Related papers: Nonlinear compressive reduced basis approximation …
Gaussian processes (GPs) based methods for solving partial differential equations (PDEs) demonstrate great promise by bridging the gap between the theoretical rigor of traditional numerical algorithms and the flexible design of machine…
Physics informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely…
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the…
Given a Banach space X and one of its compact sets F, we consider the problem of finding a good n dimensional space X_n \subset X which can be used to approximate the elements of F. The best possible error we can achieve for such an…
Traditional projection-based reduced-order modeling approximates the full-order model by projecting it onto a linear subspace. With a fast-decaying Kolmogorov $n$-width of the solution manifold, the resulting reduced-order model (ROM) can…
We study the optimal approximation of the solution of an operator equation by certain n-term approximations with respect to specific classes of frames. We study worst case errors and the optimal order of convergence and define suitable…
We consider the estimation of parameter-dependent statistics of functional outputs of elliptic boundary value problems (BVPs) with parametrized random and deterministic inputs. For a given value of the deterministic paremeter, a stochastic…
We extend stochastic basis adaptation and spatial domain decomposition methods to solve time varying stochastic partial differential equations (SPDEs) with a large number of input random parameters. Stochastic basis adaptation allows the…
In this paper, we propose a model reduction method for solving multiscale elliptic PDEs with random coefficients in the multiquery setting using an optimization approach. The optimization approach enables us to construct a set of localized…
We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with…
Partial differential equations (PDEs) form a central component of scientific computing. Among recent advances in deep learning, evolutionary neural networks have been developed to successively capture the temporal dynamics of time-dependent…
This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…
Partial differential equations (PDEs) are widely used for the description of physical and engineering phenomena. Some key parameters involved in PDEs, which represent certain physical properties with important scientific interpretations,…
It is known that standard stochastic Galerkin methods face challenges when solving partial differential equations (PDEs) with random inputs. These challenges are typically attributed to the large number of required physical basis functions…
We present a model reduction approach that extends the original empirical interpolation method to enable accurate and efficient reduced basis approximation of parametrized nonlinear partial differential equations (PDEs). In the presence of…
The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficiently learn quantities of interest of solutions to a…
In this work, we investigate the estimation of a parameter $f$ in PDEs using Bayesian procedures, and focus on posterior distributions constructed using Gaussian process priors, and its variational approximation. We establish contraction…
Neural operators (NOs) struggle with high-contrast multiscale partial differential equations (PDEs), where fine-scale heterogeneities cause large errors. To address this, we use the Generalized Multiscale Finite Element Method (GMsFEM) that…
Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions' qualitative properties seems limited, partially due to the lack of efficient…