Related papers: Quantum algorithm for partial differential equatio…
The variational principle serves as a fundamental framework for describing equilibrium states of physical systems via the minimization or extremization of an energy-like functional. While quantum algorithms have demonstrated promising…
This paper introduces Quantum Orthogonal Separable Physics-Informed Neural Networks (QO-SPINNs), a novel architecture for solving Partial Differential Equations, integrating quantum computing principles to address the computational…
Inverse problems involving partial differential equations (PDEs) with discontinuous coefficients are fundamental challenges in modeling complex spatiotemporal systems with heterogeneous structures and uncertain dynamics. Traditional…
In this work, we present a case study in implementing a variational quantum algorithm for solving the Poisson equation, which is a commonly encountered partial differential equation in science and engineering. We highlight the practical…
Deep models have achieved impressive progress in solving partial differential equations (PDEs). A burgeoning paradigm is learning neural operators to approximate the input-output mappings of PDEs. While previous deep models have explored…
Partial differential equations (PDEs) with spatially-varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in…
Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from…
Scientific and engineering problems often involve parametric partial differential equations (PDEs), such as uncertainty quantification, optimizations, and inverse problems. However, solving these PDEs repeatedly can be prohibitively…
Elliptic Partial Differential Equations (PDEs) play a central role in computing the equilibrium conditions of physical problems (heat, gravitation, electrostatics, etc.). Efficient solutions to elliptic PDEs are also relevant to computer…
Variational quantum algorithms are one of the most promising methods that can be implemented on noisy intermediate-scale quantum (NISQ) machines to achieve a quantum advantage over classical computers. This article describes the use of a…
In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded…
We introduce a unified framework -- Quantum Neural Ordinary and Partial Differential Equations (QNODEs and QNPDEs) -- which extends the continuous-time formalism of classical neural ordinary and partial differential equations into quantum…
This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear partial differential equations (PDEs). The main idea of these algorithms is to utilize the solutions on the $k$-th…
We propose quantum algorithms for complex-valued nonlinear partial differential equations in the strongly nonlinear regime, where the dynamics is governed by vortex cores, phase singularities, and nonlinear vortex interactions. Examples…
Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…
We study a class of partial differential equations (PDEs) in the family of the so-called Euler-Poincar\'e differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular…
In this work, a novel quantum Fourier ordinary differential equation (ODE) solver is proposed to solve both linear and nonlinear partial differential equations (PDEs). Traditional quantum ODE solvers transform a PDE into an ODE system via…
There is a rising interest in Spatio-temporal systems described by Partial Differential Equations (PDEs) among the control community. Not only are these systems challenging to control, but the sizing and placement of their actuation is an…
Spectral methods for solving partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) often use Fourier or polynomial spectral expansions on either uniform and non-uniform grids. However, while very widely…
In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu…