핵심 개념
Manifold-based autoencoder method using Ricci flow for learning time-dependent dynamics.
초록
The content introduces a method for learning nonlinear dynamics in time using a manifold-based autoencoder guided by Ricci flow. The methodology involves simulating Ricci flow in a physics-informed setting to evolve the latent space. The approach is applied to numerical experiments of partial differential equations, showcasing competitive error rates and generalization capabilities.
Introduction:
Data-driven techniques in exploring PDEs.
Encoder-decoder methods for approximating time-dependent PDE solutions.
Geometric dynamic variational autoencoders (GD-VAEs) show promise.
Problem Formulation:
Task: Learn the solution of parameterized PDEs.
Framework: Autoencoder with manifold collections evolving under Ricci flow.
Methodology:
Utilizes neural networks for encoding, decoding, and metric solution.
Physics-informed neural network for solving the Ricci flow equation.
Experiments:
Tested on viscous Burger's equation, diffusion-reaction equation, Navier-Stokes equation, and 2-d wave equation.
Future Work and Limitations:
Potential improvements in extrapolation data incorporation.
Acknowledgments:
Thanks to Xiaohui Chen for discussions on intrinsic vs extrinsic geometric flows.
통계
Our code is available at https://github.com/agracyk2/Ricci-flow-guided-autoencoders-in-learning-time-dependent-dynamics.