Related papers: A deep learning method for solving high-order nonl…
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the…
We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…
In this paper, we investigate the applications of operator learning, specifically DeepONet, for solving nonlinear partial differential equations (PDEs). Unlike conventional function learning methods that require training separate neural…
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new…
Deep neural networks have rightfully won the place of one of the most accurate analysis tools in high energy physics. In this paper we will cover several methods of improving the performance of a deep neural network in a classification task…
We present an approach for solving optimal Dirichlet boundary control problems of nonlinear optics by using deep learning. For computing high resolution approximations of the solution to the nonlinear wave model, we propose higher order…
This paper presents a partial differential equation framework for deep residual neural networks and for the associated learning problem. This is done by carrying out the continuum limits of neural networks with respect to width and depth.…
The moments of spatial probabilistic systems are often given by an infinite hierarchy of coupled differential equations. Moment closure methods are used to approximate a subset of low order moments by terminating the hierarchy at some order…
In this paper, we investigate neural networks applied to multiscale simulations and discuss a design of a novel deep neural network model reduction approach for multiscale problems. Due to the multiscale nature of the medium, the fine-grid…
A high-order finite element method is proposed to solve the nonlinear convection-diffusion equation on a time-varying domain whose boundary is implicitly driven by the solution of the equation. The method is semi-implicit in the sense that…
We propose a deep learning based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz method is naturally nonlinear, naturally…
In this paper, we present a novel framework to solve differential equations based on multilayer feedforward network. Previous works indicate that solvers based on neural network have low accuracy due to that the boundary conditions are not…
In this work, we propose a multi-stage training strategy for the development of deep learning algorithms applied to problems with multiscale features. Each stage of the pro-posed strategy shares an (almost) identical network structure and…
High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE…
We propose a deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) and high-dimensional forward-backward stochastic differential equations with jumps (FBSDEJs), where the jump-diffusion…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…
This work investigates the application of the Newton's method for the numerical solution of a nonlinear boundary value problem formulated through an ordinary differential equation (ODE). Nonlinear ODEs arise in various mathematical modeling…
We consider a hierarchy of nonlinear Schr\"{o}dinger equations (NLSEs) and forecast the evolution of positon solutions using a deep learning approach called Physics Informed Neural Networks (PINN). Notably, the PINN algorithm accurately…
The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of extended dynamic mode…