Dear reviewer,

thanks for taking the time to write your detailed and helpful review and for recognizing the contribution of our benchmark, given that there are scarcely any publically available in the mechanics domain.

General Comments:

The necessity of the FQO-UNet model having to interpolate [...] doesn't seem like the most adaptable solution for real life settings.

We are able to drop this constraint. Please see Section 3 of our general answer for details.

It appears that the proposed numerical plate simulations are still somewhat simplistic, as for example there is no high frequency noise present in FRFs

We agree that considering noisy data is an interesting task for applications like structural health monitoring, where sensor data is available. However, we target design space exploration / optimization of a numerical model. In this setting, we do not have noisy data, but deterministic simulation data in a high dimensional design space.

Moreover there is no investigation of difficult FRF phenomenon which may arise in real life such as the appearance of closely spaced modes [...]

Conceptually, our model directly predicts deflection shapes, which are a superposition of all modes. Even for frequency responses with closely spaced peaks, the underlying physics and governing equations apply irrespectively. For modeling frequency responses with dense modes, our frequency query approach is suitable since it allows us to sample the FRF at arbitrary resolution.

The proposed FQO-UNet does not feel like a strong original contribution. It is constructed very much from "commercial off the shelf/plug-and-play"-type NN components.

We kindly disagree on this point. Our FQO-UNet results from systematic experimentation and exploration specifically for vibration prediction. It involves established as well as unconventional components, but the combination of them into a working system is far from trivial. These are several crucial choices:

• Transformation procedure between frequency response and velocity fields: The choice of our exact transformation (Eq. 1 in paper and code) is crucial to avoid numerical issues and enable gradient flow.
• Balance compute between the shared embedding of the plate geometry and the frequency-specific decoder.
• Strategy for incorporating scalar geometry parameters and frequency query: The geometry parameters are inserted in earlier layers, to affect the shared embedding for the plate geometry. FiLM layers lead to a better generalization than concatenating parameters or sinusoidal embeddings.
• Use of self-attention: The global information exchange through self-attention support the prediction, since vibration patterns depend on the global plate geometry.

Our model clearly outperforms established architectures, like the Fourier Neural Operator and DeepONet. We believe that building on established components is indeed one of the reasons for the impressive progress in (applied) machine learning in recent years.

If you disagree with this perspective, we gladly elaborate on this.

Questions:

Are there any plans to extend the nature of the dataset? Perhaps to different plate geometries [...]? Perhaps also to include the effect of damage in the plate? [...]

We extended the G5000 setting by variable boundary conditions and force position, please see Section 1 of the general answer. In the future, we intend to move to more complicated structures, such as lightweight plate structures with additional frames and stringers. Modeling structure-fluid interactions to study sound radiation is an additional area of interest.

Damage prediction would require taking the frequency response as an input for a classifier. While this is an important application, it is not directly compatible with our model design which maps from plate geometries to frequency responses. Thus, we mainly consider the design optimization task (general answer Section 2) which is directly applicable to our existing dataset.

Limitations:

We identified two main points in the answer: transfer learning/domain shift and sample efficiency.

Transfer Learning / Domain Shift:

Perhaps also the transfer learning section can be enhanced. [...] taking a pre-trained model on bridge A, and zero/one shot applying it to bridge B [...] pre-traind model [...], can be extended to a real world model [...]

We concur on the significance of transfer learning and conduct an additional experiment assessing transfer learning between from a more constrained dataset to a more general dataset (see Section 1 general answer). We hope you understand that addressing transfer to entirely new geometries and real world data is not possible within the scope of this paper.

Sample Efficiency:

[...] It would be good to analyze what is the absolute minimal amount of computational training data [...]

We agree that data-efficiency is crucial for engineering problems and generate a new dataset consisting of 50,000 plate geometries (V5000 setting) but only 15 frequency evaluations per geometry. We train models on subsets with a fixed amount of 150,000 data points by varying the ratio of frequencies and plate geometries (Table 4, Figure 2 PDF). Our original dataset has 1.5 Mio data points.

