A comparison of PINN approaches for drift-diffusion equations on metric graphs

TL;DR

This paper compares various physics-informed neural network (PINN) methods for solving drift-diffusion equations on metric graphs, highlighting their effectiveness and differences in handling quantum graph problems.

Contribution

It provides a systematic comparison of multiple PINN approaches specifically applied to drift-diffusion equations on metric graphs, a novel focus in the field.

Findings

PINNs can effectively solve drift-diffusion equations on metric graphs.

Different PINN approaches vary in accuracy and computational efficiency.

PINNs offer flexible alternatives to traditional numerical methods for quantum graphs.

Abstract

In this paper we focus on comparing machine learning approaches for quantum graphs, which are metric graphs, i.e., graphs with dedicated edge lengths, and an associated differential operator. In our case the differential equation is a drift-diffusion model. Computational methods for quantum graphs require a careful discretization of the differential operator that also incorporates the node conditions, in our case Kirchhoff-Neumann conditions. Traditional numerical schemes are rather mature but have to be tailored manually when the differential equation becomes the constraint in an optimization problem. Recently, physics informed neural networks (PINNs) have emerged as a versatile tool for the solution of partial differential equations from a range of applications. They offer flexibility to solve parameter identification or optimization problems by only slightly changing the problem formulation used for the forward simulation. We compare several PINN approaches for solving the drift-diffusion on the metric graph.