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Revising recurrent neural networks to eliminate numerical derivatives in forming Physics-Informed loss terms with respect to time.

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Title: Revising recurrent neural networks to eliminate numerical derivatives in forming Physics-Informed loss terms with respect to time.
Authors: Jahani-nasab, Mahyar1 (AUTHOR), Bijarchi, Mohamad Ali1 (AUTHOR) bijarchi@sharif.edu
Source: Computational Mechanics. Apr2026, Vol. 77 Issue 4, p867-884. 18p.
Subjects: Recurrent neural networks, Numerical differentiation, Back propagation, Partial differential equations, Time, Inverse problems
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 novel architecture, termed the Mutual Interval RNN (MI-RNN), excels in extrapolation, long-term temporal simulation, and inverse parameter discovery tasks. The MI-RNN exhibits lower loss values during training and superior extrapolation accuracy compared to vanilla physics-informed neural network (PINN) architectures. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
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 novel architecture, termed the Mutual Interval RNN (MI-RNN), excels in extrapolation, long-term temporal simulation, and inverse parameter discovery tasks. The MI-RNN exhibits lower loss values during training and superior extrapolation accuracy compared to vanilla physics-informed neural network (PINN) architectures. [ABSTRACT FROM AUTHOR]
ISSN:01787675
DOI:10.1007/s00466-025-02691-5