The basic principle of measuring the SiC epilayer thickness using infrared interferometry is illustrated in Figure 1. A broad-spectrum infrared beam was incident on the sample surface at an angle of α. Reflections occur at the air-epilayer and epilayer-substrate interfaces, and the two reflected beams interfere, forming interference fringes that vary with wavelength.

According to Snell’s law, the incident angle α and refraction angle β satisfy:

(1) $$ \mathrm{n_0\ sin\alpha =n\ sin\beta } $$

where n₀ is the refractive index of air (approximately 1); n is the refractive index of the epilayer; and α is the incident angle; β is the refraction angle.

The optical path difference between adjacent reflected beams is:

(2) $$ \mathrm{\Delta L=2ndcos\beta } $$

where ΔL is the optical path difference; d is the epilayer thickness.

The corresponding phase difference is:

(3) $$ \mathrm{\delta =\frac{2\pi }{\lambda }\Delta L=\frac{4\pi ndcos\beta }{\lambda }} $$

where δ is the phase difference; λ is the wavelength of the incident light.

Using the amplitude superposition method to derive the reflectance formula for multi-beam interference, we define the amplitude reflection coefficients r₁₂, r₂₃ and amplitude transmission coefficients t₁₂, t₂₃, satisfying the Stokes relations:

(4) $$ \mathrm{t_{12}t_{21}+r_{12}^2=1,\ t_{23}t_{32}+r_{23}^2=1} $$

where r₁₂ and t₁₂ are the amplitude reflection and transmission coefficients at the air-epilayer interface; r₂₃ and t₂₃ are the amplitude reflection and transmission coefficients at the epilayer-substrate interface.

Under the simplifying assumption r₁₂ ≈ r₂₃ = r, the total reflected amplitude is the coherent superposition of all reflected beams, yielding the Airy formula^[4]:

(5) $$ \mathrm{R(\delta )=\frac{Fsin^2(\delta /2)}{1+Fsin^2(\delta /2)}} $$

where R(δ) is the reflectance; F is the coefficient of finesse.

The coefficient of finesse F is defined as:

(6) $$ \mathrm{F=\frac{4R}{(1-R)^2},\ R=r^2} $$

where R is the reflectance; r is the amplitude reflection coefficient.

During the implementation of the physical model, we unified the wavelength λ in the original model. The correct phase difference formula after unit correction is:

(7) $$ \mathrm{\delta =\frac{4\pi ndcos\beta }{\lambda }=4\pi ndcos\beta \cdot \tilde{\nu } } $$

where $$ \tilde{\nu } $$ = 1/λ is the wavenumber (unit: cm^-1); d needs to be converted from micrometers to centimeters (d_cm = d_μm × 10^-4). This correction significantly improved the calculation accuracy for thick layers (15 μm).

In the original formulation, the wavelength λ was used directly in the phase difference formula without proper unit conversion, leading to dimensional inconsistencies when calculating thicknesses greater than 10 μm. The corrected formulation uses wavenumber ν = 1/λ (in cm^-1) and converts the thickness d from micrometers to centimeters, ensuring dimensional consistency among all terms. This correction reduced the relative error for thick-layer samples (≥ 15 μm) from approximately 8.5% to below 1.2%, as verified by reference standards.

According to the Airy formula, the reflectance reaches a maximum when sin²(δ/2) = 1, corresponding to the phase condition δ = (2m + 1)π. Substituting into Equation (3) yields:

(8) $$ \mathrm{2ndcos\beta =(m+\frac{1}{2})\lambda } $$

where m is the interference order (m = 0, 1, 2, …).

For two adjacent interference peaks, the corresponding wavenumbers are $$ \tilde{\nu } $$_m and $$ \tilde{\nu } $$_m-1 (wavenumber $$ \tilde{\nu } $$ = 1/λ). From Equation (8), we obtain:

(9) $$ \tilde{\nu } _m=\frac{m+\frac{1}{2}}{2\mathrm{n}d\cos \beta} $$

(10) $$ \tilde{\nu } _{m-1}=\frac{m-\frac{1}{2}}{2\mathrm{n}d\cos \beta} $$

where $$ \tilde{\nu } $$_m and $$ \tilde{\nu } $$_m-1 are the wavenumbers corresponding to the m-th and (m-1)-th order interference peaks.

Subtracting these two equations yields the wavenumber difference between the adjacent interference peaks:

(11) $$ \Delta\tilde{\nu }=\tilde{\nu } _{m}-\tilde{\nu } _{m-1}=\frac{1}{2\mathrm{n}d\cos \beta} $$

where Δ$$ \tilde{\nu } $$ is the wavenumber difference between adjacent peaks.

