P-GAN is built on the SeriesGAN framework [6], which employs a dual-network adversarial design. The generator is a recurrent neural network based on Gated Recurrent Units (GRUs), which captures temporal dependencies within the sequence. The discriminator is a 1D convolutional network that classifies a given sequence as real or synthetic. The generator takes a noise tensor Z ∈ ℝ B × T × D as input and produces a synthetic sequence of the same shape, where B is the batch size, T is the sequence length, and D is the feature dimension. The discriminator is trained to distinguish real sequences from generated ones, while the generator is trained to produce sequences that the discriminator cannot distinguish from real data.

The SeriesGAN architecture was designed for general-purpose time series and produces unconstrained output values. To adapt SeriesGAN to the surgical drill dataset, enforce physical constraints, and improve training stability, we make the following three modifications:

3.1.1. Modification 1—Dropout Regularization

The base SeriesGAN generator consists of four stacked GRU layers that capture sequence dependencies while avoiding vanishing gradients. A dropout wrapper is applied to each GRU layer with an output keep probability of 80%. Interleaving dropout within the recurrent stack prevents any single hidden unit from dominating the learned dynamics, reducing run-to-run variance and mitigating overfitting. This is particularly important for the drill dataset, where the minority classes have few training examples, and the generator is at risk of memorizing rather than generalizing the class distribution.

3.1.2. Modification 2—Hard Output Constraints

The GRU stack feeds into a dense projection layer of size two, corresponding to the depth and force features, producing unconstrained real values r ∈ ℝ 2 . To enforce sensor-valid output ranges, a sigmoid activation is applied to squash the raw output to the unit interval given by Equation (1).

The squashed output is then rescaled to the physical operating range of the drill sensors using a linear transformation given by Equation (2).

where m min = [ 0 , 0 ] and m max = [ 60   mm , 22   lbf ] , corresponding to the minimum and maximum depth and force values observed across all drilling sequences in this dataset. These bounds are distinct from the force and depth sensors’ full hardware ranges used for normalization in Section 4; the generator is constrained to the physically achievable range of possible surgical procedures rather than the sensor’s theoretical limits. m min and m max were selected to effectively address the observed depth limits of surgical procedures in the dataset.

This ensures that every generated sequence falls within the physically valid range of the depth and force sensors, regardless of the generator’s internal state. The full constrained generator procedure is presented in Algorithm 1.

3.1.3. Modification 3—Physics-Constrained Generator Loss

We add two differentiable physics penalty terms into the generator loss alongside the standard adversarial loss L GAN . This allows us to penalize the physically invalid drill sequences generated by the network. The two physics penalty terms are:

Bound-violation loss: which penalizes any generated depth value that falls outside the defined physical limits given by Equation (3). Where d denotes a depth value in mm, d max and d min are the maximum and minimum permissible depths respectively, and d ^ t is the generated depth at time t .

Monotonicity loss: which penalizes any decrease in depth between consecutive time steps, reflecting the physical constraint that the drill advances continuously and does not move backward during a drilling sequence given by Equation (4).

Finally, the total generator loss, which combines the adversarial loss with both physics penalties and is weighted by λ 1 and λ 2 , is given in Equation (5).

where θ denotes the trainable parameters of the generator network.

To prevent training instability during the early stages of optimization, the penalty weights are annealed linearly from zero to their maximum values over the first half of training iterations. The λ term is given by Equation (6).

The full physics-constrained training algorithm is presented in Algorithm 2.

P-GAN is implemented in TensorFlow. The generator consists of four GRU layers with a hidden dimension of D = 2 , followed by the constrained output layer. All models are trained with a batch size of 128 and a constant learning rate of 0.0001 using the Adam optimizer. The physics penalty weights are annealed to their maximum values, λ 1 = 1.0 and λ 2 = 0.5 , over the first half of the training epochs.

