Abstract
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.
Highlights
Poisson–Boltzmann (PB) theory has been used effectively in many disciplines including biophysics, electrochemistry, and colloid chemistry [1,2,3,4,5,6]
Since normalization for PB equation (PBE) requires multiple Newton iterations, the algebraic system (Ax = b) should be solved multiple times, which makes the real-time simulation of PBE solutions almost impossible
We propose a new way of predicting solutions of PBE based on artificial neural networks (ANNs)
Summary
Poisson–Boltzmann (PB) theory has been used effectively in many disciplines including biophysics, electrochemistry, and colloid chemistry [1,2,3,4,5,6]. Biomolecular interactions and dynamics of electrons in semiconductors or plasma can be modeled via the PB equation (PBE). The electrostatic potential and energy of molecular tRNA in an ionic solution can be described via PBE. Developing numerical methods for predicting solutions of PBE is of interest in many fields. There are many FEM/FDM based algorithms for PBE (see [1,4,7,8,9,10,11,12,13] and the references therein), one observes that algorithms are rather complicated due to the nature of the governing equation. A Dirac-delta type singularity arises at the right-hand side of PBE which should be handled properly via some regularization process.
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