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

2026-06-18 · Source: Machine Learning · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

A new framework, Structure-Oriented Randomized Neural Networks (SO-RaNN), has been developed to solve the Poisson-Nernst-Planck (PNP) and Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) systems. This approach iteratively solves decoupled linearized subproblems using randomized neural networks within a space-time framework. For concentration variables, SO-RaNN employs a pointwise cut-off to ensure positivity and discrete mass-scaling factors for exact mass matching at specific correction instants, promoting approximate mass preservation. An SAV-type post-processing correction is also integrated to introduce auxiliary discrete dissipation and ensure monotonicity of the SAV auxiliary variable. In the context of the PNP-NS system, a Structure-Preserving Randomized Neural Network (SP-RaNN) is utilized for the velocity field, guaranteeing pointwise satisfaction of the incompressibility constraint. Theoretical contributions include residual-based estimates for raw RaNN solvers, a conditional local-in-time convergence result for the PNP system's Picard iteration, and analysis of the correction steps. Numerical experiments confirm approximation accuracy, value-level positivity, selected-time mass matching, and divergence-free velocity approximations.

Key takeaway

For research scientists developing neural network-based solvers for complex physical systems, particularly those involving coupled transport phenomena like the Poisson-Nernst-Planck equations, you should investigate the SO-RaNN framework. This method provides a robust approach to enforce critical physical constraints such as positivity, mass conservation, and incompressibility directly within randomized neural network solutions. Adopting this structure-oriented design can significantly improve the accuracy and physical consistency of your simulations, mitigating common issues found in purely data-driven or less constrained models.

Key insights

SO-RaNN uses randomized neural networks with structure-preserving corrections to solve complex PNP and PNP-NS systems.

Principles

• Enforce physical constraints like positivity and mass conservation.
• Decouple complex systems into iteratively solvable subproblems.
• Integrate auxiliary dissipation for solution stability.

Method

SO-RaNN iteratively solves linearized subproblems using randomized neural networks, applying pointwise cut-offs for positivity, discrete mass-scaling for mass preservation, and SAV-type post-processing for dissipation.

In practice

• Model ion transport in electrochemical systems.
• Simulate fluid dynamics coupled with charge transport.
• Ensure physical consistency in neural network solvers.

Topics

• Poisson-Nernst-Planck
• Randomized Neural Networks
• Numerical Analysis
• Computational Physics
• Fluid Dynamics Simulation
• Physical Constraints

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.