We thank the reviewer for this insightful point. Minimizing these drifts is indeed a key challenge in moving data-driven models toward operational use. Incorporating physical constraints into the training process (e.g., penalizing energy changes during level-flight labels) would force the model to respect the steady-state dynamics that BADA inherently assumes.

Change in manuscript: We have added a clarification in Section 6 regarding potential strategies to minimize these deviations. Specifically, we discuss the use of physics-based regularization, such as penalizing energy changes during level-flight segments, to enforce steady-state equilibrium conditions and reduce cruise drifts.

While lateral dynamics are intuitively less complex from a purely kinematic standpoint, our decision to focus on the vertical-longitudinal profile was motivated by the fact that it represents the most challenging modelling task. Specifically, accurately capturing the energy balance governing climb and acceleration requires learning unobservable physical dependencies, such as the aircraft’s mass and the energy share factor (how available power is partitioned between potential and kinetic energy). Lateral dynamics, although involving challenges such as turn detection and the distinction between orthodromic (great circle) and loxodromic paths, remain largely decoupled from these fundamental propulsion constraints. We therefore prioritized the vertical axis as the core complexity of trajectory reconstruction.

Change in manuscript: We have added a clarification in Section 6 explaining the deliberate focus on vertical-longitudinal dynamics due to the underlying physics of energy partitioning, which represents a higher modelling challenge than lateral kinematics.

We thank the reviewer for this suggestion. The methodology applied to QAR data in Reference 5 indeed shares the same Neural ODE architecture, but operates on fundamentally different input data. QAR (Quick Access Recorder) data provides high-frequency measurements (typically 4–25 Hz) with direct access to internal flight parameters such as fuel flow, engine thrust settings (N1), aircraft mass, and precise autopilot targets. In contrast, ADS-B and Mode S EHS data are lower-frequency (around 1 Hz or less) and require the reconstruction of control intent from noisy and irregular signals, as discussed in Section 6. This introduces significant uncertainty in variables such as selected altitude, selected speed, and vertical rate targets, which propagates into both training and evaluation.

As a result, we expect the QAR-based model to benefit from cleaner inputs and richer physical observability (mass, thrust, fuel flow), leading to higher accuracy and better interpretability. The ADS-B approach, while more scalable and globally accessible, operates under noisier conditions and lacks direct propulsion information. Despite these limitations, the Neural ODE framework demonstrates promising vertical profile reconstruction quality, suggesting that the continuous-time formulation provides sufficient regularization to absorb inconsistencies in open surveillance data. The key trade-off is therefore between data accessibility and physical interpretability: ADS-B enables large-scale trajectory modelling without proprietary data, but cannot support absolute performance metrics such as fuel consumption.

Change in manuscript: We have added a comparative discussion in Section 6 highlighting the differences between ADS-B and QAR-based approaches. We emphasize that QAR data benefits from direct access to propulsion parameters (N1, fuel flow, mass) and cleaner control targets, whereas ADS-B requires noisy intent reconstruction. Despite these differences, the Neural ODE framework achieves promising trajectory reconstruction performance, demonstrating robustness to input uncertainty.

We agree and have removed “vertical” from the sentence.

Change in manuscript: Removed “vertical” from “longitudinal dynamics” in the Discussion section.

We agree that “hidden” is more accurate and have made this change.

Change in manuscript: Changed “limited observability of control inputs” to “hidden control inputs” in the Introduction.

Corrected the citation order.

Change in manuscript: Swapped citation order in Literature Review section.

We agree that “flyability” is more specific to the aviation domain and have made this change.

Change in manuscript: Replaced “physical guarantees” with “flyability” in the Literature Review.

Corrected to use the full official ICAO terminology.

Change in manuscript: Changed to “ICAO aircraft type designators” in Section 3 (Data Sources).

We have reviewed the entire Data Sources and Methods sections to ensure consistent tense usage: present tense for general methods and past tense for our specific procedures. Three corrections were made.

Change in manuscript: Corrected tense coordination throughout Sections 3 and 4.

We have added “type” as suggested.

Change in manuscript: Changed “For each selected aircraft” to “For each selected aircraft type” in Section 3.

The geographic coverage is indeed worldwide, as ADS-B data from the OpenSky Network provides global coverage. We have clarified this in the manuscript.

Change in manuscript: Added “worldwide” to clarify geographic coverage in Section 3.

We have added the proper citation to the ERA5 reanalysis dataset.

Change in manuscript: Added citation for ERA5 in Section 3.

We have added the citation as suggested.

Change in manuscript: Added citation for fastmeteo library in Section 3.

Corrected for consistency with the terminology used throughout the paper.

Change in manuscript: Changed “EH messages” to “EHS messages” in Table 4.

We agree and have added a reference to Figure 1 in the methodology introduction.

