Related papers: Learning elliptic partial differential equations w…
We study the following problem and its applications: given a homogeneous degree-$d$ polynomial $g$ as an arithmetic circuit, and a $d \times d$ matrix $X$ whose entries are homogeneous linear polynomials, compute $g(\partial/\partial x_1,…
This paper presents an a priori error analysis of the Deep Mixed Residual method (MIM) for solving high-order elliptic equations with non-homogeneous boundary conditions, including Dirichlet, Neumann, and Robin conditions. We examine MIM…
Suppose a graph $G$ is stochastically created by uniformly sampling vertices along a line segment and connecting each pair of vertices with a probability that is a known decreasing function of their distance. We ask if it is possible to…
We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is…
Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the…
We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable…
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…
In this paper, to the best of our knowledge, we make the first attempt at studying the parametric semilinear elliptic eigenvalue problems with the parametric coefficient and some power-type nonlinearities. The parametric coefficient is…
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…
We propose a probabilistic way for reducing the cost of classical projection-based model order reduction methods for parameter-dependent linear equations. A reduced order model is here approximated from its random sketch, which is a set of…
Physics-Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss…
We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite…
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…
In this paper, we study the statistical limits in terms of Sobolev norms of gradient descent for solving inverse problem from randomly sampled noisy observations using a general class of objective functions. Our class of objective functions…
Solutions to differential equations are of significant scientific and engineering relevance. Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving differential equations, but they lack a theoretical…
In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent…
Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of $d$-dimensional second-order elliptic PDEs…
The partial differential equation (PDE) plays a significantly important role in many fields of science and engineering. The conventional case of the derivation of PDE mainly relies on first principles and empirical observation. However, the…
Partial differential equation is a powerful tool to characterize various physics systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper, we present a…
In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely,…