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

Jan Blechschmidt, Tom-Christian Riemer, Max Winkler, Martin Stoll, Jan-F. Pietschmann

TL;DR

The paper tackles solving nonlinear drift-diffusion on metric graphs and identifying unknown parameters by leveraging physics-informed DeepONets. It introduces three edge-type surrogates (inflow, inner, outflow) that are coupled at graph vertices to satisfy Kirchhoff-Neumann and boundary conditions, forming a graph-wide solver for $\mathcal{H}(\rho_e)=\partial_t\rho_e-\partial_x(\varepsilon\partial_x\rho_e-\nu_e f(\rho_e))=0$. The method combines branch and Fourier-trunk networks, physics residual losses, and an RBF-based time-varying coupling to enable efficient forward simulations and rapid inverse problems on graphs of varying topology. The results demonstrate accurate edge-wise solutions, successful recovery of initial conditions and velocities in inverse problems, and scalable performance on large graphs, highlighting potential applications in traffic and biological transport.

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.

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

TL;DR

The paper tackles solving nonlinear drift-diffusion on metric graphs and identifying unknown parameters by leveraging physics-informed DeepONets. It introduces three edge-type surrogates (inflow, inner, outflow) that are coupled at graph vertices to satisfy Kirchhoff-Neumann and boundary conditions, forming a graph-wide solver for . The method combines branch and Fourier-trunk networks, physics residual losses, and an RBF-based time-varying coupling to enable efficient forward simulations and rapid inverse problems on graphs of varying topology. The results demonstrate accurate edge-wise solutions, successful recovery of initial conditions and velocities in inverse problems, and scalable performance on large graphs, highlighting potential applications in traffic and biological transport.

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.
Paper Structure (16 sections, 6 theorems, 70 equations, 13 figures, 5 tables)

This paper contains 16 sections, 6 theorems, 70 equations, 13 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Drift-diffusion equations on metric graphs
  3. Learning surrogate models
  4. Model evaluation
  5. Inverse problems
  6. Conclusion
  7. Proof of Theorem \ref{['thm:existence']}
  8. Numerical solvers
  9. Finite volume scheme
  10. DeepONet further details
  11. Comments
  12. Discussion of strong GP Prior
  13. Further applications
  14. Computational complexity
  15. Loss plots
  16. ...and 1 more sections

Key Result

Theorem 2.2

Let the initial data $u^{\textup{init}} \in L^2(\Gamma)$ satisfy $0 \le u^{\textup{init}} \le 1$ a.e. on $\mathcal{E}$ and let nonnegative functions $u^{\textup{inflow}}_{v}, u^{\textup{outflow}}_v \in L^\infty(0,T)$, $v\in \mathcal{V}_D$ and non-negative numbers $\nu_e$, $e\in\mathcal{E}$, be given for all test functions $\varphi \in H^1(\Gamma)$ and a.a. $t\in(0,T)$. Here $L^2$ denotes the space

Figures (13)

  • Figure 1: Model graphs that were used to generate training data for physics-informed DeepONets. Green edges are used to train inflow model, blue ones for inner model and red ones for outflow model.
  • Figure 2: Illustration of random GP training data: initial condition measurements $u^{\textup{init}}$ (blue), inflow measurements $u^{\textup{inflow}}_{v}$ (green), outflow measurements $u^{\textup{outflow}}_{v}$ (orange).
  • Figure 3: Illustration of physics-informed DeepONet adapted to our setting from wang2021learning.
  • Figure 4: Model graphs that were used to verify our methodology.
  • Figure 5: Upper row: Almost indistinguishable reference solution (solid) and PI DeepONet solution (dashed) on model graph at $t=0.5$ (left) and $t=1.0$ (right). Lower row: Absolute difference between reference and PI DeepONet solution.
  • ...and 8 more figures

Theorems & Definitions (13)

  • Remark 2.1
  • Theorem 2.2
  • Lemma 1.1
  • proof
  • Theorem 1.2
  • proof
  • Proposition 1.3
  • proof
  • Lemma 1.4: Time regularity for $\rho_\tau$
  • proof
  • ...and 3 more