Revising the Structure of Recurrent Neural Networks to Eliminate Numerical Derivatives in Forming Physics Informed Loss Terms with Respect to Time

TL;DR

This paper introduces the MI-RNN, a modified recurrent neural network structure that eliminates the need for numerical derivatives in physics-informed loss functions, leading to more accurate solutions of unsteady PDEs.

Contribution

The study proposes the MI-RNN architecture with overlapping time intervals and conditional hidden states to directly compute derivatives via backpropagation, improving accuracy over traditional RNNs.

Findings

MI-RNN achieves higher accuracy than traditional RNNs in benchmark PDE problems.

In the heat conduction problem, MI-RNN's relative error was an order of magnitude lower.

The approach simplifies training by removing numerical derivative calculations.

Abstract

Solving unsteady partial differential equations (PDEs) using recurrent neural networks (RNNs) typically requires numerical derivatives between each block of the RNN to form the physics informed loss function. However, this introduces the complexities of numerical derivatives into the training process of these models. In this study, we propose modifying the structure of the traditional RNN to enable the prediction of each block over a time interval, making it possible to calculate the derivative of the output with respect to time using the backpropagation algorithm. To achieve this, the time intervals of these blocks are overlapped, defining a mutual loss function between them. Additionally, the employment of conditional hidden states enables us to achieve a unique solution for each block. The forget factor is utilized to control the influence of the conditional hidden state on the prediction of the subsequent block. This new model, termed the Mutual Interval RNN (MI-RNN), is applied to solve three different benchmarks: the Burgers equation, unsteady heat conduction in an irregular domain, and the Green vortex problem. Our results demonstrate that MI-RNN can find the exact solution more accurately compared to existing RNN models. For instance, in the second problem, MI-RNN achieved one order of magnitude less relative error compared to the RNN model with numerical derivatives.