Related papers: Error analysis for the deep Kolmogorov method
Uncovering the underlying dynamics from observed data is a critical task in various scientific fields. Recent advances have shown that combining deep learning techniques with linear multistep methods (LMMs) can be highly effective for this…
Stochastic Differential Equations (SDEs) in high dimension, having the structure of finite dimensional approximation of Stochastic Partial Differential Equations (SPDEs), are considered. The aim is to compute numerically expected values and…
Over the last few years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition,…
Deep neural networks are powerful tools for approximating functions, and they are applied to successfully solve various problems in many fields. In this paper, we propose a neural network-based numerical method to solve partial differential…
Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental…
Motivated by the advent of machine learning, the last few years have seen the return of hardware-supported low-precision computing. Computations with fewer digits are faster and more memory and energy efficient, but can be extremely…
This contribution proposes novel data-driven surrogate modeling approaches for parameterized parabolic PDEs, where the parameter dependence can be split into two parts with different decay behavior of the Kolmogorov $N$-width. Such problems…
Deep learning methods - usually consisting of a class of deep neural networks (DNNs) trained by a stochastic gradient descent (SGD) optimization method - are nowadays omnipresent in data-driven learning problems as well as in scientific…
This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and…
We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…
It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense that…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
Recent advances in deep learning makes solving parabolic partial differential equations (PDEs) in high dimensional spaces possible via forward-backward stochastic differential equation (FBSDE) formulations. The implementation of most…
A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…
We develop a new class of distance-aware error bounds that tightly characterize the approximation error of spline neural networks. Our bottom-up approach analyzes the error bound of each neuron (a spline) and then extends it to the full…
Accurate approximation of scalar-valued functions from sample points is a key task in computational science. Recently, machine learning with Deep Neural Networks (DNNs) has emerged as a promising tool for scientific computing, with…
Deep learning methods have achieved great success in solving partial differential equations (PDEs), where the loss is often defined as an integral. The accuracy and efficiency of these algorithms depend greatly on the quadrature method. We…
Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve…
In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already…
Developing efficient and accurate algorithms for chemistry integration is a challenging task due to its strong stiffness and high dimensionality. The current work presents a deep learning-based numerical method called DeepCombustion0.0 to…