffusion-wave equations and nonlinear time-fractional partial integro-differential equations. In…
Solving partial differential equations (PDEs) is an important research means in the fields of physics, biology, and chemistry. As an approximate alternative to numerical methods, PINN has received extensive attention and played an important…
Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for…
Physics-informed neural networks (PINNs) have been increasingly employed due to their capability of modeling complex physics systems. To achieve better expressiveness, increasingly larger network sizes are required in many problems. This…
Partial differential equations (PDEs) are fundamental across numerous scientific fields. As these problems scale to high dimensions, classical numerical schemes introduce severe computational bottlenecks, known as the curse of…
We present a novel deep learning-based algorithm to accelerate - through the use of Artificial Neural Networks (ANNs) - the convergence of Algebraic Multigrid (AMG) methods for the iterative solution of the linear systems of equations…
Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for…
Computer models are used as replacements for physical experiments in a large variety of applications. Nevertheless, direct use of the computer model for the ultimate scientific objective is often limited by the complexity and cost of the…
Partial Differential Equations (PDEs) are integral to modeling many scientific and engineering problems. Physics-informed Neural Networks (PINNs) have emerged as promising tools for solving PDEs by embedding governing equations into the…
We introduce a new class of circuits for constructing efficiently decodable error-correction codes, based on a recently discovered contractible tensor network. We perform an in-depth study of a particular example that can be thought of as…
We present a small computer algebra program for use in Maxwell's theory. The Maxwell equations and the energy-momentum current of the electromagnetic field are formulated in the language of exterior differential forms. The corresponding…
In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial…
Kjolstad et. al. proposed a tensor algebra compiler. It takes expressions that define a tensor element-wise, such as $f_{ij}(a,b,c,d) = \exp\left[-\sum_{k=0}^4 \left((a_{ik}+b_{jk})^2\, c_{ii} + d_{i+k}^3 \right) \right]$, and generates the…
Recent advances in deep learning have enabled us to address the curse of dimensionality (COD) by solving problems in higher dimensions. A subset of such approaches of addressing the COD has led us to solving high-dimensional PDEs. This has…
In the theory and practice of inverse problems for partial differential equations (PDEs) much attention is paid to the problem of the identification of coefficients from some additional information. This work deals with the problem of…
We propose to compute the time-dependent Dirac equation using physics-informed neural networks (PINNs), a new powerful tool in scientific machine learning avoiding the use of approximate derivatives of differential operators. PINNs search…
A linearized numerical scheme is proposed to solve the nonlinear time fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson…
For the study of reactions in High Energy Physics (HEP) automatic computation systems have been developed and are widely used nowadays. GRACE is one of such systems and it has achieved much success in analyzing experimental data. Since we…
During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey…
Solutions to differential equations are of significant scientific and engineering relevance. Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving differential equations, but they lack a theoretical…