Physics-Informed DeepONets for drift-diffusion on metric graphs: simulation and parameter identification

TL;DR

This paper introduces a physics-informed deep learning method using DeepONets to efficiently solve nonlinear drift-diffusion equations on metric graphs, enabling accurate simulations and parameter identification for complex networked systems.

Contribution

The paper presents a novel approach coupling multiple DeepONets for drift-diffusion on metric graphs, facilitating model solution and parameter identification without extensive numerical tailoring.

Findings

Effective DeepONet models for inflow, inner, and outflow edges

Accurate solution of drift-diffusion equations on graphs

Framework suitable for inverse and optimization problems

Abstract

We develop a novel physics informed deep learning approach for solving nonlinear drift-diffusion equations on metric graphs. These models represent an important model class with a large number of applications in areas ranging from transport in biological cells to the motion of human crowds. While traditional numerical schemes require a large amount of tailoring, especially in the case of model design or parameter identification problems, physics informed deep operator networks (DeepONet) have emerged as a versatile tool for the solution of partial differential equations with the particular advantage that they easily incorporate parameter identification questions. We here present an approach where we first learn three DeepONet models for representative inflow, inner and outflow edges, resp., and then subsequently couple these models for the solution of the drift-diffusion metric graph problem by relying on an edge-based domain decomposition approach. We illustrate that our framework is applicable for the accurate evaluation of graph-coupled physics models and is well suited for solving optimization or inverse problems on these coupled networks.