Continuous Convolutional Neural Networks: Coupled Neural PDE and ODE

TL;DR

This paper introduces a novel CNN variant that models physical systems through coupled neural PDE and ODE frameworks, enabling the learning of hidden dynamics directly from complex differential equations.

Contribution

It presents a new CNN-based approach that directly learns the underlying differential equations governing physical systems, bridging deep learning with PDE and ODE modeling.

Findings

Successfully applied to steady-state PDEs on irregular domains

Effective in modeling heat and Navier-Stokes equations

Demonstrates potential for physics-informed neural network development

Abstract

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.