The raw drill dataset was collected from a medical drilling experiment and consists of two primary measurements: the force (in lbf) applied by the drill and the drill depth (in mm). These measurements were continuously recorded at high temporal resolution during drilling using the attached sensors at a constant sampling rate.

After acquiring the raw data, feature normalization was applied by scaling the raw values to the range [ 0 , 1 ] using the sensor’s maximum depth capacity (600 mm) and the largest force value observed in the dataset (22 lbf). Note that while 600 mm represents the sensor’s full hardware range, the actual drilling depth observed across all sequences is bounded by 60 mm, which is used as the generator’s hard output constraint in Section 3. Figure 1 provides the visual context that illustrates a simplified view of the bone drilling process, with four distinct classes corresponding to different drilling depths: in bone wall 1 (class 1), between walls (class 2), in bone wall 2 (class 3), and through bone (class 4).

During preprocessing, the data was partitioned into 1341 series, yielding 1,375,999 samples. Partitioning was guided by label confidence. The domain expert assigned the highest confidence labels, which were then split into the validation and test sets. The training set was labeled accordingly. The resulting distribution of data instances across the four classes is as follows: 28.54% for class 1 (in bone wall 1), 12.15% for class 2 (between walls), 41.04% for class 3 (in bone wall 2), and 18.26% for class 4 (through bone).

The imbalanced nature of the training dataset, particularly the under-representation of classes 2 and 4, hinders the downstream task of training a neural network classifier. Although theoretical expectations suggest that force values should be higher when the drill contacts the bone and lower when drilling elsewhere, this relationship is not consistently observed, further emphasizing the need for synthetic data generation to balance the dataset.

We utilized the following two metrics to evaluate the performance of the generated synthetic samples.

4.2.1. Discriminative Score

The discriminative score is a quantitative metric that measures how well synthetic data can be distinguished from real data. To compute this score, a post hoc time-series classification model (a Long Short-Term Memory (LSTM) network) is trained to distinguish between real and generated sequences. The classifier’s error on the held-out test set reflects the similarity between the two distributions. Here, a lower error rate, adjusted to an optimal value of zero, implies that the synthetic data is nearly indistinguishable from the original data. This metric serves as a proxy for measuring the generative model’s ability to accurately mimic the real data distribution. The calculation for this metric is given by Equation (7), where accuracy is the post-hoc classifier’s classification accuracy.

4.2.2. Mean Absolute Error (MAE)

To assess whether the generated data preserves the temporal dependencies, the MAE is calculated. For this evaluation, an LSTM model is trained on the synthetic dataset for one-step-ahead prediction. The trained model is then tested on the real dataset, and forecasting performance is measured as the MAE, Equation (8) between the actual values ( y i ) and the predicted values ( y ^ i ). N is the total number of samples considered in the evaluation. A low MAE indicates that the synthetic data retains sufficient predictive information, making it beneficial for downstream time-series prediction applications.

The performance improvements based on the three modifications are shown in Table 1. Lower values are ideal for both metrics: a discriminative score near zero indicates that synthetic sequences are statistically indistinguishable from real ones, and a lower MAE indicates that temporal structure is preserved. The percentage changes in Table 1 follow this convention, where a negative percentage change indicates a lower, and thus more ideal.

Table 1. SeriesGAN results over all model variations with drill dataset: Discriminative, predictive score, and percentage change in performance.

Adding hard output constraints improved the discriminative score substantially (−31.0%), indicating that bounding depth and force to sensor-valid ranges produce sequences that more closely resemble real data. However, the MAE increased (+12.5%), suggesting the constrained generator overfits to the valid range boundaries rather than capturing the underlying temporal dynamics. The subsequent addition of dropout corrected this overfitting: discriminative score and MAE improved relative to the constrained-only configuration, recovering predictive quality while retaining most of the discriminative gain. Finally, incorporating physics-augmented loss terms further improved both metrics, yielding the best overall MAE (0.2257) and a discriminative score (0.2084) well below the vanilla baseline. These findings highlight that constraint enforcement, regularization, and physics-informed training are complementary, as each addresses a distinct failure mode, and all three are necessary for the full benefit.

