相关论文: Kernel Learning of PDE Solution Operators

Neural operators as novel neural architectures for fast approximating solution operators of partial differential equations (PDEs), have shown considerable promise for future scientific computing. However, the mainstream of training neural…

This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated…

Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to…

Deep operator networks (DeepONets) represent a powerful class of data-driven methods for operator learning, demonstrating strong approximation capabilities for a wide range of linear and nonlinear operators. They have shown promising…

This work establishes a rigorous bridge between infinite-dimensional delay dynamics and finite-dimensional Koopman learning, with explicit and interpretable error guarantees. While Koopman analysis is well-developed for ordinary…

We study distributed learning with the least squares regularization scheme in a reproducing kernel Hilbert space (RKHS). By a divide-and-conquer approach, the algorithm partitions a data set into disjoint data subsets, applies the least…

Physics-informed machine learning combines the expressiveness of data-based approaches with the interpretability of physical models. In this context, we consider a general regression problem where the empirical risk is regularized by a…

Accurate estimation of spatial derivatives from discrete and noisy data is central to scientific machine learning and numerical solutions of PDEs. We extend kinetic-based regularization (KBR), a localized multidimensional kernel regression…

Simultaneously solving multiple related learning tasks is beneficial under a variety of circumstances, but the prior knowledge necessary to correctly model task relationships is rarely available in practice. In this paper, we develop a…

We study a class of regularized proximal operators in Wasserstein-2 space. We derive their solutions by kernel integration formulas. We obtain the Wasserstein proximal operator using a pair of forward-backward partial differential equations…

In this work, we describe a novel approach to building a neural PDE solver leveraging recent advances in transformer based neural network architectures. Our model can provide solutions for different values of PDE parameters without any need…

Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to…

We propose an efficient nonparametric strategy for learning a message operator in expectation propagation (EP), which takes as input the set of incoming messages to a factor node, and produces an outgoing message as output. This learned…

Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…

Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDE) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations…

Industrial design evaluation often relies on high-fidelity simulations of governing partial differential equations (PDEs). While accurate, these simulations are computationally expensive, making dense exploration of design spaces…

We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of…

In the realm of computational science and engineering, constructing models that reflect real-world phenomena requires solving partial differential equations (PDEs) with different conditions. Recent advancements in neural operators, such as…

Any applied mathematical model contains parameters. The paper proposes to use kernel learning for the parametric analysis of the model. The approach consists in setting a distribution on the parameter space, obtaining a finite training…