Related papers: Efficient Shallow Ritz Method For 1D Diffusion Pro…
This paper studies the shallow Ritz method for solving one-dimensional diffusion-reaction problems. The method is capable of improving the order of approximation for non-smooth problems. By following a similar approach to the one presented…
This paper analyzes local convergence of the block Newton (BN) method introduced in [5, 6] for one-dimensional shallow neural network approximation to functions and diffusion-reaction problems. The BN method consists of the 2x2 block…
Deep neural networks (DNN) have shown unprecedented success in various computer vision applications such as image classification and object detection. However, it is still a common annoyance during the training phase, that one has to…
In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the $H^{1/2}(\Gamma_D)$ norm is introduced and analyzed for linear elasticity…
The ground state energy of a many-electron system can be approximated by an variational approach in which the total energy of the system is minimized with respect to one and two-body reduced density matrices (RDM) instead of many-electron…
We propose an adaptive diffusion mechanism to optimize a global cost function in a distributed manner over a network of nodes. The cost function is assumed to consist of a collection of individual components. Diffusion adaptation allows the…
In this paper, we propose a generalization of the Batch Normalization (BN) algorithm, diminishing batch normalization (DBN), where we update the BN parameters in a diminishing moving average way. BN is very effective in accelerating the…
In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. A dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional…
The classical Ka\v{c}anov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation,…
Diffusion models have recently achieved great success in the synthesis of high-quality images and videos. However, the existing denoising techniques in diffusion models are commonly based on step-by-step noise predictions, which suffers…
Deep Ensemble (DE) approach is a straightforward technique used to enhance the performance of deep neural networks by training them from different initial points, converging towards various local optima. However, a limitation of this…
Considering a real-valued diffusion, a real-valued reward function and a positive discount rate, we provide an algorithm to solve the optimal stopping problem consisting in finding the optimal expected discounted reward and the optimal…
In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately solving weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in…
We propose a communication- and computation-efficient distributed optimization algorithm using second-order information for solving ERM problems with a nonsmooth regularization term. Current second-order and quasi-Newton methods for this…
This paper considers a nested stochastic distributed optimization problem. In it, approximate solutions to realizations of the inner-problem are leveraged to obtain a Distributed Stochastic Cubic Regularized Newton (DiSCRN) update to the…
Deep learning methods have shown great promise in many practical applications, ranging from speech recognition, visual object recognition, to text processing. However, most of the current deep learning methods suffer from scalability…
Deep learning has been a successful model which can effectively represent several features of input space and remarkably improve image recognition performance on the deep architectures. In our research, an adaptive structural learning…
We study the problem of minimizing a sum of convex objective functions where the components of the objective are available at different nodes of a network and nodes are allowed to only communicate with their neighbors. The use of…
Stochastic optimization methods encounter new challenges in the realm of streaming, characterized by a continuous flow of large, high-dimensional data. While first-order methods, like stochastic gradient descent, are the natural choice,…
Shallow water surface flows commonly entrain sediments, resulting in scouring and/or deposition of the underlying substrate that may strongly influence the pattern of subsequent flow. These coupled phenomena, which can be investigated…