Change in manuscript: Added Figure 1 reference in Section 4 introduction.

This was an oversight. We have corrected the notation to use subscript.

Change in manuscript: Corrected $e_1(t)$ notation in Figure 1.

We agree that this section needed clarification. We have restructured Section 4 to follow the logical data flow. First, we added an itemized overview immediately after the state vector definition to explicitly describe the model’s inputs, the sequential processing through the Trajectory and Derivative layers, and the final ODE integration. Second, we reordered the detailed descriptions so that the analytical Trajectory Layer formulas appear before the neural Derivative Layer implementation, ensuring a cohesive narrative.

Change in manuscript: Restructured Section 4 to include an itemized architecture overview after Equation 1 and reordered subsections to match the logical flow (Trajectory Layer formulas before Derivative Layer details).

We fully agree with this description of the model flow. As detailed in our response to Comment 14, we have added an explicit itemized description at the beginning of Section 4 stating exactly this: inputs $x(t)$ are processed by the trajectory layer to derive $e_1(t)$, which is then passed to the derivative layer to obtain the gradients. This flow is now stated explicitly earlier in the paper, as recommended.

Change in manuscript: See changes for Comment 14 (added itemized architecture overview).

We fully agree. $u(t)$ and $e_0(t)$ are indeed exogenous inputs defined for the entire flight duration. We have added an explicit note in Section 4 (after Equation 5) to clarify that these variables are available for all $t$ and are not generated autoregressively.

Change in manuscript: Added explicit clarification in Section 4 that $u(t)$ and $e_0(t)$ are exogenous inputs available for all $t$.

We appreciate the reviewer’s perspective on this terminology. We chose the term “layer” to reflect its structural role as a functional block within a differentiable computation graph. In modern deep learning frameworks, a “layer” defines a transformation of the data, regardless of whether it contains learnable parameters. This is consistent with standard components like pooling, normalization, or fixed positional encoding layers.

In our PINN architecture, this module is implemented directly in PyTorch to ensure it is a first-class component that propagates gradients. To avoid any ambiguity, we have consistently explicitly labeled it as analytical in the manuscript (e.g., at Line 183). This nomenclature emphasizes its role as a modular architectural component while being transparent about its deterministic nature.

Change in manuscript: The terminology was maintained for architectural consistency. We have double-checked that its “analytical” and non-trainable nature is clearly specified in the model description to prevent any misunderstanding.

We have added clarification in the text that $V_z$ and $V_{\mathsf{GS}}$ are computed analytically in the trajectory layer and passed directly without transformation.

Change in manuscript: Added clarification in Section 4 about $V_z$ and $V_{\mathsf{GS}}$ identity mapping.

We agree that this distinction is crucial. We have updated the caption of Figure 1 to explicitly state that it represents the single-step derivative estimation, while the full trajectory is generated autoregressively via ODE integration.

Change in manuscript: Updated Figure 1 caption to clarify single-step estimation vs autoregressive generation.

We have corrected Figure 1 to use “Trajectory Layer” to match the text.

Change in manuscript: Changed “Trajectory Analytics” to “Trajectory Layer” in Figure 1.

Corrected as part of the tense coordination review.

Change in manuscript: Changed “we use” to “we used” in Section 4.

We agree with the reviewer that formalizing the ODE model early improves readability. We have moved Equation 5 (and its description) to adhere to the structure proposed in the introduction of Section 4, placing it immediately before the detailed breakdown of the layers. Regarding Table 4, we have added a reference to it (“Summary of features”) immediately where the state and context variables are introduced, ensuring the reader has access to the feature definitions early in the section.

Change in manuscript: Moved Equation 5 to the beginning of the methodology section and added early reference to Table 4.

This is an important clarification. The loss is computed on partial trajectories generated autoregressively. Specifically, we use sequences of $N = 60$ time steps. The ODE is integrated forward from $t_0$, and the loss is accumulated over all $N$ predicted states $x(t_1) \ldots x(t_N)$ comparing them against the ground truth. We have updated Equation 10 and the surrounding text to make this explicit.

Change in manuscript: Updated loss function description and Equation 10 to explicitly specify the summation over autoregressive sequences of $N = 60$ time steps.

We agree that this phrasing is precise and consistent with the figure. We have replaced “weights of the neural ODE” with “weights of the derivative layer” in the text.

Change in manuscript: Changed “weights of the neural ODE” to “weights of the derivative layer” in Section 4.

We have followed this recommendation. To ensure consistency throughout the text, we have typeset “derivative layer” in bold at every occurrence in Section 4.

Change in manuscript: Applied bold formatting to all instances of “derivative layer” in Section 4.

We have added the full definition. The text now reads “4th order Runge–Kutta (RK4)”.

Change in manuscript: Defined RK4 as “4th order Runge–Kutta”.