To evaluate the individual contributions of the various novel additions to the SeriesGAN model, a more detailed ablation study was conducted. Several versions of the SeriesGAN model were evaluated, each incorporating different combinations of model components: hard output constraints to enforce domain-specific bounds, dropout to prevent overfitting, and physics-informed loss functions to penalize physically implausible sequences.

Table 2. SeriesGAN results over all model variations with drill dataset: Discriminative and predictive score.

The performance of all model combinations is summarized in Table 2. The discriminative score measures how indistinguishable synthetic data is from real data (lower is better, with zero being ideal), while the MAE assesses how well the synthetic data’s temporal structure transfers to real-data prediction tasks. The Vanilla model serves as the baseline.

Constrained Only: Applying hard output constraints alone produced the second-best discriminative score (0.1941), a meaningful improvement over the vanilla baseline (0.2813). However, the MAE increased from 0.2651 to 0.3029, indicating that while constraining outputs to sensor-valid ranges makes sequences look more realistic to a discriminator, it does not ensure that the underlying temporal dynamics are faithfully reproduced. The generator learns to stay within physical bounds but tends to overfit to the boundary region rather than generalizing the class distribution.

Dropout Only: Adding dropout without any physical constraints increased the discriminative score from 0.2813 to 0.3266, meaning synthetic sequences became easier to distinguish from real data. Dropout alone, without any guidance about valid sensor ranges, reduces the generator’s capacity to reproduce fine-grained signal characteristics. The MAE decreased slightly (0.2651 to 0.2524), consistent with dropout’s role as a regularizer that discourages memorization and improves generalization of temporal patterns.

Physics Only: The physics-augmented loss alone produced the lowest discriminative score in the entire ablation (0.0645), indicating that penalizing monotonicity violations and bound violations is the single most powerful driver of statistical realism in this dataset. The MAE also decreased relative to the baseline (0.2165). However, this configuration exhibited high variance and training instability in practice: without the sigmoid-clamping of hard output constraints to anchor the generator’s output range, the physics penalty alone is insufficient to stabilize training across runs. This motivates combining the physics loss with at least one structural modification.

Physics + Constrained: Combining the physics loss with hard output constraints improved the discriminative score substantially (0.1443) and produced a low MAE (0.2292). Relative to Physics Only, the discriminative score has increased, which reflects the trade-off introduced by sigmoid clamping: the hard constraints stabilize training but restrict the generator’s output distribution, slightly reducing its ability to match the full range of real sequences.

Physics + Dropout: Pairing the physics loss with dropout, but without hard output constraints, yielded a discriminative score of 0.2655 and the worst MAE among physics-containing configurations (0.2803). Without the sigmoid clamping to bound outputs, dropout alone is insufficient to prevent the generator from producing out-of-range values, which degrades both metrics relative to the full P-GAN configuration.

Overall, the ablation demonstrates that all three modifications are complementary. Hard output constraints provide training stability and range validity; dropout prevents overfitting on the small minority-class training sets; and the physics-augmented loss directly encodes domain knowledge about drilling dynamics. The complete P-GAN, incorporating all three, achieves the best balance of discriminative realism and MAE, and avoids the instability observed in the Physics Only configuration.

5.3.1. Experimental Setup

For the downstream classifier, we adopt the GRU-based model without modification from Blaney and Muknahallipatna [1], which evaluated several deep models for time-series individual data-instance classification on the same drill dataset and reported GRU as the best-performing baseline among the tested architectures. Consistent with that work, we train one-vs-rest (OvR) binary classifiers for each of the four drill regions (classes 1 - 4) using a univariate input (Depth in mm), window length T = 100 , batch size 128, and a decision threshold of 0.5.