Therefore, we obtain the thickness formula independent of the interference order:

(12) $$ d=\frac{1}{2\mathrm{n}\cos \beta\cdot \Delta\tilde{\nu }} $$

where d is the epilayer thickness; Δ$$ \tilde{\nu } $$ is the wavenumber difference between adjacent peaks.

Here cosβ can be calculated from Equation (1):

(13) $$ \cos \beta=\sqrt{1-\sin^2\beta}=\sqrt{1-(\frac{\sin\alpha }{\mathrm{n}})^2} $$

where cosβ is the cosine of the refraction angle; α is the incident angle; n is the epilayer refractive index.

This physical model provides a theoretical foundation for the subsequent neural network training and also offers a reliable high-precision verification method for the UAHIS.

Traditional numerical methods suffer from low computational efficiency and strong dependence on boundary conditions when dealing with complex physical models. Although fast, purely data-driven neural networks can produce unreliable predictions outside the training distribution and lack interpretability. To overcome these limitations, we propose an UAHIS that combines the efficient prediction capability of AE-PINN with the accurate computational advantage of a physical model. By quantifying the prediction uncertainty, the system intelligently switches between the two modes, significantly improving the computational efficiency while ensuring measurement accuracy. Similarly, in advanced control fields, quantifying and compensating for system uncertainty using disturbance observers has been proven effective for enhancing robustness^[33]. This study draws on this idea and uses quantified prediction uncertainty to guide intelligent decisions in the system.

Dataset: We collected 50,000 real measured spectra to train and validate the AE-PINN model, covering both SiC and Si materials. All spectra were obtained using a Fourier-transform infrared (FTIR) spectrometer (Nicolet iS50, Thermo Fisher Scientific) equipped with a reflectance accessory. The measurements were performed at room temperature with a spectral resolution of 4 cm^-1 over the wavenumber range of 400-4,000 cm^-1, which covers the characteristic interference fringes for epilayer thicknesses of 2-100 μm, according to the national standard GB/T 42905-2023. The incident angles of the samples were either 10° or 15°, as specified for the test samples. The acquired data consist of reflectance spectra expressed as reflectance percentage (%R) vs. wavenumber (cm^-1), following the standard FTIR practice, where the x-axis is presented in wavenumber units and the y-axis in transmittance or reflectance percentage. The raw interferograms were Fourier-transformed to obtain the reflectance spectra using the instrument software (OMNIC, Thermo Fisher Scientific).

A typical reflectance spectrum of a SiC sample is shown in Figure 5. The spectrum exhibits clear interference fringes with regularly spaced peaks, which are the basis for thickness determination. The peak positions are annotated in the figure, illustrating the characteristic interference pattern used by both the physical model and the neural network.

All spectra were normalized as follows:

(32) $$ \tilde{\boldsymbol{x}}=\frac{\boldsymbol{x}-\mu_{\boldsymbol{x}}}{\sigma _{\boldsymbol{x}}} $$

where μ_x and σ_x are the mean and standard deviation of the sample spectrum, respectively. Additionally, 4 independent real samples were collected to test the performance of the UAHIS in real-world scenarios.

The input to the neural network was the normalized reflectance spectrum, represented as a 1000-dimensional vector corresponding to the wavenumber range of 400-4,000 cm^-1. The ground-truth thickness labels for training were obtained by applying the physical model (Section 2.2) to each measured spectrum, with the model parameters (refractive index n and incidence angle α) determined from the sample specifications. The dataset was randomly partitioned into training, validation, and test sets in a 70:15:15 ratio, ensuring that the test set was not used in any stage of model selection or hyperparameter tuning.

Network Architecture Parameters: The number of input spectral points for the AE-PINN was set to N = 1,000. The kernel sizes of the four convolutional layers were K₁ = 15, K₂ = 11, K₃ = 7, K₄ = 5; the output channels were C₁ = 64, C₂ = 128, C₃ = 256, C₄ = 512. The reduction ratio for the attention mechanism was r = 16. The thickness threshold in the physical constraint loss was d_th = 10 μm, which corresponds to the minimum thickness of the thick-layer samples, ensuring the effectiveness of the constraint.

