Physics-Informed Graph Neural Network for Inverse Design of Integrated Photonic Biosensors

1. Problem Statement

Integrated photonic biosensors, particularly those based on microring resonators, are critical for compact, label-free biochemical detection. However, their inverse design (determining the physical geometry required to achieve specific spectral characteristics) faces significant challenges:

• Complex Physics: The relationship between geometric parameters (radius, waveguide width/height, coupling gap) and spectral response (resonance wavelength, sensitivity, Q-factor) is highly nonlinear and involves strong coupling effects.
• Computational Cost: Conventional design relies on Finite-Difference Time-Domain (FDTD) simulations or adjoint optimization, which are computationally expensive and limit the exploration of high-dimensional design spaces.
• Physical Consistency: Purely data-driven AI models often struggle to satisfy strict physical constraints (e.g., resonance conditions, energy conservation), leading to designs that are mathematically optimal but physically invalid or difficult to fabricate.

2. Methodology

The authors propose a Physics-Informed Graph Neural Network (PI-GNN) framework to address these challenges. The methodology consists of three core components:

A. Graph Representation of the Device

Instead of treating device parameters as independent scalar inputs (as in standard Fully Connected Networks), the microring sensor is modeled as a graph:

• Nodes: Represent functional regions: the microring, bus waveguide, coupling region, surrounding analyte medium, and substrate.
• Edges: Encode physical interactions such as optical coupling, field confinement, and evanescent overlap.
• Features: Nodes contain geometric dimensions and material properties (refractive indices); edges characterize interaction strengths.
• Mechanism: The model uses message-passing to propagate information across the graph, allowing local geometric variations to collectively influence the global spectral prediction while respecting the device's topological connectivity.

B. Physics-Informed Learning Objective

The core innovation is the integration of physical laws directly into the loss function during training. The total loss ($L_{total}$) combines:

1. Data Loss ($L_{data}$): Mean Squared Error (MSE) between predicted and target spectral quantities (resonance wavelength $\lambda_r$ and spectral shift $\Delta\lambda$).
2. Physics-Informed Loss ($L_{physics}$): A constraint enforcing the microring resonance condition:
   $$m\lambda_r = n_{eff} 2\pi R$$
   where $m$ is the resonance order, $n_{eff}$ is the effective index, and $R$ is the ring radius. This ensures that the predicted geometry inherently satisfies the electromagnetic phase condition required for resonance.

C. Training and Inverse Design Workflow

• Datasets: Two datasets were generated:
  • Analytical Dataset: Based on coupled-mode theory for fast sampling.
  • FDTD Dataset: High-fidelity full-wave simulations capturing non-idealities (bending loss, higher-order effects).
• Forward Stage: The GNN maps geometric parameters $\to$ spectral response.
• Inverse Stage: The trained model is embedded in an optimization loop. Given a target spectrum, the model iteratively adjusts geometric parameters to minimize the discrepancy between predicted and target responses, guided by the physics-informed loss.

3. Key Contributions

• Novel Architecture: Introduction of a PI-GNN specifically tailored for microring biosensor inverse design, bridging graph representation (structural connectivity) with physics-informed constraints (electromagnetic laws).
• Enhanced Stability: By embedding resonance conditions into the loss function, the model avoids "physically inconsistent" solutions that often plague pure data-driven approaches.
• Hybrid Data Utilization: The framework effectively leverages both fast analytical models and high-fidelity FDTD simulations to improve generalization and accuracy.
• Efficiency: The approach significantly reduces reliance on repeated, costly FDTD simulations during the design optimization phase.

4. Results

The model was benchmarked against a target design ($R=200\mu m, w=2\mu m, h=2\mu m, g=0.25\mu m$) targeting a resonance at $\approx 1.55 \mu m$.

• Accuracy:
  • The PI-GNN achieved a Root-Mean-Square Relative Geometric Error (RMS-RGE) of 3.46% when trained on FDTD data and 5.14% on analytical data.
  • The inverse design converged to $R=196.62\mu m, w=2.054\mu m, h=1.973\mu m, g=0.226\mu m$, yielding a resonance of $1550.009 nm$ and a spectral shift of $0.1602 nm$.
• Comparison with Diffusion Models (DM):
  • PI-GNN demonstrated lower overall aggregate error compared to Diffusion Models, particularly for ring radius and waveguide dimensions.
  • While DMs showed competitive performance specifically for the coupling gap ($g$), PI-GNN provided superior stability across the full parameter set.
• Ablation Study:
  • Comparing PI-GNN against a standard GNN (without physics constraints) revealed that the standard GNN suffered from overfitting (lower training loss but higher test error) and poor generalization.
  • The physics-informed term was crucial for achieving monotonic convergence and ensuring the model adhered to governing resonance physics.

5. Significance and Future Outlook

• Impact: This work demonstrates that embedding domain-specific physical knowledge into graph-based neural networks is a powerful strategy for accelerating photonic inverse design. It offers a pathway to design high-performance biosensors with reduced computational overhead and higher physical reliability.
• Limitations & Future Work: The current model focuses on geometric and spectral properties. Future iterations aim to incorporate fabrication constraints (e.g., minimum feature sizes, sidewall roughness) and biochemical selectivity (antibody-antigen modeling) to transition from simulation-ready designs to experimentally realizable, application-ready sensors.