A physics-informed neural network surrogate for multi-band reflection-loss prediction of carbon-fiber-reinforced polymer / Ti-6Al-4V ELI radar absorbing materials for UAV airframe integration

Technical Summary: A Physics-Informed Neural Network Surrogate for Multi-Band Reflection-Loss Prediction of CFRP/Ti-6Al-4V ELI RAM

Problem Statement
The design of Radar Absorbing Materials (RAM) for Unmanned Aerial Vehicles (UAVs) typically relies on analytical or numerical solvers, such as the Transfer Matrix Method (TMM), Finite-Difference Time-Domain (FDTD), or Method of Moments (MoM). While accurate, these methods become computationally burdensome when exploring large design spaces involving multiple frequency bands, incidence angles, and material thicknesses. This is particularly critical for UAVs, where low-observability (stealth) must be balanced with structural integrity and mass constraints. Specifically, Carbon-Fiber-Reinforced Polymer (CFRP) composites backed by aerospace-grade Ti-6Al-4V ELI titanium offer a dual function of load-bearing structure and electromagnetic absorption, but optimizing their thickness for multi-band performance requires rapid, repeated evaluations that traditional solvers cannot efficiently provide.

Methodology
The author proposes a Physics-Informed Neural Network (PINN) surrogate model to predict the Reflection Loss (RL) of a metal-backed CFRP absorber across the 2–18 GHz frequency band and 0–80° Transverse-Electric (TE) incidence angles.

1. Data Generation: A reference dataset of 6,400 samples was generated using a custom Python implementation of the TMM. The input parameters included frequency ($f$), incidence angle ($\theta$), and absorber thickness ($d$). The material system was modeled as a homogeneous magneto-dielectric CFRP layer over a Perfect Electric Conductor (PEC) representing the Ti-6Al-4V ELI substrate.
2. Network Architecture: The PINN consists of a Fourier feature embedding layer followed by a six-layer Multi-Layer Perceptron (MLP) with 256 neurons per hidden layer and hyperbolic tangent activations.
   • Spectral Bias Mitigation: To address the tendency of standard MLPs to learn low-frequency components first (spectral bias), the input space is lifted using a random Fourier feature embedding with a Gaussian projection matrix.
   • Physics Regularization: The loss function includes explicit physics-based constraints:
     • Monotonicity: Enforcing that reflection loss improves (becomes more negative) with frequency in the absorbing regime.
     • Angular Boundary: Enforcing monotonic degradation of absorption with increasing incidence angle for TE polarization.
     • Smoothness: Tikhonov regularization to ensure smooth gradients.
3. Training: The model was trained using the Adam optimizer with cosine annealing learning rate scheduling over 600 epochs on a single CPU thread, completing in 42 seconds.

Key Contributions

• Surrogate Development: The creation of a PINN surrogate that achieves sub-decibel accuracy against the analytical TMM baseline without requiring GPU acceleration.
• Architecture Innovation: The integration of random Fourier feature embedding with specific physics-based regularizers (monotonicity and angular constraints) to ensure physically plausible predictions across the full parameter space.
• Multi-Band Optimization: Utilization of the trained surrogate to perform rapid, exhaustive thickness optimization across S-, X-, and Ku-bands to identify quarter-wave resonance optima.
• Structural Integration Assessment: A quantitative evaluation of the mass overhead required to integrate the optimized CFRP/Ti-6Al-4V ELI bilayer onto medium-altitude long-endurance (MALE) UAV airframes.

Results

• Accuracy: The surrogate model achieved a Root-Mean-Square Error (RMSE) of 0.22 dB and a Mean Absolute Error (MAE) of 0.12 dB against the TMM reference. 95% of evaluation points fell within an error margin of 0.5 dB. The largest deviations occurred only in grazing incidence regions ($\theta > 75^\circ$) where absorption is inherently weak.
• Efficiency: The training process took 42 seconds on a single CPU. The inference speedup over direct TMM evaluation is approximately 50-fold, enabling rapid parametric studies.
• Optimization Outcomes: The multi-band thickness optimization identified the following quarter-wave optima for normal incidence:
  • S-band (2–4 GHz): 3.9 mm thickness ($RL = -4.64$ dB).
  • X-band (8–12 GHz): 1.9 mm thickness ($RL = -8.93$ dB).
  • Ku-band (12–18 GHz): 1.5 mm thickness ($RL = -11.31$ dB).
• Structural Impact: For a representative MALE UAV (MQ-9A Reaper), the Ku-band optimized coating (1.5 mm) adds approximately 60.8 kg of mass, resulting in a 1.3% mass overhead. Across a broader MALE class envelope, the overhead ranges from 1.5% to 8.5%, which the author deems acceptable for radar-absorbing variants.

Significance and Claims
The paper positions this work as a computationally efficient pathway for coupled material and structural UAV design. The author claims that while the specific single-layer problem could theoretically be solved via direct analytical optimization, the proposed surrogate offers three distinct advantages for broader engineering applications:

1. Differentiability: The trained network is fully differentiable with respect to all inputs, allowing for the analytical calculation of sensitivities (e.g., $\partial RL/\partial \theta$) in a single backward pass, which is essential for gradient-based airframe optimization loops.
2. Scalability: The architecture is designed to scale to complex, analytically intractable problems (e.g., graded multilayer absorbers or frequency-dependent dispersion) where direct numerical solvers would be too slow.
3. Inverse Design Capability: Once trained, the network supports inverse queries, mapping target reflection loss specifications back to admissible material or thickness combinations.

The study validates the "surrogate-first" philosophy by demonstrating that a physics-informed approach can achieve engineering-grade accuracy (sub-0.5 dB error) on standard hardware, providing a foundation for future multi-disciplinary design optimization (MDO) frameworks that integrate stealth and structural performance simultaneously.