Learning to Predict Structural Vibrations
Authors: Jan van Delden
Julius Schultz
Christopher Blech
Sabine C. Langer
Timo Lüddecke
Venue: NeurIPS 2024 poster (2024)
Keywords: Vibrations
Engineering
Acoustics
Vibroacoustics
Surrogate Modeling
Primary area: machine_learning_for_physical_sciences
TL;DR: We present a dataset and method for predicting vibration patterns of plates in the context of noise reduction.

Abstract: In mechanical structures like airplanes, cars and houses, noise is generated and transmitted through vibrations. To take measures to reduce this noise, vibrations need to be simulated with expensive numerical computations. Deep learning surrogate models present a promising alternative to classical numerical simulations as they can be evaluated magnitudes faster, while trading-off accuracy. To quantify such trade-offs systematically and foster the development of methods, we present a benchmark on the task of predicting the vibration of harmonically excited plates. The benchmark features a total of 12,000 plate geometries with varying forms of beadings, material, boundary conditions, load position and sizes with associated numerical solutions. 
To address the benchmark task, we propose a new network architecture, named \modelname, which predicts vibration patterns of plate geometries given a specific excitation frequency. Applying principles from operator learning and implicit models for shape encoding, our approach effectively addresses the prediction of highly variable frequency response functions occurring in dynamic systems. To quantify the prediction quality, we introduce a set of evaluation metrics and evaluate the method on our vibrating-plates benchmark. Our method outperforms DeepONets, Fourier Neural Operators and more traditional neural network architectures and can be used for design optimization.
Code, dataset and visualizations: https://github.com/ecker-lab/Learning_Vibrating_Plates

Learning to Predict Structural Vibrations
Jan van Delden"", Julius Schultz?, Christopher Blech?, Sabine C. Langer', and Timo Lüddecke
Institute of Computer Science, University of Göttingen Institute for Acoustics and Dynamics, Technische Universität Braunschweig "Correspondence: jan. vandelden unni-goettingen.de
Abstract
mechanical structures like airplanes, cars and houses, noise generated and transmitted through vibrations. To take measures to reduce this noise, vibra- tions need to simulated with expensive numerical computations. Deep learn- ing surrogate models present promising alternative to classical numerical sim- ulations as they can be evaluated magnitudes faster, while trading-off accuracy. To quantify such trade-off's systematically and foster the development of meth- ods, we present a benchmark on the task of predicting the vibration harmon- ically excited plates. The benchmark features total of 12,000 plate geome- tries with varying forms beadings, material, boundary conditions, load posi- tion and sizes with associated numerical solutions. To address the benchmark task, we propose a new network architecture, named Frequency-Query Opera- tor, which predicts vibration patterns of plate geometries given specific excita- tion frequency. Applying principles from operator learning and implicit models for shape encoding, our approach effectively addresses the prediction of highly variable frequency response functions occurring in dynamic systems. To quan- tify the prediction quality, we introduce benchmark. Our method outperforms Deep-
of evaluation metrics and evaluate
the method on our vibrating-plan ONets, Fourier Neural Operators and more traditional neural network architec- tures and can be used for design optimization. Code, dataset and visualizations:
www//ckerr -otra//aaeee-eeL-LLaaaannnn Vabrating
Introduction
Humans are exposed to noise in everyday life, which unpleasant and unhealthy in the long term [I]. Therefore, designers and engineers work reducing noise that occurs, for example, in cars, airplanes, and houses. this work, we specifically consider vibrations mechanical structures as source of sound. Vibrating structures radiate sound into the surrounding air. For example in car, the engine causes the chassis to vibrate, which then radiates sound into the interior of the car. By reducing the vibration energy of the chassis, the noise can be reduced.
