Continuous Convolutional Neural Networks: Coupled Neural PDE and ODE
This addresses the problem of modeling physical systems for researchers in computational physics and machine learning, but it appears incremental as it builds on existing deep learning approaches for ODEs and PDEs.
The paper tackled learning hidden dynamics of physical systems by proposing a variant of Convolutional Neural Networks that models systems as differential equations, achieving results in solving steady-state PDEs like heat and Navier-Stokes equations on irregular domains.
Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden dynamics of a physical system using ordinary differential equation (ODEs) systems (ODEs) and Partial Differential Equation systems (PDEs). Instead of considering the physical system such as image, time -series as a system of multiple layers, this new technique can model a system in the form of Differential Equation (DEs). The proposed method has been assessed by solving several steady-state PDEs on irregular domains, including heat equations, Navier-Stokes equations.