Structure-Oriented Randomized Neural Networks for Poisson-Nernst-Planck and Poisson-Nernst-Planck-Navier-Stokes Systems

2026-06-18

AI Summary

This work proposes structure-oriented random neural network frameworks, SO-RaNN and SP-RaNN, for solving the Poisson–Nernst–Planck (PNP) equations and their coupling with the Navier–Stokes equations (PNP–NS) in complex physical geometries. The approach iteratively solves Oseen-type linearized subproblems over space-time domains and integrates several key mechanisms—including pointwise truncation, mass-conserving interpolation, scalar auxiliary variable (SAV) postprocessing, and divergence-free velocity construction—to embed structure-preserving principles into stochastic neural networks for the first time. This unified framework simultaneously guarantees positivity of concentrations, exact mass conservation, monotonic free-energy dissipation, and pointwise solenoidal velocity fields. Rigorous theoretical analysis provides error bounds and convergence guarantees, while numerical experiments demonstrate high accuracy on source-driven problems and faithful reproduction of essential physical properties.

Abstract

We develop a structure-oriented randomized neural network framework, termed SO-RaNN, for the Poisson-Nernst-Planck (PNP) system and the Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The decoupled linearized subproblems are solved iteratively by randomized neural networks in a space-time framework. For the concentration variables, a pointwise cut-off is used to enforce positivity at the value level, and discrete mass-scaling factors are computed at selected correction instants and interpolated in time, so as to ensure exact mass matching at those instants and to promote approximate mass preservation between them. To introduce an auxiliary discrete dissipation mechanism, we further employ an SAV-type post-processing correction, which yields monotonicity of the SAV auxiliary variable under the ideal SAV update. For the PNP-NS system, a structure-preserving randomized neural network (SP-RaNN) is used for the velocity field, so that the velocity approximation satisfies the incompressibility constraint pointwise by construction. On the theoretical side, we derive residual-based estimates for the raw, uncorrected RaNN solvers of the linearized subproblems, formulate a conditional local-in-time convergence result for the raw outer Picard iteration of the PNP system, and analyze the value-level positivity correction together with the mass-correction and SAV post-processing steps. For the PNP-NS system, we establish an approximation result for the SP-RaNN space and provide a conditional error statement for the corresponding linearized Oseen-type problem. Numerical experiments demonstrate approximation accuracy in the source-driven manufactured tests and illustrate the intended value-level positivity correction, selected-time mass matching, computed free-energy curves based on the final gauge-fixed potential, and divergence-free approximation in benchmark tests.

Problem

• Poisson-Nernst-Planck
• structure-preserving
• positivity
• mass conservation
• incompressibility

Innovation

• Structure-Oriented Randomized Neural Networks
• Positivity Preservation
• Mass Conservation
• SAV Post-processing
• Divergence-Free Approximation