Physics-Informed Liquid Neural Network Emulator for CRTM with Atmospheric-Layer Jacobian Capability

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper proposes CRTM-LNN, a physics-informed Community Radiative Transfer Model (CRTM) emulator that integrates Liquid Neural Networks (LNN) with an analytic radiative transfer solver. By predicting layer-wise optical depth increments instead of directly fitting brightness temperatures, the framework preserves the physical pathway of radiative transfer and Jacobian computation capability. Its accuracy and computational acceleration are validated over the IASI 15 μm CO₂ absorption band. The research topic addresses practical operational bottlenecks in hyperspectral satellite data processing; the hybrid architecture features distinct physical innovation, the experimental dataset is of sufficient scale, and the conclusions hold application reference value. However, the manuscript still has significant room for improvement.

Detailed revision comments are provided below:

1. Clarify the novelty and marginal contributions of the LNN-LOD architecture by explicitly distinguishing it from existing RNN-based and physics-informed RTM emulators, and highlight its unique technical advantages.
2.  Supplement key implementation details of LNN-LOD (hidden state dimension, network structure of *f*(·), adaptive gate formulation, hyperparameters) and add ablation studies to justify architectural choices for better reproducibility.
3.  Provide statistical characteristics and sample distribution of the dataset, ensure identical training configurations for the MLP baseline for fair comparison, and add benchmarking against state-of-the-art RTM emulators.
4.  Expand Jacobian evaluation beyond temperature to water vapor, ozone and CO₂ with quantitative metrics, to fully support the "atmospheric-layer Jacobian capability" claimed in the title.
5.  Specify complete hardware/software configurations (CPU/GPU models, CRTM compilation settings, batch size) for speedup benchmarks, and report model parameter count and memory footprint.
6.  Add direct physical consistency checks, including non-negativity of Δτ, valid range of transmittance, and weighting function normalization, and test robustness under extreme atmospheric profiles.
7.  Quantify training computational cost compared with the MLP baseline, and further discuss technical challenges and feasible pathways for extending to all-sky, scattering and surface emissivity scenarios.

Author Response

We sincerely thank the reviewer for the careful evaluation of our manuscript and for recognizing the potential value of the proposed CRTM-LNN framework. We appreciate the reviewer’s positive assessment of the layer-optical-depth learning strategy, the explicit differentiable radiative-transfer pathway, the scale of the evaluation dataset, and the potential operational relevance of the work.

We agree that additional implementation details, quantitative Jacobian analyses, physical-consistency checks, computational information, and discussion of the present limitations are necessary. The manuscript has therefore been substantially revised. Our point-by-point responses are provided attached.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript presents a physics-informed Liquid Neural Network (LNN) emulator designed to accelerate the Community Radiative Transfer Model (CRTM). By combining an ODE-inspired LNN for layer-by-layer optical-depth evolution with an analytical radiative-transfer solver, the proposed approach preserves physical consistency while substantially reducing computational cost. The demonstrated results are encouraging, particularly the improved vertical coherence of the Jacobians compared with a conventional MLP emulator.

However, the validation is currently limited to clear-sky conditions, where the radiative-transfer problem has a relatively simple recursive structure: atmospheric transmittance can be consistently represented through cumulative optical depth, and this formulation is naturally compatible with the ODE-like hidden-state evolution of an LNN. It therefore remains an open question whether the demonstrated advantages will persist when the emulator is extended to cloudy-sky conditions. In particular, cloud scattering introduces substantially more complex interactions among radiance, atmospheric layers, and propagation directions, which may affect both the vertical coherence of the resulting Jacobians and the suitability of the current hidden-state formulation.

I recommend publication after minor revision, but suggest that the authors acknowledge this limitation and discuss the challenges associated with extending the approach to cloudy-sky applications. In addition, the reported computational advantage should be placed in the context of the specific applications demonstrated in the manuscript. If the actual acceleration relative to the physical CRTM is approximately 5× for the clear-sky case, despite an 18× speedup reported for the forward simulation, the authors should clarify the practical significance of the computational gain for end users. In particular, it would be valuable to provide examples of specific applications for which the computational cost of the physical CRTM becomes a significant limiting factor, but for which CRTM-LNN could provide a sufficiently accurate and computationally efficient alternative. Such discussion would help demonstrate the practical value of the proposed emulator beyond its computational performance in the benchmark experiments.