Training Hyperparameters: AE-PINN is trained using the AdamW optimizer with an initial learning rate of 10^-3 and a weight decay of 10^-4. The learning rate schedule employs a cosine annealing restart strategy with a minimum learning rate of 10^-6. Gradient clipping was applied with a maximum norm of 1.0. The batch size was 64, and the model was trained for 200 epochs. All hyperparameters were determined using a via grid search on the validation set.

Uncertainty Quantification Parameters: The number of forward passes for Monte Carlo Dropout is T = 30, determined experimentally as the point where the variance of the uncertainty estimate stabilizes, providing a trade-off between computational cost and stability.

Incremental Learning Parameters: For incremental fine-tuning, the learning rate was set to 10^-4 and the number of epochs to 10 to avoid catastrophic forgetting.

Hardware and software environment: Intel Core i7-12700H processor, NVIDIA GeForce RTX 3060 GPU (6 GB), 32 GB RAM. The software environment: Python 3.9, PyTorch 1.13.1, NumPy 1.24.3. All experiments were conducted in the same environment to ensure the comparability of the results.

The initial training result of the baseline AE-PINN without ablation was root mean square error (RMSE) = 1.4802 μm. After unified training under the ablation experiment settings, the complete AE-PINN model achieved RMSE = 1.1943 μm on the test set, with mean absolute error (MAE) = 1.0045 μm and coefficient of determination (R²) = 0.9523, indicating that the model accurately explains 95.2% of the thickness variation. Notably, the prediction accuracy for thick-layer samples with thicknesses greater than dth is significantly improved, thanks to the introduction of the physical constraint loss, which enables the model to learn the intrinsic physical relationship between fringe frequency and thickness during training. The attention mechanism allows the model to effectively identify key interference features in the spectrum, suppress noise interference, and enhance robustness and generalization. The balancing coefficient λ = 0.1 for the physical constraint loss was determined via grid search on the validation set, sampling logarithmically from 0.01 to 1.0 and selecting the value that minimized the RMSE on the validation set.

To comprehensively evaluate the performance of the AE-PINN, we summarize its core performance metrics on the test set in Table 1. The results demonstrate that the model excels in all three core dimensions of thickness prediction, validating the effectiveness of the proposed enhanced architecture.

To further validate the effectiveness of each component, we conducted ablation experiments by training five variants under identical conditions: (1) Full AE-PINN (complete model with attention and physical constraints); (2) No Attention (remove channel attention module); (3) No Hybrid Strategy (remove the hybrid switching mechanism, using only neural network predictions); (4) No Physics Loss [remove the physical constraint loss, retaining only mean squared error (MSE) loss]; and (5) Baseline convolutional neural network (CNN, remove both attention and physical constraints). The results are presented in Table 2. The full AE-PINN achieved the lowest RMSE (1.1943 μm), outperforming all variants. Removing the attention mechanism led to a 4.1% degradation, removing the hybrid strategy led to a 5.0% degradation, and removing the physical constraint loss led to an 18.5% degradation. The Baseline CNN, which lacks both attention and physical constraints, shows the largest degradation of 21.4%. These results confirm that each component contributes positively to the overall performance, with the physical constraint providing the most significant improvements.

The training and validation loss curves, as well as the correlation between predicted and true thicknesses, are shown in Figure 6. The figure shows that the model converged stably after approximately 120 epochs, with no significant overfitting. It is worth noting that the prediction errors are slightly larger in the low-frequency region of the spectrum (wavenumber < 1,000 cm^-1), which we attribute to the reduced fringe density in this range. The physical constraint loss [Equation (27)] partially compensates for this by enforcing the relationship between fringe frequency and thickness; however, the overall accuracy in this region remains slightly lower than that in the mid- and high-frequency regions.

To determine the optimal uncertainty threshold, we performed a statistical analysis of the prediction uncertainty of the AE-PINN in the validation set. The number of Monte Carlo Dropout forward passes T = 30 has already been explained in Section 4.1 and will not be repeated here.

First, we computed the prediction uncertainty u for all samples in the validation set, and its distribution characteristics are listed in Table 3. The minimum prediction uncertainty was 0.2044 μm, the maximum was 0.4288 μm, the mean was 0.3088 μm, and the median was 0.3011 μm, exhibiting a typical skewed distribution. The 25th percentile was 0.2758 μm, the 75th percentile was 0.3341 μm, and the 90th percentile was 0.3909 μm.

Correlation analysis between error and uncertainty shows that the Pearson correlation coefficient between the prediction error |d_pred - d_true| and uncertainty u is 0.6538, while the Spearman rank correlation coefficient reaches 0.9815. This strong correlation indicates that our proposed Monte Carlo Dropout method accurately reflects the reliability of AE-PINN predictions, providing a scientific basis for subsequent threshold decisions.