Vibrations of mechanical structures depend on the frequency of the excitation force (e.g. by the engine). A special case occurs when the excitation frequency matches eigenfrequency of given structure. In this case, the external force adds energy in phase with the structure's 's natural vibration and amplifies the motion with each cycle. This continues until the energy added equals the energy lost due to damping, resulting in large vibration amplitudes. This effect is called resonance and leads characteristic resonance peaks in the dynamic response of the system. At resonance frequencies, due to the higher vibration amplitudes, more noise is emitted. A second distinctive feature of struc- tural vibrations is the vibration pattern, i.e. the spatial field of vibration velocity amplitudes. With increasing frequency, these vibration patterns become more complex and exhibit more local maxima and minima (Figure left) 2
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Velocity feids: maximum velocity per location and frequency
Hz
H20 Hz
280 Hz
Finito elements
predict
Σ Spotial average
numerical simulation
Hamonic exciaations
Model S
predict
3001 wWW
Uff
100 HE
200/6
200 for
(anureerry matteree)
Frequency response function
Input
Figure I: Left: We introduce the Vibrating Plates dataset of 12,000 samples for predicting vibration patterns based on plate geometries. harmonic force excites the plates, causing them vibrate. The vibration patterns of the plates are obtained through numerical simulation. Diverse architectures are evaluated on the dataset. Right: Beadings are indentations and used in many vibrating technical systems. Here, on an oil filter, a washing machine and disk drive. They increase the structural stiffness and alter the vibration.
To reduce noise, the vibration patterns of a mechanical structure can be influenced through modifica- tions to its design. One method is the placement of damping elements, that absorb vibrational energy and thereby reduce sound emission, but this adds weight and requires space. Another approach is introducing beadings, which are indentations in plate-like structures (Figure [ right). Beadings in- shift in the structure's 's eigenfrequencies and
crease the local stiffness of a structure, resulting subsequent resonance peaks. When they well-placed, beadings for a range of excitation frequencies by shifting the resonance peaks out of the range. Reducing the vibration energy for specific range of frequencies is goal in many applications, e.g. in automotive design, where a motor excites vibrations in a range of frequencies 3
reduce the vibration energy
In this work, focus crucial prerequisite for targeted modifications to design: Computing its vibrational behavior. The finite element method (FEM) is an established approach for numerically solving partial differential equations. The geometry of design is discretized into small elements and the solution of the PDE approximated by simple functions, e.g. polynomial functions, defined This method enables the numerical simulation of vibration patterns, but computationally expensive. With increasing frequency and decreasing wavelength, finer meshes required to accurately resolve the vibrations. This leads high increase computational load and limits the number of designs and value of the frequencies that can be evaluated. Deep learning surrogate models could accelerate the evaluation of design candidates by several magnitudes.
on these elements. 4 3
Related work on predicting the solution of partial differential equations with deep learning has mostly focused time-domain problems [e.g. 6 7 (8. In contrast, for our problem the change time is not of interest. Instead, we predict steady-state vibration patterns in the frequency domain. Steady-state refers to the fact that the system vibrates harmonically and the amplitude and frequency remain constant over time since the system in a dynamic equilibrium. Despite being practically relevant in acoustics and structural dynamics in general this problem is far under-explored by machine learning research.
Contributions. To explore the potential of vibration prediction with deep learning methods, we (1) introduce a benchmark and define evaluation metrics it, (2) evaluate range of machine learning methods on the benchmark and (3) introduce our own method.
Our novel benchmark dataset consists of 12,000 instances of an exemplary structural mechanical system, a plate excited by a harmonic force, and their numerically computed vibrations given range of excitation frequencies. Given plate instance, the task is to predict the vibration patterns and frequency response. We vary material properties and the boundary conditions of the plate as well as the geometry by adding beadings. Plates with beadings are abundant in technical systems

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(Figure I right). Plates are also often a component of more complex mechanical systems and their vibrational behavior on their own similar to more complex systems [9] TO], making them a well- posed and scalable initial benchmark problem for deep learning methods.
To address the benchmark task, propose a novel network architecture named Frequeney- Operator (FQO). This model is trained to predict the resulting vibration pattern from plate geome- tries together with an excitation frequency query. This approach is inspired by work on operator learning for predicting the solution to partial differential equations [II and implicit models for shape representation [e.g. I2 instead of a fixed grid. In our case, this enables predictions for any excitation frequency, includ- ing those not seen during training. On our vibrating poprating-plates benchmark, the proposed FQO can accurately predict the highly variable resonances occurring in vibration patterns and outperforms DeepONet [1], Fourier Neural Operators [5] and other baselines.
