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Abstract The reaction-diffusion system is naturally used in chemistry to represent substances reacting and diffusing over the spatial domain. Its solution illustrates the underlying process of a chemical reaction and displays diverse spatial patterns of the substances. Numerical methods like finite element method (FEM) are widely used to derive the approximate solution for the reaction-diffusion system. However, these methods require long computation time and huge computation resources when the system becomes complex. In this paper, we study the physics of a two-dimensional one-component reaction-diffusion system by using machine learning. An encoder-decoder based convolutional neural network (CNN) is designed and trained to directly predict the concentration distribution, bypassing the expensive FEM calculation process. Different simulation parameters, boundary conditions, geometry configurations and time are considered as the input features of the proposed learning model. In particular, the trained CNN model manages to learn the time-dependent behaviour of the reaction-diffusion system through the input time feature. Thus, the model is capable of providing concentration prediction at certain time directly with high test accuracy (mean relative error <3.04%) and 300 times faster than the traditional FEM. Our CNN-based learning model provides a rapid and accurate tool for predicting the concentration distribution of the reaction-diffusion system.
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This content will become publicly available on June 1, 2026
Solving spatial-temporal PDEs with arbitrary boundary conditions using physics-constrained convolutional recurrent neural networks
The inception of physics-constrained or physics-informed machine learning represents a paradigm shift, addressing the challenges associated with data scarcity and enhancing model interpretability. This innovative approach incorporates the fundamental laws of physics as constraints, guiding the training process of machine learning models. In this work, the physics-constrained convolutional recurrent neural network is further extended for solving spatial-temporal partial differential equations with arbitrary boundary conditions. Two notable advancements are introduced: the implementation of boundary conditions as soft constraints through finite difference-based differentiation, and the establishment of an adaptive weighting mechanism for the optimal allocation of weights to various losses. These enhancements significantly augment the network's ability to manage intricate boundary conditions and expedite the training process. The efficacy of the proposed model is validated through its application to two-dimensional phase transition, fluid dynamics, and reaction-diffusion problems, which are pivotal in materials modeling. Compared to traditional physics-constrained neural networks, the physics-constrained convolutional recurrent neural network demonstrates a tenfold increase in prediction accuracy within a similar computational budget. Moreover, the model's exceptional performance in extrapolating solutions for the Burgers' equation underscores its utility. Therefore, this research establishes the physics-constrained recurrent neural network as a viable surrogate model for sophisticated spatial-temporal PDE systems, particularly beneficial in scenarios plagued by sparse and noisy datasets.
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- Award ID(s):
- 2347658
- PAR ID:
- 10628129
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Neurocomputing
- Volume:
- 634
- Issue:
- C
- ISSN:
- 0925-2312
- Page Range / eLocation ID:
- 129917
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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