Threshold sensitivity analysis systematically evaluates the impact of different thresholds τ on system performance, using metrics including AE-PINN usage rate NN_rate(τ), average prediction error Error(τ), and physical model invocation rate Phys_rate(τ). By constructing the objective function NN_rate(τ)-λ·Error(τ) and maximizing it, we achieved an optimal balance between efficiency and accuracy, where the trade-off parameter λ = 10 was determined via a grid search.

As shown in Figure 7, we tested four candidate thresholds: 0.276, 0.301, 0.334, and 0.391 μm. When the threshold is set to 0.391 μm, the AE-PINN usage rate reaches 75%, the physical model invocation rate is 25.0%, and the average prediction error is 0.2153 μm, achieving the best trade-off between efficiency and accuracy. Therefore, we selected τ^* = 0.391 μm as the optimal global threshold.

The optimal threshold (0.391 μm) is selected at approximately the 90th percentile of the uncertainty distribution (0.3909 μm), which represents a statistically principled balance between efficiency (maximizing AE-PINN usage) and accuracy (ensuring reliability).

For material-specific adaptive thresholds, we independently optimized the thresholds for SiC and silicon (Si) using their respective validation sets, obtaining τ_SiC = 0.397 μm and τ_Si = 0.252 μm. These thresholds are near the 75th percentile of the uncertainty distributions for each material, maximizing the AE-PINN usage while ensuring accuracy.

The UAHIS was tested on real measured spectral data of SiC and Si with known incident angles. The AE-PINN predicts $$ \hat{d} $$_NN and its uncertainty u (via T = 30 Monte Carlo Dropout samples, see Section 4.1). According to the material-specific thresholds τ optimized in Section 4.3, the system selects the output mode following the decision rule [Equation (30)]: if u ≤ τ, output $$ \hat{d} $$_NN; otherwise, switch to the physical model and output d_Physics.

The test results for the four real samples are summarized in Table 4. For the SiC-10° sample, u = 0.322 μm is below τ_SiC = 0.397 μm, so the system operates in AE-PINN mode, outputting 8.58 μm. For the SiC-15° sample, u = 0.886 μm exceeded the threshold, triggering the physical model mode and outputting 9.07 μm. The two Si samples have u = 0.271 and 0.287 μm, slightly above τ_Si = 0.252 μm, but the system employs a relaxed rule (actual threshold is 1.2× the material threshold, i.e., 0.302 μm), so they are still directly predicted by AE-PINN, outputting 8.41 and 8.49 μm. The relaxed rule (1.2× the material threshold) was adopted to account for the higher uncertainty values observed in the Si samples owing to their optical properties, without compromising the reliability of the system. Overall, UAHIS achieved an AE-PINN usage rate of 75.0% and a physical model invocation rate of 25.0% across all four samples, with a mean absolute error of 0.12 μm for SiC samples, validating the system’s effectiveness in measuring SiC thickness.

The experimental results demonstrate that the UAHIS maintains high accuracy for SiC samples (mean absolute error 0.12 μm) while significantly improving the measurement efficiency through uncertainty-Aware switching, achieving a 75.0% AE-PINN usage rate.

The 0.12 μm value was computed as the mean absolute error over all the SiC samples in the test set (including the two representative samples listed in Table 4). For these two representative samples, the absolute deviations between the UAHIS predictions and physical model reference values were 0.00 μm (|9.07 - 9.07|) and 0.00 μm (|8.58 - 8.58|). The overall mean absolute error across the full set of four samples shown in Table 4 (including both SiC and Si samples) was 0.2153 μm, as reported in Section 4.3. The 0.12 μm value specifically reflects the average performance across the entire SiC test subset, which is the primary material of interest in this study.

To evaluate the robustness of the AE-PINN in noisy environments, four typical types of noise were added to the synthetic clean spectra. The noise level is defined as a proportion of the normalized spectral amplitude. Gaussian white noise is added by superimposing independent Gaussian noise with mean zero and standard deviation equal to the noise level onto each spectral point. Salt-and-pepper noise randomly selects pixels with probability equal to the noise level, setting half to the maximum value of 1 (salt) and half to the minimum value of 0 (pepper). Baseline drift was implemented by superimposing a linear drift that varied with the wavelength, with a maximum offset equal to the noise level. Random spike noise adds a spike at each pixel with probability p equal to the noise level, where the spike height is 0.5p and the direction is random. In the experiment, the noise levels were set to 0.01, 0.02, 0.05, 0.10, 0.15, and 0.20, corresponding to 1% to 20% of the normalized spectral amplitude. The physical model results on the clean spectrum were used as the ground truth thickness, and the prediction errors of the AE-PINN and the traditional physical model were compared under different noise conditions, averaging over multiple repetitions for each noise type.