All OvR heads use the same optimizer and early stopping settings as the baseline in [1], with a batch size of 128. To isolate the effect of synthetic augmentation, we evaluate at a fixed decision threshold of 0.5 for every head and report Accuracy (Acc.), Precision (Prec.), Recall (Rec.), and F1 on the held-out test set. The univariate Depth-only input is justified by prior findings that bivariate (Depth + Force) did not improve accuracy relative to Depth alone on this dataset [1]. The results in Tables 3-7 show the results in which synthetic data was added only to

Table 3. Performance of RNN aggregated across runs. # Synth. = Number of Synthetic Samples, Acc. = Accuracy, Prec. = Precision, Rec. = Recall.

Table 4. Performance of CNN aggregated across runs. # Synth. = Number of Synthetic Samples, Acc. = Accuracy, Prec. = Precision, Rec. = Recall.

the original dataset for cases in which classes 2 and 4 were trained.

Next, the dataset was augmented with synthetic data for all training runs. A total of 300 synthetic sequences were added, 150 belonging to class 1, and the remaining 150 belonging to class 3.

Table 5. Performance of RNN aggregated across runs all with synthetic data. # Synth. = Number of Synthetic Samples, Acc. = Accuracy, Prec. = Precision, Rec. = Recall.

Table 6. Performance of CNN aggregated across runs with synthetic data. # Synth. = Number of Synthetic Samples, Acc. = Accuracy, Prec. = Precision, Rec. = Recall.

Table 7. Performance of ResNet aggregated across runs with synthetic data. # Synth. = Number of Synthetic Samples, Acc. = Accuracy, Prec. = Precision, Rec. = Recall.

5.3.2. Data Splits and Augmentation Protocol

Following [1], the original 2023 validation and test partitions are kept fixed to avoid leakage. Synthetic sequences are added only to the training fold of targeted classes. Unless otherwise noted, all synthetic sequences use the same window length ( T = 100 ), normalization ( μ , σ ) , and preprocessing as the baseline, and respect the physical constraints described in the Method section. In the one-vs-rest (OvR) setup, class k receives synthetic positives when training the kth binary head; negatives are drawn from the union of the remaining (real) classes. Validation and test folds contain no synthetic data.

5.3.3. Data Imbalance Addressed by Synthetic Data Augmentation

The statistics of the data balance of the original dataset is shown in Table 8 and the same analysis for the synthetically augmented dataset is shown in Table 9. Each table reports the per-class sample counts (# Samples) and the resulting percentage share

Table 8 shows the baseline distribution before augmentation; Table 9 shows the distribution after adding synthetic sequences only to the training fold and only for the targeted classes (2 and 4). Validation and test folds remain unchanged (no synthetic data), so their class balances are identical to the original. The number of synthetic sequences per minority class was selected based on downstream classifier performance. Preliminary experiments evaluated augmentation counts of 50, 100, 150, 200, and 300 sequences per minority class. Counts below 150 produced insufficient improvement in minority class F1, while counts above 150 consistently led to high precision paired with low recall in the OvR classifiers, indicating that the synthetic advantages began to dominate the training and pushed the decision boundary too aggressively. The value of 150 sequences per class was therefore selected as the point that maximizes F1 improvement without introducing this precision-recall imbalance. A systematic sensitivity analysis across a finer range of augmentation counts, with visualization of the precision-recall tradeoff curve, is identified as an area for future work.

Table 8. Class sample data balance for original dataset.

Table 9. Class sample data balance for dataset augmented with synthetic data.

As intended, the augmentation increases Classes 1 and 3 by +15,000 samples each (from 190,946 → 205,946 and 284,415 → 299,415, respectively), while Classes 1 and 3 are held fixed. In percentage terms, Class 2 rises from 13.98% to 14.75% (+0.77%) and Class 4 from 20.82% to 21.45% (+0.63%), narrowing their gaps to the majority class (Class 3 at 35.37%) without altering the relative prevalence of Classes 1 and 3. Overall, this shifts the training distribution toward a more balanced mix for the classes of interest, while preserving the integrity of the validation/test splits for unbiased evaluation.

