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

2025-05-07

AI Summary

This work addresses the forward solution and parameter inversion of nonlinear drift-diffusion equations on metric graphs, supporting physical modeling of complex network phenomena such as cellular transport and human mobility. We propose a physics-informed deep operator network (PI-DeepONet) framework based on edge decomposition, introducing for the first time a cooperative coupling mechanism among three specialized edge-type operators—input, interior, and output edges—that explicitly embed physical constraints such as mass conservation and enable differentiable parameter identification in an integrated manner. By synergistically combining edge-based domain decomposition, automatic differentiation–driven optimization, and physics-informed neural network (PINN) priors, the method achieves high-accuracy forward simulation and robust inverse problem solving across multiscale metric graphs. It significantly improves generalization capability and computational efficiency compared to existing approaches.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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.

Problem

• Solving nonlinear drift-diffusion equations on metric graphs
• Incorporating parameter identification in PDE solutions
• Accurate evaluation of graph-coupled physics models

Innovation

• Physics-informed DeepONets for drift-diffusion equations
• Edge-based domain decomposition approach
• Versatile tool for parameter identification