Reducing the frequencies per geometry drastically increases the data efficiency of our method. With a tenth of data points, the MSE metric approaches the value of the original dataset, with slightly less favorable results for the other metrics. When training with the whole new dataset, half the size of V5000, the MSE is less than a third of our original model. Note, this strategy is only compatible with a frequency-query approach.

Table 4

Dear reviewer,

thank you for taking the time to write your thoughtful review. We appreciate that you recognize our dataset and neural network architecture as novel, as the area of engineering and more specifically vibration prediction is not well established in the ML community. Please also take a look at our general answer, where we detail some additional results and discuss extensions to our dataset.

General Comments:

The study is limited in scope regarding the forcing terms, material properties, and boundary conditions.

Please note that the G5000 setting already consists of variable material properties (see Tab. 4 in the appendix). However, we agree with your assessment that a flexible model is desirable and extended the G5000 setting by variable boundary conditions and force position. At this complexity level of our dataset, there are already several challenging methodological questions in e.g. data efficiency, design optimization and transfer learning specifically for frequency response data. We believe developing an understanding of these issues is easier with our extended dataset than a more complex dataset (please see Section 1 general answer for further discussion).

Possible misunderstandings and lack of clarity:

It is unclear how the beading patterns are integrated into the Mindlin differential equation.

Technically, beading patterns define the mid-surface of the global plate geometry. This geometry is discretized via the FEM method using shell elements. Shell theory combines a plate formulation (Mindlin differential equation) and a disk formulation for in-plane loads. Thus, the beading patterns define the global position and orientation of the shell elements. Hence, the beading patterns do not have to be incorporated explicitly in the Mindlin differential equation, since the equation models the local behavior of a Shell element.

The practical significance of predicting vibration patterns is not well-justified. In many applications, the frequency response of the plate is more critical than the detailed vibration modes.

We agree that from a practical standpoint, the frequency response function (FRF) is an important quantity. Because of this, we evaluate our method only on FRFs (metrics). However, for training, a clear result from our experiments is that FRF predictions become more accurate when predicting velocity fields and then directly calculating the FRF from the fields. We hypothesize that velocity field prediction acts as a regularizer and prevents (some) physically impossible solutions. Furthermore, from an acoustics perspective, the vibration pattern is crucial to predict the sound radiation characteristics. The normal velocity field of a vibrating structure induces pressure waves in the surrounding fluid. Assuming a weak coupling of the fluid-structure interaction, our DL model could be used to model the Neumann boundary condition of an adjacent fluid domain. This is only possible if we predict the velocity field.

The motivation for the study is weak and lacks clear justification.

As acknowledged by other reviewers (1y6b, 75jo), vibration prediction is an important problem. Specifically, panel structures, which can be simulated using our approach, are used in mobility, civil engineering, renewable energy technologies, household appliances and many more. By providing a method for faster vibration mode prediction, we can contribute to the noise-reduced designs. Our benchmark dataset and method represents a first step in applying deep learning based surrogate models to these vibration prediction problems.

Thank you for drawing our attention to these points. This will enable us to improve the relevant sections of our paper.

Questions:

Can the authors please justify their comparisons to the baselines? Why are these specific architectures chosen? How do they differ? Why do these networks have gaps in performance?

There have been some prominent and successful works in the machine learning for differential equations community, where we position our work in a broader sense. Fourier Neural Operator and DeepONet are arguably among the most well-known methods in this space, which is why we choose these methods as baselines. The k-Nearest Neighbor approach serves as a tool to gauge the complexity of the learning task: we expect a deep learning-based method to perform significantly better than k-nearest neighbors. The remaining architectures were constructed to investigate central architecture components based on our analysis of the problem (our paper, Section 3 Q1 to Q3). The reasoning why we assumed e.g. that the frequency-query approach is better than predicting the response for all frequencies at once is that there is intense variation between consecutive frequencies and with frequency queries the neural network can focus on single frequencies for one forward pass. In contrast to the baseline methods, our specific architectures are designed for the problem at hand, and have useful inductive biases such as the frequency query approach. Feel free to ask if you have further questions regarding our choice of baselines.

