相关论文: Numerical Solution of the Parametric Diffusion Equ…

Traditional Monte Carlo integration using uniform random sampling exhibits degraded efficiency in low-regularity or high-dimensional problems. We propose a novel deep learning framework based on deterministic number-theoretic sampling…

Deep neural networks, when optimized with sufficient data, provide accurate representations of high-dimensional functions; in contrast, function approximation techniques that have predominated in scientific computing do not scale well with…

In this Master Thesis, we study the approximation capabilities of Neural Networks in the context of numerical resolution of elliptic PDEs and Approximation Theory. First of all, in Chapter 1, we introduce the mathematical definition of…

We propose a learning paradigm for numerical approximation of differential invariants of planar curves. Deep neural-networks' (DNNs) universal approximation properties are utilized to estimate geometric measures. The proposed framework is…

We show how the success of deep learning could depend not only on mathematics but also on physics: although well-known mathematical theorems guarantee that neural networks can approximate arbitrary functions well, the class of functions of…

A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface…

Machine learning is at the heart of managing the real-world problems associated with massive data. With the success of neural networks on such large-scale problems, more research in machine learning is being conducted now than ever before.…

We are concerned with the numerical solution of a class integro-differential equations, known as Neural Field Equations, which describe the large-scale dynamics of spatially structured networks of neurons. These equations have many…

Nonlocal neural networks have been proposed and shown to be effective in several computer vision tasks, where the nonlocal operations can directly capture long-range dependencies in the feature space. In this paper, we study the nature of…

Rational and neural network based approximations are efficient tools in modern approximation. These approaches are able to produce accurate approximations to nonsmooth and non-Lipschitz functions, including multivariate domain functions. In…

The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…

The thesis focuses on various techniques to find an alternate approximation method that could be universally used for a wide range of CFD problems but with low computational cost and low runtime. Various techniques have been explored within…

I provide an introduction to the application of deep learning and neural networks for solving partial differential equations (PDEs). The approach, known as physics-informed neural networks (PINNs), involves minimizing the residual of the…

We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning. We suppose that the phenomenon of interest can be modeled by a parametrized partial…

We present a deep transformation model for probabilistic regression. Deep learning is known for outstandingly accurate predictions on complex data but in regression tasks, it is predominantly used to just predict a single number. This…

This work proposes a machine-learning framework for constructing statistical models of errors incurred by approximate solutions to parameterized systems of nonlinear equations. These approximate solutions may arise from early termination of…

Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…

Optimization problems are ubiquitous in our societies and are present in almost every segment of the economy. Most of these optimization problems are NP-hard and computationally demanding, often requiring approximate solutions for…

Machine learning, and in particular neural network models, have revolutionized fields such as image, text, and speech recognition. Today, many important real-world applications in these areas are driven by neural networks. There are also…

Neural network models and deep models are one of the leading and state of the art models in machine learning. Most successful deep neural models are the ones with many layers which highly increases their number of parameters. Training such…