Vibroacoustic Frequency Response Prediction with Query-based Operator Networks

We appreciate your thoughtful review of our paper and are grateful for your positive remarks on the organization of our code, our Related Work section and on including a traditional machine learning model in our experiments.

Response to suggestions, concerns and questions:

1. Repository Documentation:

We acknowledge the issues you raised regarding our repository’s documentation. We will incorporate the suggested improvements by updating our readme with instructions, adding version numbers to the requirements.txt and adding an explanation for Weights and Biases setup.

2. Single Test Set:

We follow the common paradigm in deep learning of splitting the data into training/validation and a single test set. To obtain an estimate of the variation we performed cross validation by six trainings with different randomly sampled training and validation splits for the FQO-UNet on the V-5000 setting. The statistics are presented in the following table.

Table: Six models were trained on random splits in training and validation sets (4500 and 500 samples respectively). The results are denoted as mean (standard deviation).

As we see, the variation across cross-validation runs is reasonably small. When evaluating on the fixed test set containing 1000 samples the variation decreases further.

3. Synthetic Dataset and Real-World Application:

Real-world experiments are beyond the scope of this work due to prohibitive costs for manufacturing and measurements of field solutions (a measurement would require laser scanning vibrometry). However, previous research validated the use of the Mindlin model for the plate and errors of the finite element method are well understood in the engineering community [1].

Please, let us know whether our rebuttal addresses your concerns and if there are further questions.

[1]: Altenbach, Holm, et al. Analysis of Shells, Plates and Beams. Springer Nature, Switzerland, 2020.

We thank you for your recognition of our paper's contributions. Your positive feedback on our paper’s problem, finite element methodology, literature review, clarity, and discussion of limitations is greatly appreciated.

Response to suggestions, concerns and questions:

1. Absence of Noise in Data: Our work concerns learning models, that predict the frequency response of a vibrating plate. The primary goal is to learn a faithful surrogate model. All experiments evaluate scores on the test set, which consists of unseen plate geometries. Hence, they measure generalization and not just how well the data is fitted. Nonetheless, we agree that robustness to noise is important and particulary relevant for real world transfer (for example when manufacturing is imperfect) and should be addressed by follow-up work.

2. Details for Finite Element Method and Solver: We agree that the details of the finite element method and linear solver are crucial. We employ a specialized FEM implementation for acoustics [1]. The setup involves shell elements with 9 nodes, 6 degree of freedom and quadratic ansatz functions for discretization. The direct solver MUMPS [2] is employed. More comprehensive descriptions will be included in the final manuscript.

3. Aggregated Frequency Response Justification: Calculating the aggregated frequency response by taking the mean squared velocity is a common choice in vibroacoustic simulation. The mean squared velocity is an important measure for the radiated sound power. Therefore, acoustic engineers use this quantity to evaluate structural design measures [3]. We will clarify this in the paper.

4. Model Size and Hyperparameter Tuning: We agree on the relevance of model and training details. Model specifications are reported in Appendix C and ablation studies that document our hyperparameter tuning are listed in Appendix D. We will add the number of parameters for the different models to Appendix C. In our experiments, naively increasing the model size did not lead to better performance.

5. Question: Dataset Generation Settings: Thank you for highlighting this aspect. In our dataset, the zero displacement condition is applied exclusively at the plate's edges. This means the central part of the plate can move (vibrate) freely, while the edges are fixed. This modeling choice is common in structural analysis, especially in engineering applications where plates are often components of more complex assemblies. To prevent further misunderstandings, we clarify this in our manuscript.

6. Single-Point Vibration Assumption: A point force excitation is one scenario among many (e.g. spatially distributed load or plane wave excitation). There are good reasons to choose a point force excitation: In real world applications, a point force can represent the force application point commonly defined by the bearing of a structure. Additionally, point actuators are used to excite the structure for experimental validation. Therefore, this setting is often employed in vibroacoustics, see e.g. [4] and [5]. We also choose an asymmetric excitation point to excite the important Eigenmodes. We will clarify this in the paper.

7. Accuracy of FEM and Solver: Generally, the finite element approach in combination with linear direct solvers converges to the exact solution [6]. The accuracy of our solutions in relation to equation (1) is assured by a suitable discretization. In particular, we have chosen the mesh size such that the smallest occurring bending wavelength is discretized with at least 10 nodes. The smallest bending wavelength occurs at the highest frequency and can be analytically calculated for thin plate structures. This approach is common practice for choosing the mesh size of vibroacoustic models [7].

