Related papers: Kernel-based Discretisation for Solving Matrix-Val…
We consider the problem of model reduction of parametrized PDEs where the goal is to approximate any function belonging to the set of solutions at a reduced computational cost. For this, the bottom line of most strategies has so far been…
Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this article, we propose a new reproducing kernel-based approach for estimation and inference of ODE given noisy observations.…
We explore how the analysis of the Carleman linearization can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. We demonstrate the well-posedness and convergence of the truncated Carleman…
We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional…
Traditionally, kernel methods rely on the representer theorem which states that the solution to a learning problem is obtained as a linear combination of the data mapped into the reproducing kernel Hilbert space (RKHS). While elegant from…
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate,…
Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However,…
We present a new framework for computing fine-scale solutions of multiscale Partial Differential Equations (PDEs) using operator learning tools. Obtaining fine-scale solutions of multiscale PDEs can be challenging, but there are many…
Finite-element (FE) discretisations have emerged as a powerful real-space alternative to large-scale Kohn-Sham density functional theory (DFT) calculations, offering systematic convergence, excellent parallel scalability, while…
A new differential equation is derived for an object ${\widehat S}(E,E^\prime,x)$, which when integrated over the appropriate range in $x$, yields the kernel $K(E,E^\prime)$ with which $n$-point correlation functions can be computed in a…
We propose kernel-based collocation methods for numerical solutions to Heath-Jarrow-Morton models with Musiela parametrization. The methods can be seen as the Euler-Maruyama approximation of some finite dimensional stochastic differential…
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…
This article proposes an efficient numerical method for solving nonlinear partial differential equations (PDEs) based on sparse Gaussian processes (SGPs). Gaussian processes (GPs) have been extensively studied for solving PDEs by…
We describe a fast method for solving elliptic partial differential equations (PDEs) with uncertain coefficients using kernel interpolation at a lattice point set. By representing the input random field of the system using the model…
We introduce a novel numerical approach for a class of stochastic dynamic programs which arise as discretizations of backward stochastic differential equations or semi-linear partial differential equations. Solving such dynamic programs…
Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural…
The space mapping technique is used to efficiently solve complex optimization problems. It combines the accuracy of fine model simulations with the speed of coarse model optimizations to approximate the solution of the fine model…
Modern Bayesian optimization and adaptive sampling methods increasingly rely on nonlinear parametric models, yet theoretical guarantees for such models under adaptive data collection remain limited. Existing analyses largely focus on…
We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for input and output function distributions. We introduce the concepts of \textit{numerical operator…
We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of partial difference operators and show that these solutions always have to be exponential polynomials. The respective polynomial space in…