Dear reviewer,

thanks for taking the time to write your informative and helpful review. Your recognition of the contributions of our work as well as finding it well written and organized is highly appreciated!

Please also take a look at our general answer, where we detail some additional results and discuss extensions to our dataset. There, we also discuss our reasoning for keeping our plate geometries comparatively simple. As you say, to extend the dataset to more complex geometries, costly simulations would need to be run. Our research currently focuses on data-efficiency, including active learning methods to keep the simulation costs reasonable, while extending our approach to more complex geometries.

Questions:

How does the current architecture scale to larger grids (finer FEMs), which would be used for instance to capture/represent smaller beadings, both in terms of accuracy and runtime ?

Predicting the velocity fields at a higher resolution requires adding additional upscaling layers, which would cause moderate increases in runtime. The speed-up of one neural network prediction compared to one FEM simulation will increase, since the computational cost of the FEM increases drastically with increased number of degrees of freedom. In terms of accuracy, we would not expect large changes in prediction quality. As the velocity fields are spatially smooth, no additional information would be added. In the investigated frequency range it is expected that very small beadings also only have very small effects on the vibration patterns and therefore not affect the accuracy of prediction much.

Would adding a physics loss based on the equation be feasible and yield better accuracy ?

Adding a physics based loss and setting up a physics informed neural network (PINN) is a compelling idea and might yield better accuracy. However, using a PINN for shell structures is not straightforward, since it requires solving the shell equations on a non-Euclidean domain. Current research has investigated such PINNs for solving a single Shell model and compares it to classical FEM. However, extending the approach to varying geometries, like in our case the different beading patterns, remains an open challenge.

We will add this discussion to the paper.

Dear reviewer,

thank you for taking the time to write your thoughtful review. We are pleased that you also consider structural vibrations to be an interesting and important problem in the engineering domain. We discuss generalization and domain shift in more detail in Section 2 of our general answer.

Questions:

Does the network learn the vibrations physics or just learn the mapping from the input data to the output? Does it learn to solve a second order vibration ODE or it is imitating the input training data?

Our numerical model can be considered a mapping from the input space to resulting vibrations, which is approximated by our network. Our results in generalization, e.g. transfer learning and finding new beading patterns designs with properties outside the training data distribution (see General Answer Section 2), suggest that some physical knowledge is acquired.

Can the authors explain how the surrogate model can predict the structural vibration of structures that has not been seen in the training phase?

Conceptually, analogously to other application fields, with the right inductive biases (e.g. predicting velocity fields, simplicity bias of SGD) a neural network learns a function that approximates the ground truth function (in our case numerical simulation). Empirically, we find our model to generalize to unseen beading patterns within distribution (Table 1 general answer) and out-of-distribution (Table 2).

How the can one use the model for scaling predicting for higher scale geometries? Have the authors utilized the model for extrapolations? How would the model behave in presence of domain shift of the input data?

We agree, that these are interesting and challenging questions. In Section 4 of the paper, we split our dataset (based on the number of mesh elements that are part of a beading or not) and trained either on the 'more beadings' or 'less beadings' subset while evaluating on the other set. Although the performance decreased, our networks were still able to perform better than several baseline methods despite never having seen data from this distribution. Further, we report results from taking a pre-trained model on V5000 and fine-tune it on G5000 (Table 2 in Section 2 general answer). As G5000 includes a substantially larger design space than V5000, the gain in performance is promising for the potential of fine-tuning a model under domain-shift. Without fine-tuning, we believe large extrapolations will fail with the current setup.

After the model is built, Can the model be used for frequency response prediction of a different system such as rotor dynamics systems or a ball bearing system?

To extend our model to applying to systems like ball bearings and rotors, we would have to construct it differently. Our network depends on the specific input space defined by the beading pattern and scalar parameters of a plate. A true foundational model capable of simultaneously modeling plates, rotors and ball bearings would need to map all these systems to a common input space. Unfortunately, we are not yet at this point. However, our method could be used to simulate plate components in such systems, where rotating components can be modeled as harmonic excitations.