Below are some specific questions and comments:

1. The LNN still employs an analytical, differentiable radiative-transfer solver. Is this solver the same as, or based on, the radiative-transfer solver used in CRTM? Please clarify the relationship between the solver used in the proposed framework and the physical CRTM solver, including any modifications or simplifications introduced for the LNN emulator.
2. From what perspective do the authors characterize the proposed framework as “GPU-native”? Nowadays, most major AI-based frameworks are designed to support GPU acceleration, making the term “GPU-native” somewhat ambiguous and potentially less meaningful without further clarification. In addition, the physical CRTM itself could potentially be reformulated or optimized for execution on GPU architectures. The authors should therefore clarify what specific architectural, algorithmic, or computational features distinguish the proposed framework as GPU-native and what advantages it offers over existing GPU-accelerated or AI-based implementations.

 

1. In Figures 7–10, how is the variability represented by the shaded areas derived? It is not clear to the reader how the statistical variability or variance is quantified, particularly because the comparisons between CRTM-LNN and CRTM appear to be based on individual cases. Please clarify what the shaded regions represent and how they were calculated, including the number of cases or samples used to derive the reported variability.

 

1. The authors provide a comprehensive description of the optimization objective, with the total loss comprising the optical-depth reconstruction loss, brightness-temperature reconstruction loss, and vertical smoothness regularization term. However, they do not sufficiently elaborate on how these individual terms are defined and quantified. In addition, the loss-function diagram is not clearly connected to a well-illustrated LNN architecture. A more detailed explanation, together with a clearly illustrated LNN structure and its connection to the individual loss components (preferably supported by illustrative diagrams), is needed to clarify the underlying methodology and strengthen the justification for the proposed approach. This would also help readers better understand the fundamental operation of the LNN and how it is used to construct the CRTM emulator.
2. The authors state that “The layer-by-layer state-evolution architecture provides an extensible foundation for incorporating surface emissivity, clouds, aerosols, and scattering processes, although further development is needed to improve weaker trace-gas Jacobians and extend the framework to full-spectrum and all-sky applications.” As a reader, I would recommend that the authors make this statement more cautious, particularly regarding the extension of the proposed architecture to scattering processes associated with clouds and aerosols. Incorporating scattering into the radiative-transfer formulation can make the physical interpretation of the LNN hidden state substantially less straightforward. In the presence of cloud or aerosol scattering, the radiative-transfer equation involves angular coupling among radiances propagating in different directions, as well as multiple scattering and interactions between upward- and downward-propagating radiation. Consequently, the radiative state can no longer be represented simply as a sequentially attenuated and emitted radiance along a single propagation path. The authors should therefore discuss more explicitly the challenges associated with extending the current layer-by-layer state-evolution formulation to scattering media and avoid implying that such an extension would be straightforward based solely on the current clear-sky results.

 

 

 

 

Author Response

We sincerely thank the reviewer for the positive assessment of the manuscript and for the thoughtful and constructive comments. We particularly appreciate the reviewer’s recognition of the value of combining ODE-inspired layer-by-layer optical-depth modeling with an explicit differentiable radiative-transfer pathway, as well as the encouraging improvement in the vertical coherence of the resulting Jacobians relative to the conventional MLP baseline.

In response to the reviewer’s concern regarding cloudy-sky applications, we have revised the Abstract, Discussion, and Conclusions to define the demonstrated scope of the present study more clearly. The current framework is evaluated for clear-sky, absorption-only thermal infrared radiative transfer, and the results do not establish that the same accuracy, computational efficiency, or Jacobian coherence will necessarily persist under cloudy or scattering conditions. We have added a detailed discussion explaining that an all-sky extension would require representation of additional cloud and aerosol optical properties, angular coupling between upward- and downward-propagating radiances, multiple-scattering interactions, and potentially an augmented neural-state formulation coupled with a differentiable multi-stream or multiple-scattering solver. The revised manuscript therefore presents all-sky development as an important but technically substantial direction for future research rather than a straightforward extension of the current framework.

We have also clarified the computational benchmarks and their practical interpretation. The reported speedups correspond to different computational tasks and hardware configurations and should not be interpreted as alternative estimates of the same operation. For forward simulation, CRTM-LNN reduces the processing time from approximately 90 to 7.5 min in the CPU-to-CPU comparison, corresponding to an approximately 12× speedup, and achieves an approximately 18× wall-clock speedup when the emulator is executed on a single GPU relative to CPU-based CRTM. For Jacobian generation, automatic differentiation with one and two GPUs provides approximately 2.5× and 5× system-level speedups, respectively, relative to the CPU-based CRTM tangent-linear calculation. We now explicitly distinguish the hardware-comparable CPU-to-CPU benchmark from the CPU-to-GPU comparisons and report the model configuration, parameter count, training cost, hardware, software, batch size, and distributed-computing settings used in this study.