13 14, both techniques enable evaluating any point in the domain
2 Dataset and Benchmark Construction
2.1 Vibrating Plates Dataset
We introduce a dataset consisting of instances of aluminum plate geometries and their vibration patterns. The plates are simply supported, i.e. the edges cannot move up and down. Depending on the dataset setting, the rotational stiffness at the boundary varied, which corresponds to free rotation or clamped edges. The plate is excited by harmonic point force at varying positions with the excitation frequency varied between and 300 Hz. While the specific setting in other mechanical engineering design tasks may differ, this setup functions as an exemplary engineering design problem. Analogous problems are the design of an air-conditioning enclosure [[6], a washing machine [17 or parts of differences that allow for a comparatively easy experimental real world validation of the computed vibration patterns and do not change typical vibrational characteristics: First, exciting the plate with a point force is a common experimental setup, where a plate is excited via a shaker. Second, the condition of no rotational stiffness at the edges in comparison to clamped edges does not introduce additional uncertainty and parameters into the measurement and mirrors e.g. a bonnet of a car that rests on the chassis. Other typical types of fixation include screws or welding. In the following, we describe the specific quantity of interest of the vibration patterns, how the vibration patterns of the plate obtained via numerical simulation and how the plate geometry and parameters are varied.
chassis (3). Compared to these problems, our plate setup has two
Vibration patterns and frequency response function. Our benehmark is designed to address a vibroacoustic engineering design problem. Therefore, the goal is to predict a quantity that best reflects the noise emitted by a mechanical structure. For a plate, natural choice is the maximum velocity field V2 (1, y) for a specific frequency f- Here, U2 (2r, yff") represents the component of the velocity field orthogonal the plate surface (in the following ((f) denotes the velocity field on the discrete grid). This component closely relates to how much sound is radiated, but specific details about where the velocity on the plate is highest are superfluous. Therefore, we use the mean of the squared velocity as a more compact representation and express it in a frequency response function F, which a function of the excitation frequency:
F(f) = 10 log "2/
(1)
The square velocity is proportional to the kinetic energy and is therefore closely related to how strongly the vibration couples into a surrounding fluid and can then be perceived as airborne sound. In the above expression, A is the plate area over which the velocity is averaged. The result is scaled by a reference value r and converted to a decibel scale.
Numerical simulation. Historically, plate structures have been the subject of intense research regarding their vibrational behavior [e.g. [88 [9]. A common approach in plate modeling is to reduce the model to two-dimensional problem with the goal to accurately describe the vibrational behavior while being computationally efficient 220 To model the vibrational behavior of plates in this work, we use a shell formulation based on Mindlin's plate theory [8]. This theory is applicable for moderately thin plates and represents the plate using mid-plane with constant thickness. Mindlins
3

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(a)
IZA
(b)
V-5000
4
6
10
Number of peaks
(d)
Amplitude
50
Pcak count
2000
25
1000
left plate
-25
right plate
0
100
200
300
100
200
300
100
200
300
Prequency
Frequency
Frequency
Figure 3: Dataset analysis. (a) shows two discretized plate geometries with their corresponding frequency response, the red crosses mark the detected peaks. (b) shows the mean plate design and frequency response. (c) shows number of peaks in different dataset settings. (d) shows the distribution the peaks over the frequencies.
plate theory standard choice many engineering applications and has been experimentally validated [21]
We apply the finite element method to solve the shell formulation and simulate the vibrational
Finite element mash
Field solution maps frequency query
Parmank trequency
behavior of the plate (Figure ( This in-
volves partitioning the plate geometry into dis-
crete elements and approximating the solution on these elements by simple ansatzfunctions. By choosing a sufficiently large number of ele- ments, the solution converges the exact solu- tion of the model [23]. We discretize the plate with regular grid and use triangular elements in the domain to allow a flexible representation of beadings. The discretization is sufficient to re- solve wave lengths in the plate structure, but limits the detail that can be represented with the beading patterns. After discretizing the plate, the PDE is integrated over the elements and a linear system of equations is derived. This linear system describes the dynamics of the discretized structure and solved with a direct solver. We perform the computations with a specialized FEM software for acoustics [24]. Further details the setup and mechanical model are given Appendix A.]
Numerical integration,
Fland digdacament.
system assembiing.
solving of lincar system
Figure 2: Process of the finite element solution in
frequency domain in order to compute the velocity field at each frequency query.
