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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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Stochastic Simulation of Continuous Time Random Walks: Minimizing Error in Time-Dependent Rate Coefficients for Diffusion-Limited Reactions
A reaction limited by standard diffusion is simulated stochastically to illustrate how the continuous time random walk (CTRW) formalism can be implemented with minimum statistical error. A step-by-step simulation of the diffusive random walk in one dimension reveals the fraction of surviving reactants P(t) as a function of time, and the time-dependent unimolecular reaction rate coefficient K(t). Accuracy is confirmed by comparing the time-dependent simulation to results from the analytical master equation, and the asymptotic solution to that of Fickian diffusion. An early transient feature is shown to arise from higher spatial harmonics in the Fourier distribution of walkers between reaction sites. Statistical ‘shot’ noise in the simulation is quantified along with the offset error due to the discrete time derivative, and an optimal simulation time interval t0 is derived to achieve minimal error in the finite time-difference estimation of the reaction rate. The number of walkers necessary to achieve a given error tolerance is derived, and W = 10^7 walkers is shown to achieve an accuracy of ±0.2% when the survival probability reaches P(t) ∼ 1/3 . The stochastic method presented here serves as an intuitive basis for understanding the CTRW formalism, and can be generalized to model anomalous diffusion-limited reactions to prespecified precision in regimes where the governing wait-time distributions have no analytical solution.
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- Award ID(s):
- 1912694
- PAR ID:
- 10524200
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Journal of Statistical Theory and Practice
- Volume:
- 17
- Issue:
- 3
- ISSN:
- 1559-8608
- Page Range / eLocation ID:
- 46
- Subject(s) / Keyword(s):
- Diffusion limited reaction Diffusion controlled reaction Random walk Transients Noise analysis CTRW
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s42519-023-00343-6
