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Recent work on Path-Dependent Partial Differential Equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using…
Variational methods are employed in situations where exact Bayesian inference becomes intractable due to the difficulty in performing certain integrals. Typically, variational methods postulate a tractable posterior and formulate a lower…
In this paper, we investigate the application of radial basis functions (RBFs) for the approximation with collocation of the Stokes problem. The approximate solution is constructed in a multi-level fashion, each level using compactly…
Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…
In this paper, we propose and study neural network based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two…
This paper addresses the problem of approximating a function of bounded variation from its scattered data. Radial basis function(RBF) interpolation methods are known to approximate only functions in their native spaces, and to date, there…
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…
An efficient proximal-gradient-based method, called proximal extrapolated gradient method, is designed for solving monotone variational inequality in Hilbert space. The proposed method extends the acceptable range of parameters to obtain…
Real-world problems of operations research are typically high-dimensional and combinatorial. Linear programs are generally used to formulate and efficiently solve these large decision problems. However, in multi-period decision problems, we…
We present a proximal gradient method for solving convex multiobjective optimization problems, where each objective function is the sum of two convex functions, with one assumed to be continuously differentiable. The algorithm incorporates…
One of the core problems in variational inference is a choice of approximate posterior distribution. It is crucial to trade-off between efficient inference with simple families as mean-field models and accuracy of inference. We propose a…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…
A framework is presented for fitting inverse problem models via variational Bayes approximations. This methodology guarantees flexibility to statistical model specification for a broad range of applications, good accuracy and reduced model…
In the paper we consider the problem of multivariate function approximation in polynomial basis. In order to solve this problem, we adjust the least squares method (LSM) by adding information about derivatives of the function. This…
Radial Basis Function Networks (RBFNs) are used primarily to solve curve-fitting problems and for non-linear system modeling. Several algorithms are known for the approximation of a non-linear curve from a sparse data set by means of RBFNs.…
Given a finite set of sample points, meta-learning algorithms aim to learn an optimal adaptation strategy for new, unseen tasks. Often, this data can be ambiguous as it might belong to different tasks concurrently. This is particularly the…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide…
The problem of individualized prediction can be addressed using variants of conformal prediction, obtaining the intervals to which the actual values of the variables of interest belong. Here we present a method based on detecting the…
A method for approximating continuous functions $\mathbb{Z}_{p}^{n}\rightarrow\mathbb{Z}_{p}$ by a linear superposition of continuous functions $\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ is presented and a polynomial regression model is…