Physics-Informed Neural Network-Based Solution for the Poisson-Boltzmann Equation in an Independent Double-Gate MOSFET

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a promising framework for solving partial differential equations (PDEs) by incorporating physical constraints into deep learning models. However, their convergence remains a challenge, particularly in problems where the governing equations are stiff or insufficiently constrained, leading to non-unique or unstable solutions.

This work investigates whether augmenting the standard PINN framework with a small amount of labeled data—specifically, electrostatic potential values at selected collocation points within the domain—can improve convergence. The goal is not to introduce a new hybrid method, but to assess the effectiveness of combining existing physics-informed and data-driven elements in the context of a specific semiconductor modeling problem.

We focus on solving the Poisson–Boltzmann Equation (PBE) for electrostatic potential distribution in an Independent Double-Gate MOSFET. A vanilla PINN struggles to converge despite extensive training, while a purely supervised model performs better numerically but fails to capture the underlying physics. In the hybrid setup, the data-driven loss term initially assists training and is gradually reduced, allowing the model to increasingly rely on the physical laws. An adaptive sampling strategy is also employed, with training points initially concentrated near the boundaries and later distributed more uniformly.

Numerical experiments are conducted to evaluate whether this hybrid setup improves convergence and accuracy compared to standard PINNs and purely data-driven models. Generalization is further tested on unseen values of gate voltages and oxide thicknesses. The findings suggest that even a small amount of labeled data, when used effectively, can aid the convergence of PINNs in complex physical systems.