Related papers: An optimal Petrov-Galerkin framework for operator …
Operator learning techniques have recently emerged as a powerful tool for learning maps between infinite-dimensional Banach spaces. Trained under appropriate constraints, they can also be effective in learning the solution operator of…
We propose a new deflation strategy to accelerate the convergence of the preconditioned conjugate gradient(PCG) method for solving parametric large-scale linear systems of equations. Unlike traditional deflation techniques that rely on…
Neural networks have emerged as powerful surrogates for solving partial differential equations (PDEs), offering significant computational speedups over traditional methods. However, these models suffer from a critical limitation: error…
Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE…
Machine learning, especially deep learning is gaining much attention due to the breakthrough performance in various cognitive applications. Recently, neural networks (NN) have been intensively explored to model partial differential…
Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but…
We propose PO-CKAN, a physics-informed deep operator framework based on Chunkwise Rational Kolmogorov--Arnold Networks (KANs), for approximating the solution operators of partial differential equations. This framework leverages a Deep…
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…
We formulate a stabilized quasi-optimal Petrov-Galerkin method for singularly perturbed convection-diffusion problems based on the variational multiscale method. The stabilization is of Petrov-Galerkin type with a standard finite element…
Partial differential equations (PDEs) with inputs that depend on infinitely many parameters pose serious theoretical and computational challenges. Sophisticated numerical algorithms that automatically determine which parameters need to be…
Design and optimal control problems are among the fundamental, ubiquitous tasks we face in science and engineering. In both cases, we aim to represent and optimize an unknown (black-box) function that associates a performance/outcome to a…
In recent years, deep learning methods, exemplified by Physics-Informed Neural Networks (PINNs), have been widely applied to the numerical solution of differential equations. However, these methods may suffer from limited accuracy, high…
The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…
This paper is concerned with the numerical approximation of quantities of interest associated with solutions to parametric elliptic partial differential equations (PDEs). The key novelty of this work is in its focus on the quantities of…
In the numerical solution of partial differential equations (PDEs), a central question is the one of building variational formulations that are inf-sup stable not only at the infinite-dimensional level, but also at the finite-dimensional…
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…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We…
Physics-Informed Neural Networks (PINNs) have revolutionized the computation of PDE solutions by integrating partial differential equations (PDEs) into the neural network's training process as soft constraints, becoming an important…
Operator learning has become a powerful tool in machine learning for modeling complex physical systems governed by partial differential equations (PDEs). Although Deep Operator Networks (DeepONet) show promise, they require extensive data…