Related papers: Quantum framework for parameterizing partial diffe…
Quantum linear system (QLS) algorithms offer the potential to solve large-scale linear systems exponentially faster than classical methods. However, applying QLS algorithms to real-world problems remains challenging due to issues such as…
We propose a machine learning framework to accelerate numerical computations of time-dependent ODEs and PDEs. Our method is based on recasting (generalizations of) existing numerical methods as artificial neural networks, with a set of…
Partial-differential-equation (PDE)-constrained optimization is a well-worn technique for acquiring optimal parameters of systems governed by PDEs. However, this approach is limited to providing a single set of optimal parameters per…
The cost of data input can dominate the run-time of quantum algorithms. Here, we consider data input of arithmetically structured matrices via block encoding circuits, the input model for the quantum singular value transform and related…
The discretized Poisson equation matrix (DPEM) in 1D has been shown to require an exponentially large number of terms when decomposed in the Pauli basis when solving numerical linear algebra problems on a quantum computer. Additionally,…
Matrices with the displacement structures of circulant, Toeplitz, and Hankel types as well as matrices with structures generalizing these types are omnipresent in computations of sciences and engineering. In this paper, we present efficient…
Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at…
Simulations of complex physical systems are typically realized by discretizing partial differential equations (PDEs) on unstructured meshes. While neural networks have recently been explored for surrogate and reduced order modeling of PDE…
Solving linear ordinary differential equations (ODE) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed non-unitary…
Discrete Laplacian operators arise ubiquitously in scientific computing and frequently appear in quantum algorithms for tasks such as linear algebra, Hamiltonian simulation, and partial differential equations. Block encoding provides the…
Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation.…
Partial differential equations (PDEs) are ubiquitous in science and engineering. Prior quantum algorithms for solving the system of linear algebraic equations obtained from discretizing a PDE have a computational complexity that scales at…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order…
Quantum signal processing and quantum singular value transformation are powerful tools to implement polynomial transformations of block-encoded matrices on quantum computers, and has achieved asymptotically optimal complexity in many…
We present a quantum solver for partial differential equations based on a flexible matrix product operator representation. Utilizing mid-circuit measurements and a state-dependent norm correction, this scheme overcomes the restriction of…
Mixed dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such equations naturally arise in a wide range of scientific fields including geology,…
Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require…
A general approach to provide approximate parameterizations of the "small" scales by the "large" ones, is developed for stochastic partial differential equations driven by linear multiplicative noise. This is accomplished via the concept of…
Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, applied mathematics and data science, remain formidable on a classical computer. For uncertainties that arise in partial differential equations…