FD-Net with Auxiliary Time Steps: Fast Prediction of PDEs using Hessian-Free Trust-Region Methods

TL;DR

This paper introduces FD-Net with auxiliary time steps, a neural network framework inspired by finite differences, to efficiently learn and predict the evolution of PDEs using trust-region methods for optimization.

Contribution

It proposes a novel finite-difference inspired CNN framework for PDE discovery and compares second-order trust-region methods with first-order optimizers for training efficiency.

Findings

TRCG outperforms ADAM in training efficiency

The framework accurately predicts PDE solutions

Finite difference-inspired filters improve PDE learning

Abstract

Discovering the underlying physical behavior of complex systems is a crucial, but less well-understood topic in many engineering disciplines. This study proposes a finite-difference inspired convolutional neural network framework to learn hidden partial differential equations from given data and iteratively estimate future dynamical behavior. The methodology designs the filter sizes such that they mimic the finite difference between the neighboring points. By learning the governing equation, the network predicts the future evolution of the solution by using only a few trainable parameters. In this paper, we provide numerical results to compare the efficiency of the second-order Trust-Region Conjugate Gradient (TRCG) method with the first-order ADAM optimizer.