The reviewer is correct to point out this ambiguity. This refers to the first data point available in the ADS-B recording for a given flight, which serves as the initialization point for the integration. We have clarified the text to read: “applied at the first available point of the recorded trajectory (depending on ADS-B coverage)”.

Change in manuscript: Clarified that the starting point corresponds to the first available point of the recorded trajectory.

We have reordered the figures to ensure correct sequential referencing.

Change in manuscript: Reordered figures in Section 5.

This is a crucial point indeed. The evaluation is performed autoregressively on the full trajectory (from $t = 0$ to $T$). The errors reported are the accumulated errors over the entire flight, which is a much more rigorous test than point-by-point prediction. We have added a sentence in Section 5 (Results) to make this explicit.

Change in manuscript: Clarified in the Results section that evaluation metrics are computed on full autoregressive trajectories ($t = 0$ to $T$).

We deliberately show raw, unfiltered ADS-B data to demonstrate the model’s robustness to measurement noise and quantization artifacts inherent in open surveillance data. This transparency is important for reproducibility and shows that the model can handle real-world data quality.

Change in manuscript: No change (justified choice to show raw data).

Indeed there is a lot of information to bring together and we tried many combinations, to no avail. We think though that this is not critical on this plot.

Change in manuscript: No change.

Since this change is purely stylistic and does not affect clarity or interpretation of the figure, we have chosen to keep the current formatting, consistent with other figures.

Change in manuscript: No change.

We agree that these two paragraphs discuss the same results. We have merged them into a single paragraph to improve the flow of the text.

Change in manuscript: Merged the paragraph starting with “Overall…” with the preceding one.

We appreciate the reviewer’s observation. Figure 3 is intentionally designed as a comparative visualization to emphasize regions where the predictive model performs worse than BADA, rather than to allow direct reading of absolute numerical errors. The numerical values referenced in Lines 265–272 are provided in Table 3 for clarity.

Change in manuscript: No change.

We agree with the reviewer that these paragraphs interpret the results and discuss potential causes of discrepancy (BADA methodology, selected altitude noise, mass assumptions) rather than presenting raw findings. We have moved this discussion to Section 6, grouping it under a new subsection “Interpretation of Benchmarking Results”. We also merged the discussion on selected altitude noise with the corresponding limitation paragraph to avoid redundancy.

Change in manuscript: Moved the interpretative discussion from the Results section to the Discussion section.

This is a fundamental point. We chose Neural ODEs to bridge the gap between rigid physics-based models (BADA) and black-box sequence models (LSTMs). Unlike BADA, our model learns latent operational factors (mass, airline policy) from data. Unlike LSTMs, the continuous-time formulation naturally handles the irregular sampling inherent to ADS-B data and allows for the seamless integration of physical constraints (kinematic equations). We have added an explicit justification at the end of Section 2.

Change in manuscript: Added a paragraph in the Related Work section explaining the necessity of Neural ODEs for handling irregular sampling and integrating physical constraints compared to sequence-based and purely physics-based methods.

The primary contribution is methodological: we demonstrate that enforcing analytical kinematic constraints within a Neural ODE framework acts as a sufficient regularizer to reconstruct physically consistent dynamics even when key variables (mass, thrust) are unobservable. This goes beyond engineering by establishing “hybrid modelling” as a valid theoretical approach to recover high-fidelity dynamics from low-fidelity, incomplete open data. We have explicitly stated this contribution in the Introduction.

Change in manuscript: Added a statement in the Introduction clarifying the methodological contribution regarding hybrid modelling as a compensation for unobservable states.

We fully agree. To address this, we have added a dedicated subsection “Interpretation of Benchmarking Results” in Section 6 (Discussion). This section explicitly acknowledges the sources of bias, including the noise in Mode S inputs (selected altitude), the difference in mass assumptions (fixed for BADA vs learned for Neural ODE), and the fact that BADA is a generic model not designed for individual trajectory reproduction with noisy controls.

Change in manuscript: See changes for Reviewer 2 (creation of “Interpretation of Benchmarking Results” subsection).

We appreciate the concern for reproducibility. Most of these details were already present in Section 4:

• Network Configuration: 3-layer backbone (48 neurons) + 2-layer heads (48 neurons), ReLU activations (Section 4).

• ODE Solver: Explicit Euler scheme, as RK4 was deemed unnecessary for this sampling rate (Section 4).

• Loss Weighting: Inverse standard deviation of each feature (Eq. 10 and text below).

• Training: AdamW optimizer ($lr = 10^{-4}$), Batch size 512, 40k steps (Section 4).

Furthermore, to ensure absolute transparency and reproducibility, we have released the full source code, pre-trained weights, and preprocessing scripts on GitHub.

Change in manuscript: No definition added as the text already contained all specified parameters and GitHub repo available.