Papers
Topics
Authors
Recent
Detailed Answer
Quick Answer
Concise responses based on abstracts only
Detailed Answer
Well-researched responses based on abstracts and relevant paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses
Gemini 2.5 Flash
Gemini 2.5 Flash 45 tok/s
Gemini 2.5 Pro 54 tok/s Pro
GPT-5 Medium 22 tok/s Pro
GPT-5 High 20 tok/s Pro
GPT-4o 99 tok/s Pro
Kimi K2 183 tok/s Pro
GPT OSS 120B 467 tok/s Pro
Claude Sonnet 4 39 tok/s Pro
2000 character limit reached

Discrete Gradient Flow Approximations of High Dimensional Evolution Partial Differential Equations via Deep Neural Networks (2206.00290v1)

Published 1 Jun 2022 in math.NA and cs.NA

Abstract: We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose discrete gradient flow approximations based on non-standard Dirichlet energies for problems involving essential boundary conditions posed on bounded spatial domains. The imposition of the boundary conditions is realized weakly via non-standard functionals; the latter classically arise in the construction of Galerkin-type numerical methods and are often referred to as "Nitsche-type" methods. Moreover, inspired by the seminal work of Jordan, Kinderleher, and Otto (JKO) \cite{jko}, we consider the second class of discrete gradient flows for special classes of dissipative evolution PDE problems with non-essential boundary conditions. These JKO-type gradient flows are solved via deep neural network approximations. A key, distinct aspect of the proposed methods is that the discretization is constructed via a sequence of residual-type deep neural networks (DNN) corresponding to implicit time-stepping. As a result, a DNN represents the PDE problem solution at each time node. This approach offers several advantages in the training of each DNN. We present a series of numerical experiments which showcase the good performance of Dirichlet-type energy approximations for lower space dimensions and the excellent performance of the JKO-type energies for higher spatial dimensions.

Citations (11)

Summary

We haven't generated a summary for this paper yet.

List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

Lightbulb On Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.