Dataset variations. The plate instances are varied in two settings: For the V-5000 setting, we generate random beading patterns consisting of 1 lines and 0 ellipses. Also, the width of the srading-element is randomly varied. The size of the plates as well as material, boundary and loading parameters are fixed. For the G-5000 setting, we apply the same beading pattern variation and additionally vary the plate geometry (length, width and thickness) as well as the damping loss factor, rotational stiffness at the boundary and forcing position. For each setting, 5000 instances for training and validation are generated. 1000 further instances are generated test set and are not used during training or to select a model. Further details are given Appendix A.2
Dataset analysis. The at the plate's edge (see Figure (p). With a greater proportion of beaded area in given plate, the number of peaks tends decrease (see Figure 3)). This is due additional beadings stiffening the plates, and it represents an interesting trait specific to our problem. The density of peaks is related the frequency. As the frequency increases, so does the peak density. Starting from around 120 Hz the peak density plateaus (see Figure Bd). The average number of peaks in the G-5000 setting is smaller than in the V-5000 setting. This is influenced by the on average smaller plates being stiffer and therefore having less peaks in the frequency range (see Figure 3t).
plate design shows close to uniform distribution, with margin
2.2 Evaluation
Before computing our metrics, we perform the following preprocessing steps to address numerical issues as well as facilitate an easier interpretation of the evaluation metrics. We normalize the fre-

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quency response and the velocity fields. To do this, we first take the log of the velocity fields, to align with the dB-scale of the frequency response. Then, we subtract the mean per frequency over all samples (depicted in Figure p for frequency response) and then divide by the overall standard devi- ation across all frequencies and samples. Small changes in the beading pattern can cause frequency shifts, potentially pushing peaks out of the considered frequency band. To reduce the effect of such edge cases, we predict frequency responses between and 300 Hz but evaluate on the frequency band between I and 250 Hz.
We propose three complementary metrics measure the quality of the frequency response predic- tions.
Mean squared error. The mean squared error (MSE) is a well-known regression error measure: For the global deviation we compare the predicted F(f) and numerically computed frequency re- sponse F(f) by the MSE error EMSE = (F(.)) F(
Earth mover distance. The earth mover distance [25] 26] expresses the work needed to transmute distribution P into another distribution Q. As a first step, the optimal flow + is identified. Based on fy the earth mover distance is expressed as follows:
EDDD FEMD((.Q) (P,Q) Eis 9rg -dy
with ? min
Yij dij
Ei, 9tz
8.J
where di is the distance between bins i and j in ls and Q. Correspondingly, 7ij is the flow between bins and We calculate the εEMD based on the original amplitudes in m/ss that have not been transformed to the log-scale (dB) and normalize these amplitudes with the sum over all frequencies. As consequence and unlike the MSE, εEMD is invariant to the mean amplitude and only considers the shape of the frequency response. In this form, our metric is equivalent to the W, Wasserstein metric 22 (8))
Peak frequency error. To specifically address the prediction of resonance peaks, which are par- ticularly relevant for noise emission, we introduce third metric called peak frequency error. The metric answers two questions: (1) Does the predicted frequency response contain the same number of resonance peaks as the true response? (2) How far are corresponding ground truth and prediction peaks shifted against each other? To this end, we set up an algorithm that starts by detecting a set of peaks K in the ground truth and a set of peaks K in the prediction using the find_peaks function in scipy (29] (examples in Appendix B). Then, we match these peaks pairwise using the Hungarian algorithm (30 based on the distance between the frequencies of the peaks &p. This allows us to
determine the ratio between predicted and actual peaks
[K|
and
To equally penalize predicting [
|K|
K
too many and too few peaks we consider the minimum of both ratios: ÊPEAKS = 1 min
K
K K
3 Predicting Vibrations with Neural Networks
We propose method to predict the frequency response, Fg. for plates characterized by their geometry g (influenced by beading patterns) and scalar parameters m (height, width, thickness, damping loss factor, rotational stiffness at boundary, loading position). This process involves two steps: (1) First, the input and m are encoded by an encoder 4. Because g is defined on a regular grid, standard image processing architectures are suitable. (2) Frequency response predictions are generated for specific excitation frequencies f by a decoder V (Figure 4). The computation can then be expressed as:
( m), f) = F
This problem formulation, training a neural network predict a function and evaluating this fune- tion, given some input values, is a common paradigm in operator learning [II] It allows for the evaluation of any frequency query f, even if it has not been part of the training data. In contrast, predicting frequencies on a fixed grid only allows for the evaluation of those frequencies. This for- mulation shares similarities with implicit models, for instance by 13 in the context of 3d shape prediction. Based on this, we investigate the following central aspects of our architecture:
5

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Mesh Reometry
Feature volume
Geometry encoder
Query
decoder
Velocity field atf
Frequercy
Scalar properties
Query
Figure 4: Frequency- Queery Operator method. The geometry encoder takes the mesh geometry and the scalar properties as input. The resulting feature volume along with a frequency query is passed to the query decoder, that either predicts a velocity field or directly a frequency response. The velocity field is aggregated to arrive the frequency response at the query frequency f.