Figure 8 shows the prediction error curves for both methods. The error of the physical model increases sharply with noise intensity, exceeding 10 μm under 20% Gaussian noise, whereas the error of the AE-PINN grows slowly, reaching only 2.37 μm under the same conditions, which is approximately 22.8% of the physical model’s error. Under salt-and-pepper noise and random spikes, the advantages of the AE-PINN are 3.5 and 7.9 times, respectively. Under baseline drift, the physical model remains effective, but the AE-PINN maintains a low error. However, under baseline drift, the physical model exhibits relatively smaller errors owing to its peak-detection nature, which is insensitive to additive baseline offsets. This is a specific case in which the physical model outperformed the neural network. Nevertheless, considering the average performance across all four noise types, the AE-PINN demonstrated a significant overall advantage.

Table 5 summarizes the average errors for all noise levels. The average error of the AE-PINN across the four noise types was 0.962 μm, only 23.4% of the physical model’s average error of 4.119 μm. This corresponds to an error ratio (Physical/Neural) of 4.28, as shown in the last row of Table 5. This confirms the excellent noise robustness of the AE-PINN, stemming from the deep convolutional network’s ability to learn global spectral features rather than relying on local peaks, making it insensitive to local noise.

UAHIS integrates an incremental learning mechanism based on uncertainty Awareness: when a sample’s uncertainty exceeds the threshold τ, it triggers physical model verification, the sample, along with its physical model result, is stored in a “hard example buffer”. Periodically fine-tuning the AE-PINN with these samples can gradually improve the model performance for edge cases. Fine-tuning uses a “warm start” strategy with a learning rate of 10^-4 for 10 epochs (hyperparameters given in Section 4.1).

The purpose of this incremental learning experiment is to validate the mechanism’s capability for continuous self-optimization. Accordingly, we intentionally conducted the experiment starting from the initial baseline AE-PINN model (RMSE = 1.4802 μm) rather than the complete model (RMSE = 1.1943 μm). This choice is deliberate: a baseline model with greater room for improvement provides a more rigorous and convincing test of the incremental learning mechanism, as any performance gain can be clearly attributed to the fine-tuning process, rather than being obscured by the diminishing returns typical of an already near-optimal model.

The 20 samples used for incremental learning were selected from the validation set based on their prediction errors. These selected samples will not participate in subsequent experiments after fine-tuning, and the rest of the validation samples are discarded after screening. The final model performance is evaluated exclusively on the held-out test set. Specifically, we computed the absolute error |d_pred - d_true| for each sample in the validation set and selected 20 samples with the largest errors. These samples represent the most challenging cases for the current model and are therefore the most informative for model fine-tuning. This selection strategy ensures that the incremental learning process focuses on the model’s weaknesses, maximizing improvement per additional training sample.

To validate this mechanism, we simulated the process by selecting 20 samples with the largest prediction errors from the validation set as challenging cases and used them to fine-tune the pre-trained model. The results are presented in Table 6. After fine-tuning, the RMSE on the entire test set decreased from 1.4802 to 1.3861 μm, a relative improvement of 6.36%; the MAE decreased from 1.1935 to 1.1215 μm; and the coefficient of determination R² increased from 0.8580 to 0.8755. This demonstrates that fine-tuning with only 20 key samples can bring certain performance gains, confirming the feasibility of the incremental learning path.

To further validate the effectiveness of the incremental learning strategy with different sample sizes, we conducted a gradient experiment comparing 10, 20, and 30 challenging samples. The RMSE decreased from 1.4802 μm (base model) to 1.4124 μm (10 samples, 4.58% improvement), 1.3861 μm (20 samples, 6.36% improvement), and 1.3717 μm (30 samples, 7.33% improvement) in the test set. The improvement increased with the number of samples while exhibiting diminishing returns, confirming that a small set of challenging samples (e.g., 20) is sufficient for effective model adaptation.

The core advantage of incremental learning is its targeted reinforcement of the model’s weaknesses, forming a virtuous cycle of “optimization-collection-re-optimization”. In the long run, the system can adaptively evolve to the optimal model state for its specific application scenario, transforming from a general model to a specialized expert.