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Physics-Informed DeepONets for drift-diffusion on metric graphs: simulation and parameter identification
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Builds physics-informed DeepONet surrogates for inflow, inner and outflow edges of drift-diffusion on metric graphs, couples them via edge-based domain decomposition for simulation, and identifies initial conditions and edge velocities from noisy sensor data.
Problem
Solving nonlinear drift-diffusion equations on metric graphs and solving parameter identification (inverse) problems on these graphs efficiently, where traditional numerical schemes need substantial tailoring.
Approach
Train three edge-type DeepONets (inflow, inner, outflow) with physics-informed losses (PDE, initial, edge flux conditions) once, then couple them on arbitrary graphs by optimizing interface values (Kirchhoff and continuity losses); inverse problems only add data-misfit terms.
Key results
- Largest model (width 200, 20K data) reaches relative space-time L2 errors of about 1e-2 on the test graphs (G1 1.06e-02, G2 1.20e-02, G3 1.30e-02).
- Inverse problem with 10% measurement noise recovers initial conditions and velocities with absolute L2 errors around 0.04 to 0.19 on test graphs, improving as noise decreases.
- Increasing training data from 5K to 20K and width from 100 to 200 consistently reduces validation loss and coupled-graph error.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| increase training data from 5K to 20K solves (ablation) | Physics-informed DeepONet, width 100 · Edge validation (inflow, inner, outflow) | validation loss: 4.34e-03 -> 1.09e-03; 1.42e-03 -> 5.67e-04; 3.63e-03 -> 1.05e-03 | Helped |
| increase training data from 5K to 20K solves (ablation) | Physics-informed DeepONet, width 200 · Edge validation (inflow, inner, outflow) | validation loss: 8.09e-03 -> 6.64e-04; 1.42e-03 -> 2.62e-04; 6.07e-03 -> 6.30e-04 | Helped |
| increase width from 100 to 200 (ablation) | Physics-informed DeepONet with 5K training solves · Inflow edge validation | validation loss: 4.34e-03 -> 8.09e-03 | Hurt |
| increase width from 100 to 200 (ablation) | Physics-informed DeepONet with 20K training solves · Edge validation (inflow, inner, outflow) | validation loss: 1.09e-03 -> 6.64e-04; 5.67e-04 -> 2.62e-04; 1.05e-03 -> 6.30e-04 | Helped |
| increase training data from 5K to 20K solves and use width 200 (ablation) | Physics-informed DeepONet · G1, G2, and G3 | absolute space-time L2 error: G1: 2.87e-02 -> 4.68e-03; G2: 1.71e-02 -> 5.62e-03; G3: 2.35e-02 -> 5.81e-03 | Helped |
| increase training data from 5K to 20K solves and use width 200 (ablation) | Physics-informed DeepONet · G1, G2, and G3 | relative space-time L2 error: G1: 5.48e-02 -> 1.06e-02; G2: 3.45e-02 -> 1.20e-02; G3: 4.79e-02 -> 1.30e-02 | Helped |
| reduce measurement noise from ε1 to ε3 | Physics-informed DeepONet inverse identification · G1, G2, and G3 | absolute L2 error of recovered density: G1: 4.42e-02 -> 1.83e-02; G2: 6.73e-02 -> 1.65e-02; G3: 4.27e-02 -> 1.70e-02 | Helped |
Limitations
- Only one-dimensional edges and a single drift-diffusion model with f(rho)=rho(1-rho) tested
- Larger noise yields substantially larger velocity errors on some edges away from chain ends
- Software and data release promised only upon acceptance
Experimental setup
Datasets: synthetic drift-diffusion solutions from finite volume solver with Gaussian process random data
Benchmarks: finite volume reference solutions on three test graphs chain graph with 7 edges
Metrics: absolute space-time L2 error relative space-time L2 error validation loss L2 error of initial condition and velocity recovery
Baselines: finite volume method
Builds on: DeepONet physics-informed neural networks XPINNs finite volume method
Closest papers by meaning
- PEGNet: A Physics-Embedded Graph Network for Long-Term Stable Multiphysics Simulation (AAAI 2026 · similarity 0.87)
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.87)
- Text2PDE: Latent Diffusion Models for Accessible Physics Simulation (ICLR 2025 · similarity 0.87)
- Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks (ICLR 2025 · similarity 0.87)
- Prometheus: Out-of-distribution Fluid Dynamics Modeling with Disentangled Graph ODE (ICML 2024 · similarity 0.86)
- SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases (ICLR 2024 · similarity 0.86)
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