相关论文: Deep learning for gradient flows using the Brezis-…

The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…

Gradient descent, or negative gradient flow, is a standard technique in optimization to find minima of functions. Many implementations of gradient descent rely on discretized versions, i.e., moving in the gradient direction for a set step…

Inspired by Gauss-Newton-like methods, we study the benefit of leveraging the structure of deep learning objectives, namely, the composition of a convex loss function and of a nonlinear network, in order to derive better direction oracles…

Real-time simulation of elastic structures is essential in many applications, from computer-guided surgical interventions to interactive design in mechanical engineering. The Finite Element Method is often used as the numerical method of…

We demonstrate how deep convolutional neural networks can be trained to predict 2+1 D hydrodynamic simulation results for flow coefficients, mean-transverse-momentum and charged particle multiplicity from the initial energy density profile.…

In recent years, deep learning for modeling physical phenomena which can be described by partial differential equations (PDEs) have received significant attention. For example, for learning Hamiltonian mechanics, methods based on deep…

This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional…

We are concerned with the study of the well-posedness of a nonlinear diffusion equation with a monotonically increasing multivalued time-dependent nonlinearity derived from a convex continuous potential having a superlinear growth to…

Deep learning has shown great potential for modeling the physical dynamics of complex particle systems such as fluids. Existing approaches, however, require the supervision of consecutive particle properties, including positions and…

We present our deep learning framework to solve and accelerate the Time-Dependent partial differential equation's solution of one and two spatial dimensions. We demonstrate DiffusionNet solver by solving the 2D transient heat conduction…

Motivated by important applications in image processing, we study a class of second-order geometric quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems…

We develop a numerical framework, the Deep Tangent Bundle (DTB) method, that is suitable for computing solutions of evolutionary partial differential equations (PDEs) in high dimensions. The main idea is to use the tangent bundle of an…

In recent years, deep learning technology has been used to solve partial differential equations (PDEs), among which the physics-informed neural networks (PINNs) emerges to be a promising method for solving both forward and inverse PDE…

We derive the system of differential equations for the gradient flow characterizing the training process of linear in-context learning in full generality. Next, we explore the geometric structure of the gradient flows in two instances,…

A long-standing problem at the interface of artificial intelligence and applied mathematics is to devise an algorithm capable of achieving human level or even superhuman proficiency in transforming observed data into predictive mathematical…

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…

We propose a deep backward regression-based (DBR) scheme for solving high-dimensional nonlinear parabolic partial differential equations. Building on the DBDP method of Hur\'e, Pham, and Warin~\cite{HCPHWX20}, the proposed method…

In this work, we extend deep learning-based numerical methods to fully coupled forward-backward stochastic differential equations (FBSDEs) within a non-Markovian framework. Error estimates and convergence are provided. In contrast to the…

In this paper, we propose a new adaptation of the D-iteration algorithm to numerically solve the differential equations. This problem can be reinterpreted in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the…

Deep kernel learning is a promising combination of deep neural networks and nonparametric function learning. However, as a data driven approach, the performance of deep kernel learning can still be restricted by scarce or insufficient data,…