To demonstrate the practical significance of these gains, we have expanded the Discussion to describe specific applications in which repeated physical-CRTM calculations can become computationally limiting. These include large-volume observation-minus-background monitoring, rapid quality assessment, iterative atmospheric retrievals, channel-selection and information-content studies, ensemble sensitivity experiments, and variational or ensemble-based data assimilation requiring repeated forward and Jacobian evaluations. We also clarify that the reported acceleration applies to the tested radiative-transfer workloads; the end-to-end benefit within an operational system will additionally depend on data input/output, preprocessing, quality control, minimization, and communication costs. These revisions place the computational results in a more realistic application context while demonstrating the potential value of CRTM-LNN as an efficient, differentiable surrogate observation operator.

Specific responses are provided attached.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Attached as annex.

Comments for author File: Comments.pdf

Author Response

We sincerely thank the reviewer for the careful and constructive evaluation of our manuscript and for recognizing the scientific value of combining machine-learning-based layer-optical-depth prediction with an explicit, differentiable radiative-transfer pathway. We particularly appreciate the reviewer’s positive assessment of the large evaluation dataset, the agreement with CRTM in brightness temperature and intermediate optical quantities, the automatic-differentiation-based Jacobian capability, and the potential practical relevance of the computational acceleration.

We agree that the original manuscript did not always distinguish sufficiently clearly among capabilities that are structurally enforced by the framework, results demonstrated within the present experiments, and possible extensions proposed for future work. The manuscript has therefore been substantially revised to calibrate the conclusions to the actual scope of the study. We now state explicitly that the current results apply to 111 IASI channels in the 15-μm CO₂ absorption band, under clear-sky, absorption-only conditions and with unit surface emissivity. We have also replaced potentially overstated descriptions of continuous dynamics and post-training adaptability with the more precise characterization of an ODE-inspired, input-conditioned, layer-by-layer state-evolution model on the fixed ECMWF91L grid. Similarly, “physical consistency” is now described in terms of the explicit optical-depth-to-radiance pathway and the physical constraints enforced under the assumptions of the present solver, rather than as equivalence to the complete CRTM. Extensions to broader spectral regions, realistic surface emissivity, clouds, aerosols, precipitation, and multiple scattering are now clearly presented as substantial future developments rather than straightforward consequences of the current clear-sky results.

The primary objective of this manuscript is not to conduct a general comparison or establish a universal ranking among LNN, MLP, and RNN architectures. Rather, our goal is to develop and evaluate a differentiable CRTM-LNN emulator that predicts layer optical-depth increments, retains physically interpretable intermediate radiative quantities, and supports atmospheric-state Jacobian generation through automatic differentiation. The MLP is included as a supporting reference baseline, not as a parameter-matched architectural ablation. Both models use the same source training dataset, thereby avoiding differences caused by atmospheric sampling, but we now clarify that the comparison evaluates the two complete emulator configurations rather than isolating the effect of the recurrent architecture alone. The Introduction has also been expanded to position CRTM-LNN relative to direct-output, recurrent, Jacobian-constrained, optical-property, and differentiable-physics emulators without presenting cross-study RMSE or speed values as a universal quantitative ranking.

To strengthen methodological clarity and reproducibility, we added dedicated subsections describing atmospheric-state and auxiliary-input preprocessing, the state and auxiliary-input encoders, and the active model configuration and training hyperparameters. Appendix A now provides the mathematical definitions of the nonlinear hidden-state tendency, bounded adaptive gate. The revised architecture figure connects preprocessing, encoding, LNN-LOD state evolution, layer-optical-depth prediction, the analytic radiative-transfer solver, and the individual training losses within a single computational pathway. We also report the active parameter count, parameter breakdown, optimizer, learning-rate schedule, batch size, number of epochs, numerical precision, distributed-training configuration, software versions, hardware, and total training time.

We have also strengthened the quantitative Jacobian evaluation. Tables 6–8 now report Jacobian RMSE, correlation, absolute peak-pressure error, absolute centroid-pressure error, and vertical-roughness mismatch. A controlled ablation of the vertical-smoothing loss, , shows that its contribution is modest and channel dependent; models trained with and without this term retain generally similar Jacobian structures. The revised manuscript therefore no longer attributes the improved Jacobian coherence primarily to the smoothing penalty or to any single architectural component. Instead, the results are interpreted as the joint outcome of LNN-LOD vertical state evolution, supervised layer-optical-depth prediction, cumulative optical-depth constraints, and analytic radiative-transfer coupling.

