Comparing spectral-robust training objectives in physics-consistent dual-wavelength diffractive networks

Technical Summary: Comparing Spectral-Robust Training Objectives in Physics-Consistent Dual-Wavelength Diffractive Networks

Problem Statement
Diffractive deep neural networks (D2NNs) offer potential advantages in parallel wave propagation and low-latency optical inference. However, their physical robustness is contingent on numerous factors, including wavelength dependence, propagation sampling, material dispersion, and layer alignment. While robust learning strategies such as worst-case optimization and average-risk training are established in general machine learning, it remains unclear whether sample-wise worst-case spectral training provides a reproducible performance advantage over average multi-shift training in physically coupled, dual-wavelength diffractive architectures. Specifically, the study questions if worst-case objectives yield superior robustness when architecture, physical models, optimization budgets, and propagation sampling are strictly controlled.

Methodology
The study conducted a controlled numerical comparison using a dual-wavelength differential D2NN architecture. The system utilized a shared bounded physical thickness profile (fused silica) where wavelength-dependent phase was determined by Malitson Sellmeier dispersion rather than independent trainable masks. The evaluation focused on two primary objectives:

1. Sample-wise worst-case training: Minimizing the maximum loss across a set of wavelength offsets for each sample.
2. Average multi-shift training: Minimizing the mean cross-entropy across the same set of offsets.

The experimental design included three distinct phases to ensure rigorous comparison:

• Broad Benchmark: Eight configurations trained on a 64 × 64 propagation grid and re-evaluated on a 128 × 128 grid using frozen device parameters.
• Direct High-Resolution Optimization: Ten paired independent training runs for each objective, initialized randomly and trained end-to-end directly on a 128 × 128 grid (4 µm pitch).
• Prespecified Physical-Domain Analysis: A confirmation study spanning variations in layer spacing (2–20 mm), maximum relief thickness (1.0–2.0 µm), and training drift ranges (±10 to ±40 nm).

Statistical units were defined as complete independent training runs (seeded 42–51), not individual test images. Performance was measured using the Balanced Wavelength-Robustness Score (Balanced-WRS) Area Under the Curve (AUC). The primary metric was the paired difference (worst-case minus average-loss) with bootstrap 95% confidence intervals and exact two-sided sign-flip p-values.

Key Results
The study found no statistically reliable advantage for sample-wise worst-case training over average multi-shift training under the tested conditions:

• Broad Benchmark (64-grid training / 128-grid frozen re-evaluation): The paired difference in Balanced-WRS AUC was +0.375 percentage points (95% CI: −0.546 to 1.279; p = 0.4648).
• Direct 128-grid Training: The point estimate reversed sign, showing a difference of −0.537 percentage points (95% CI: −2.080 to 1.143; p = 0.5469).
• Prespecified ±20 nm Confirmation: The difference was −0.557 percentage points (95% CI: −2.131 to 0.833; p = 0.4980).

In all confirmatory comparisons, the confidence intervals crossed zero, indicating that neither objective demonstrated a superior worst-case advantage. Furthermore, the sign change between the frozen-device resampling and direct high-resolution optimization suggests that evaluation protocols significantly influence observed outcomes.

Additional failure-boundary analyses revealed:

• Geometric Robustness: Independent per-layer displacement (even at 1 propagation pixel) reduced performance toward chance levels (approx. 25% for a 4-class task), indicating that spectral robustness does not imply geometric alignment robustness.
• Input Corruption: Corruption-augmented training produced only small, corruption-dependent changes, failing to establish general joint input-and-spectral robustness.
• Performance Baseline: The optical models achieved approximately 27–29% balanced accuracy, substantially lower than a compact electronic CNN (75.03%) on the same task, highlighting the current limitations of the low-capacity optical model.

Significance and Claims
The paper explicitly positions itself as a "negative result" and a methodological contribution rather than a demonstration of a new universal theory or a hardware validation. Its primary claims are:

1. No Established Superiority: Under the specific physical model, OCTMNIST task, and optimization budget used, sample-wise worst-case training does not provide a reproducible benefit over average multi-shift training.
2. Protocol Sensitivity: The reversal of effect signs between low-resolution resampling and direct high-resolution optimization demonstrates that frozen-device evaluation cannot substitute for direct high-resolution training.
3. Methodological Rigor: The study advocates for a stricter evaluation workflow for spectral-robust training, emphasizing the distinction between frozen-device resampling and direct optimization, the use of complete independent runs as statistical units, and the reporting of paired effects with uncertainty.
4. Scope Limitations: The results are confined to scalar angular-spectrum simulations without fabricated devices, measured source spectra, or polarization effects. The study does not claim clinical or deployable-system viability, noting that current optical performance is insufficient for such applications.

In summary, the work provides an uncertainty-aware numerical comparison that challenges the assumption of inherent superiority for worst-case spectral training in dual-wavelength D2NNs and highlights the critical importance of evaluation protocols in physical neural network research.