A weighted adaptive least-squares finite element method for the Poisson–Boltzmann equation

2012 ◽  
Vol 218 (9) ◽  
pp. 4892-4902 ◽  
Author(s):  
Jehanzeb Hameed Chaudhry ◽  
Stephen D. Bond ◽  
Luke N. Olson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boltzmann Equation ◽  
Least Squares ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Element Method

scholarly journals A Deep Neural Network based on ResNet for Predicting Solutions of Poisson–Boltzmann Equation

Electronics ◽  
2021 ◽  
Vol 10 (21) ◽  
pp. 2627
Author(s):  
In Kwon ◽  
Gwanghyun Jo ◽  
Kwang-Seong Shin
Keyword(s):  
Neural Network ◽  
Finite Element Method ◽  
Finite Element ◽  
Boltzmann Equation ◽  
Mean Squared Error ◽  
Immersed Finite Element Method ◽  
Immersed Finite Element ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Element Method

The Poisson–Boltzmann equation (PBE) arises in various disciplines including biophysics, electrochemistry, and colloid chemistry, leading to the need for efficient and accurate simulations of PBE. However, most of the finite difference/element methods developed so far are rather complicated to implement. In this study, we develop a ResNet-based artificial neural network (ANN) to predict solutions of PBE. Our networks are robust with respect to the locations of charges and shapes of solvent–solute interfaces. To generate train and test sets, we have solved PBE using immersed finite element method (IFEM) proposed in (Kwon, I.; Kwak, D. Y. Discontinuous bubble immersed finite element method for Poisson–Boltzmann equation. Communications in Computational Physics 2019, 25, pp. 928–946). Once the proposed ANNs are trained, one can predict solutions of PBE in almost real time by a simple substitution of information of charges/interfaces into the networks. Thus, our algorithms can be used effectively in various biomolecular simulations including ion-channeling simulations and calculations of diffusion-controlled enzyme reaction rate. The performance of the ANN is reported in the result section. The comparison between IFEM-generated solutions and network-generated solutions shows that root mean squared error are below 5·10−7. Additionally, blow-ups of electrostatic potentials near the singular charge region and abrupt decreases near the interfaces are represented in a reasonable way.

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