The MLP comparison is now supported by quantitative results rather than metrics described as “not shown.” For temperature, CRTM-LNN generally provides lower RMSE, higher correlation, smaller centroid-pressure errors, and substantially lower vertical-roughness mismatch than CRTM-MLP, although peak-pressure improvements are not uniform across all channels. We further expanded the analysis beyond temperature by adding CO₂-Jacobian results. CRTM-LNN substantially reduces CO₂-Jacobian RMSE and roughness mismatch relative to CRTM-MLP and generally improves correlation; nevertheless, the remaining low correlations (<0.25) and inconsistent vertical-localization improvements clearly show that CO₂ sensitivities remain challenging. We now discuss that this limitation likely reflects both the weak CO₂ sensitivity and the restricted CO₂ variability associated with the cap in the current CRTM configuration.

Although a complete component-by-component ablation involving an MLP-Δτ model, comparable GRU or residual-RNN configurations, alternative gates, and multiple loss combinations would provide additional architectural insight, such a study would constitute a broader neural-architecture benchmark beyond the central objective of the present work. To avoid unsupported causal attribution, we have revised the manuscript so that the demonstrated performance is assigned to the integrated CRTM-LNN framework, rather than claiming that the liquid recurrence alone is responsible for every improvement. The current  ablation directly addresses the concern that smoother Jacobians might arise mainly from explicit vertical regularization, while a broader component-wise ablation is identified as an important direction for future work.

The computational evaluation has also been documented and interpreted more carefully. The revised manuscript distinguishes the approximately 12× CPU-to-CPU forward-model speedup from the GPU-versus-CPU comparisons. The approximately 18× forward speedup obtained with one GPU and the approximately 2.5× and 5× Jacobian speedups obtained with one and two GPUs, respectively, are now described as practical system-level accelerations under the tested configurations rather than hardware-normalized algorithmic comparisons. Model configuration, parameter count, training cost, hardware, software, batch size, and distributed-computing details have been added to the revised manuscript. We also acknowledge that the layer-by-layer recurrence and two-stage update introduce sequential operations along the 91-layer grid, limiting vertical parallelism relative to fully feed-forward architectures.

Additional direct physical-consistency checks have been incorporated. The inverse output transformation guarantees nonnegative layer optical-depth increments; cumulative optical depth is therefore monotonic, transmittance remains within (0,1], and the corresponding layer weighting factors remain nonnegative. Robustness within the evaluated clear-sky domain is further examined using six temporally independent days and by stratifying the optical-property comparisons into four cumulative-optical-depth regimes, from nearly transparent to strongly opaque conditions. We now state explicitly that these tests demonstrate robustness within the atmospheric domain represented by the current datasets and do not constitute unrestricted out-of-distribution extrapolation.

Finally, the Discussion and Conclusions have been reorganized to reduce repetition and distinguish demonstrated results from future possibilities. Surface-emissivity treatment is described as an extension of the surface boundary condition, whereas all-sky applications are recognized as substantially more complex because they require additional optical properties, angular coupling, interactions between upward- and downward-propagating radiance, multiple scattering, an augmented state representation, and a differentiable scattering solver. The conclusions are therefore restricted to the clear-sky, absorption-only spectral domain evaluated here.

We believe that these revisions substantially strengthen the methodological justification, quantitative baseline evaluation, reproducibility, physical interpretation, and computational documentation of the study while preserving its central contribution: the development of a compact, differentiable CRTM-LNN emulator that combines ODE-inspired layer-optical-depth modeling with an explicit analytic radiative-transfer pathway.

Our point-by-point responses attached.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript is well organized, the research question is clearly defined, methods and results are sound, and the conclusions are supported by the data. The paper meets the requirements for publication in this journal. I recommend accepting this manuscript as is.

Author Response

Response: We sincerely thank the reviewer for the careful evaluation of our manuscript and the recommendation for acceptance without revision. We greatly appreciate the positive assessment of the manuscript’s clarity, organization, and scientific soundness, and we are grateful for the time devoted to reviewing our work.

Reviewer 3 Report

Comments and Suggestions for Authors

Please find the attachment.

Comments for author File: Comments.pdf

Author Response

please see the attached for our responses to all 33 comments. Thanks.

Author Response File: Author Response.docx