Related papers: Computing Anti-Derivatives using Deep Neural Netwo…
The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging…
This paper addresses the problem of modeling and estimating dynamic multi-valued mappings. While most mathematical models provide a unique solution for a given input, real-world applications often lack deterministic solutions. In such…
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 derive computable error estimates for finite element approximations of linear elliptic partial differential equations (PDE) with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that…
We present a neural network based calibration method that performs the calibration task within a few milliseconds for the full implied volatility surface. The framework is consistently applicable throughout a range of volatility models…
In this paper, we propose an adaptive finite difference scheme in order to numerically solve total variation type problems for image processing tasks. The automatic generation of the grid relies on indicators derived from a local estimation…
We present a novel deep learning-based algorithm to accelerate - through the use of Artificial Neural Networks (ANNs) - the convergence of Algebraic Multigrid (AMG) methods for the iterative solution of the linear systems of equations…
Artificial Neuronal Networks are models widely used for many scientific tasks. One of the well-known field of application is the approximation of high-dimensional problems via Deep Learning. In the present paper we investigate the Deep…
To sidestep the curse of dimensionality when computing solutions to Hamilton-Jacobi-Bellman partial differential equations (HJB PDE), we propose an algorithm that leverages a neural network to approximate the value function. We show that…
Adversarial attacks are usually expressed in terms of a gradient-based operation on the input data and model, this results in heavy computations every time an attack is generated. In this work, we solidify the idea of representing…
In this paper, we propose a random gradient-free method for optimization in infinite dimensional Hilbert spaces, applicable to functional optimization in diverse settings. Though such problems are often solved through finite-dimensional…
This paper provides a new approach to derive various arbitrary high order finite difference formulae for the numerical differentiation of analytic functions. In this approach, various first and second order formulae for the numerical…
Deep neural networks as image priors have been recently introduced for problems such as denoising, super-resolution and inpainting with promising performance gains over hand-crafted image priors such as sparsity and low-rank. Unlike learned…
In this article we propose a new deep learning approach to approximate operators related to parametric partial differential equations (PDEs). In particular, we introduce a new strategy to design specific artificial neural network (ANN)…
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…
This paper presents the first discrete-time distributed algorithm to track the tightest ellipsoids that outer approximates the global dynamic intersection of ellipsoids. Given an undirected network, we consider a setup where each node…
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear…
We present a theory for simultaneous approximation of the score function and its derivatives, enabling the handling of data distributions with low-dimensional structure and unbounded support. Our approximation error bounds match those in…
Recent theoretical work on automatic differentiation (autodiff) has focused on characteristics such as correctness and efficiency while assuming that all derivatives are automatically generated by autodiff using program transformation, with…
Deep learning method is of great importance in solving partial differential equations. In this paper, inspired by the failure-informed idea proposed by Gao et.al. (SIAM Journal on Scientific Computing 45(4)(2023)) and as an improvement, a…