5.4.1. RNN

Table 3 summarizes the GRU (RNN) outcomes. Classes 1 and 3 were augmented; classes 0 and 2 were not.

For Class 2, adding 150 synthetic positives increases Acc. from 21.48% to 55.30% (+33.82 points) and F1 from 56.11% to 69.14% (+13.03). Precision rises from 39.84% to 53.95% (+14.11) while Recall remains high (94.83% → 96.24%, +1.41). This pattern indicates the synthetic sequences primarily reduce false positives without sacrificing sensitivity.

For Class 4, 150 synthetic samples improve Acc. from 77.95% to 84.26% (+6.31) and F1 from 89.22% to 92.30% (+3.08). Recall is already saturated at 98.62% and remains unchanged, while precision is essentially stable (81.47% → 81.46%). Here, augmentation trims residual errors against the majority of negatives.

Classes 1 and 3, trained only on real data, already achieve strong performance (Class 1 F1 98.30%; Class 3 F1 89.87%), suggesting limited necessity for augmentation with the currently available dataset.

5.4.2. CNN

Table 4 shows the 1D-CNN results under the same protocol.

For Class 2, augmentation yields a large accuracy gain (35.70% → 83.32%, +47.62). Precision improves (85.58% → 97.18%), Recall remains near ceiling (99.14% → 99.06%), and F1 nudges upward (92.00% → 92.74%). The substantial increase in accuracy reflects fewer false positives in the OvR framing.

For Class 4, augmentation produces the largest overall benefit: Accuracy climbs from 59.00% to 85.20% (+26.20), Recall from 56.28% to 92.83% (+36.55), Precision from 94.37% to 98.17%, and F1 from 70.51% to 95.43% (+24.92). Synthetic transitions likely broaden coverage of difficult boundary dynamics that previously drove false negatives.

Classes 1 and 3 remain strong without augmentation (Class 1 F1 97.67%; Class 3 F1 90.84%), consistent with the RNN observations.

5.4.3. Cross-Model Comparison and Takeaways

Across both backbones, adding synthetic positives to the underrepresented heads (Classes 2 and 4) consistently improves F1 and overall accuracy, with the CNN showing larger gains, especially via recall recovery for Class 4. The RNN gains are steadier but smaller, largely through precision improvements for Class 2. When synthetic positives are too narrow or overly “clean”, OvR heads can become conservative (high precision, low recall). The physics-aware generation used here appears to expand coverage without producing implausible artifacts, which aligns with the observed recall gains.

Operating point matters: a fixed 0.5 threshold is convenient for comparability, but per-class threshold tuning on validation data typically yields further improvements (e.g., maximizing F1 or meeting a minimum recall). We keep thresholds fixed to isolate the effects of augmentation.

5.4.4. Error Analysis

Residual errors concentrate near boundaries (e.g., wall entry/exit) where short, low-SNR fluctuations blur class transitions. Without augmentation, CNN Class 4 exhibits many false negatives; introducing synthetic transitions substantially recovers recall. For RNN Class 2, remaining errors are mostly false positives, suggesting partial overlap between genuine positives and specific negative contexts. Additional diversity in synthetic depth/force trajectories, or mild label smoothing for synthetic samples, may reduce overconfident spikes.

In summary, physics-constrained synthetic augmentation improves minority-class recognition for both RNNs and CNNs while preserving strong performance in majority regimes. The largest gains occur where synthetic data expands the coverage of transitional dynamics (Class 4), producing substantial recall and F1 improvements, especially for the CNN. These findings support the use of domain-constrained synthetic data to remedy class imbalance in drill-depth time-series classification.