相关论文: An optimal Petrov-Galerkin framework for operator …

The neural operator (NO) framework has emerged as a powerful tool for solving partial differential equations (PDEs). Recent NOs are dominated by the Transformer architecture, which offers NOs the capability to capture long-range…

In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a…

In computational physics, a longstanding challenge lies in finding numerical solutions to partial differential equations (PDEs). Recently, research attention has increasingly focused on Neural Operator methods, which are notable for their…

A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a…

The numerical approximation of partial differential equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and…

Mixed optimal stopping and stochastic control problems define variational inequalities with non-linear Hamilton-Jacobi-Bellman (HJB) operators, whose numerical solution is notoriously difficult and lack of reliable benchmarks. We first use…

This work introduces a paradigm for constructing parametric neural operators that are derived from finite-dimensional representations of Green's operators for linear partial differential equations (PDEs). We refer to such neural operators…

Option pricing often requires solving partial differential equations (PDEs). Although deep learning-based PDE solvers have recently emerged as quick solutions to this problem, their empirical and quantitative accuracy remain not well…

We develop a class of data-driven generative models that approximate the solution operator for parameter-dependent partial differential equations (PDE). We propose a novel probabilistic formulation of the operator learning problem based on…

In this work we give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space…

Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…

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…

The study of neural operators has paved the way for the development of efficient approaches for solving partial differential equations (PDEs) compared with traditional methods. However, most of the existing neural operators lack the…

The deep Kolmogorov method is a simple and popular deep learning based method for approximating solutions of partial differential equations (PDEs) of the Kolmogorov type. In this work we provide an error analysis for the deep Kolmogorov…

Solving nonlinear optimal control problems is a challenging task, particularly for high-dimensional problems. We propose algorithms for model-based policy iterations to solve nonlinear optimal control problems with convergence guarantees.…

We develop a fast and scalable computational framework to solve large-scale and high-dimensional Bayesian optimal experimental design problems. In particular, we consider the problem of optimal observation sensor placement for Bayesian…

Physics-informed neural networks and Physics-informed DeepONet excel in solving partial differential equations; however, they often fail to converge for singularly perturbed problems. To address this, we propose two novel frameworks,…

We propose an evolutionary Multi-objective Optimization for Replica-Exchange-based Physics-informed operator-learning Networks (Morephy-Net) to solve parametric partial differential equations (PDEs) in noisy data regimes, for both forward…

We develop a Petrov-Galerkin stabilization method for multiscale convection-diffusion transport systems. Existing stabilization techniques add a limited number of degrees of freedom in the form of bubble functions or a modified diffusion,…

Traditional numerical methods, such as the finite element method and finite volume method, adress partial differential equations (PDEs) by discretizing them into algebraic equations and solving these iteratively. However, this process is…