8. Performance on Untrained Frequencies: We recognize the importance of our model's generalization capabilities, which we've explored for different amounts of beadings as detailed in our transfer learning section. Extending this to unseen frequency ranges is a great idea! We conducted this experiment and present our results in the general answer.

We would be interested to hear, whether our rebuttal addressed your concerns.

We are deeply grateful for your evaluation of our paper and appreciate your recognition of its contribution to the field. Your endorsement, especially given your experience in this domain, is highly encouraging.

Response to suggestions, concerns and questions:

1. Plate Geometries Acquisition: We acknowledge your concern regarding the practical aspects of producing plates with beading patterns. Adding beadings to plates to increase stiffness is an established method (see for example corrugations of tin cans) which can be conducted by stamping the plate. To actually produce the plates described in our work, stamps would have to be manufactured for the different beading patterns. The soft beading edges in our samples ensure manufacturability. We will include a more detailed discussion in the final manuscript.

2. Real-World Data Simulations: Our focus in this work is on establishing a data-driven benchmark dataset. Real-world experiments are beyond the scope of this work since special equipment for manufacturing and measurement would be needed. However, previous research provides good evidence for the validity of the Mindlin model and FEM in real world settings (see general answer). Generally, we are highly interested in real-world applicability and follow-up work should address this.

3. Mesh Geometries Representation: The mesh geometries are displayed (e.g. in Fig. 3) in their original discretized (an $81 \times 121$ grid). We did not apply interpolation or other image improvements to have a faithful presentation. This will be explained in the figure caption.

4. Baseline Performances in Table 3: We agree that baselines are important and follow your suggestion of adding new baselines to the revised manuscript for a more comprehensive comparison of transfer learning performance.

5. Visualization in Figure 6: We appreciate your feedback on Figure 6 and suggestions for further visualizations. Regarding Figure 6, we want to make the point that despite visible errors in the field solution prediction, the frequency response (=average) is well estimated. We agree that providing more visualizations of the model results is a good idea and will add them. For now, we provide several gifs showcasing the predicted and ground truth field solutions of the FQO-UNet. https://drive.google.com/drive/folders/1EUeeAtPm0ZFdC6oIBsxJQbsOhhfXVUk4?usp=sharing

6. Mechanical Model in Main Text: We agree on the importance of linking the mechanical model to our methodology and dataset construction. This will be changed in the revised manuscript.

7. Grid-Based Models and FNO: By grid-based models we refer to models, that can only make predictions for predefined, equidistant frequencies (i.e. a 1-dimensional grid). As you correctly point out, this holds for our implementation of the FNO method as well. The Grid-UNet and Grid-RN18 are ablation variations of the FQO-UNet and FQO-RN18, where we removed the frequency query mechanism. We understand that the distinction between baseline methods and the ablation variations of our proposed method needs to be clarified. We will do so in our updated manuscript.

Please let us know, if there is still something left unclear!

It's, in general, a good pioneering work. I'll discuss this with other reviewers, and if no further concerns are raised, I'll keep my score unchanged.

Thank you for your insightful review of our paper. We are grateful for your recognition of the interesting nature of the problem we address, the adequacy of our model choices, and the sufficiency of our experimental results.

Response to suggestions, concerns and questions:

1. Concerns on Audience and Contribution: We would like to emphasize the position of our research in the established fields of Machine learning for physics and science. While machine learning applications to engineering are a fairly new topic in the machine learning community, there is great potential for impactful research. Applications of machine learning to other domains were highly successful, e.g. AlphaFold in biology [1]. Our work seeks to bridge the gap between the communities by challenging data-driven approaches with a mechanical engineering problem. While most existing research models the temporal evolution of dynamic systems, our work's novelty is to directly predict solutions in the frequency domain. To address this task, we contribute a benchmark dataset, scores of relevant baselines, and the new FQO-UNet model.

2. Question on Baselines vs. Proposed Method: As baselines, we selected methods that were proposed in the literature for related physics-based problems. In contrast, our proposed FQO-UNet is specifically designed to address the unique challenges of vibroacoustic frequency response prediction. For instance, the prediction of resonance peaks is addressed by the frequency-query approach. However, we agree that the distinction between baseline methods and the ablation variations of our proposed method needs to be emphasized and we will do so in our updated manuscript.

We would be grateful if you report whether our answer addressed your concerns or if additional questions have come up.

[1] Jumper et al.: "Highly accurate protein structure prediction with AlphaFold", Nature (2021)

We thank all reviewers for their constructive feedback. Following your suggestions, we have uploaded an updated version of our paper. The additions have been marked red for clarity. We have also added new videos comparing the field solution predictions of different methods as supplementary material and to the google drive folder.