Q1 Frequency-quee approach: Vibrations are dominated by resonance peaks at specific fre- quencies. The resonance frequencies vary strongly across instances. An implicit operator learning approach has been shown be able to deal with high variation better in other contexts. In the con- text of vibration prediction, frequency-quee query approach could be employed to generate predictions for one specific frequency.
Q2 - VIT encoder: Image processing architectures based on convolutions encode local features. In contrast, vision transformers have a global receptive field size from early layers. As vibrations are determined by the full geometry, we expect vision transformers to perform better.
Q3 Velocity field prediction: We can train networks either directly predict the aggregate frequency response F or to predict the velocity field V and compute F from V via Equation For predicting the velocity field, much richer training data available, since it describes a field over the plate instead of the scalar frequency response. Most of this information is not represented in the frequency response.
In the following, describe architectural variations explored for these aspects.
3.1 Geometry Encoder cb
To parse the plate geometry into feature vector, employ three variants: ResNet18 31 RN18), vision transformer 32 33 ViT) and the encoder part of a UNet (34). For the RN18, we replace hatch normalization with layer normalization [35], as we found this to work substantially better. Compared to the CNN-based RN18, the ViT architecture supports interactions across different image regions in early layers. For both, the RNI8 and the ViT encoder, we obtain feature vector by average pooling the last feature map. Since the UNet generates velocity fields, no pooling is applied.
FiLM conditioning. For including the scalar parameters m, we introduce film layer (36). The film layer first encodes the scalar parameters with linear layer. The resulting encoding is then multiplied element-wise with the feature of the encoder and a bias added. This operation is applied before the last layer of the geometry encoder (UNet) or after it (RN18, ViT).
3.2 Decoder V
FQO-RN18 and FiT-VIT: Predicting F(f) directly. Having obtained an encoding of the plate geometry and properties x, a decoder now takes this well as frequency query input and maps them towards prediction. For the RN18 and ViT geometry encoders, the decoder is implemented by an MLP taking both x and scalar frequency value f as input to predict the response for that specific query frequency, i.e. (x E R. The frequency query is merged to x by a film layer (3). By querying the decoder with all frequencies individually, we obtain results for the frequency band between 1 and 300 Hz. The MLP has six hidden layers with 512 dimensions each and ReLU activations.
FQO-UNet: Predicting F(f) through the velocity field V(f). To incorporate physics-based contraints and take advantage of the larger amount of available data, we employ UNet to pre- dict the velocity fields, V(f). From V(f), we derive the frequency response F(f) (analogous to Equation []. frequency query, introduced via FiLM layer after the encoder, enables frequency- specific predictions. To reduce the memory and computation demands per geometry during training, we select random subset of & frequency queries per geometry batch, with k < 300. If not otherwise specified, & is set to 50.

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Table 1: Test results for frequency response prediction. Column VF indicates if F is indirectly predicted through the velocity field (Q3), column FQ indicates if frequency queries (Q1) are used. Q1 to Q3 refer to the model components described in Section
V-5000
G-5000
FQ
VF
EMSE
ΣLMD
ÊPEAKS
E.
ÊMSE
ELMD
ÊPEAKS
Ep
Baselines
K-NN RNI8 + FNO
0.63
21.50
0.45
8.7
0.88
32.48
0.68
21.0
0.42
10.76
0.34
0.28
14.12
0.21
6.1
DeepONet
0.49
16.91
0.48
0.44
23.05
0.57
9.9
FNO (velocity field)
0.47
13.10
0.36
6.3
0.49
21.16
0.39
10.7
Grid-RN18 FQO-RNI8 (Q1)
0.44
13.29
0.36
5.4
0.30
14.95
0.26
6,5
0.32
10.70
0.17
5.3
0.24
13.51
0.13
5.1
FQO-ViT (Q2)
0.68
20.96
0.54
7.1
0.52
24.34
0.49
11.5
Grid-UNet
0.19
7.57
0.24
2.7
0.17
9.41
0.14
4.6
FQO-UNet
0.08
4.24
0.07
1.7
0.11
7.47
0.08
3.1
Grid-Unet and Grid-RN18: Predicting F for fixed grid of frequencies. To ablate the frequency-query approach, we employ two variations the FQO-RNI8 and FQO-UNet architec- tures, that do not employ frequency queries. They instead generate predictions for 1-300 Hz once. This is done by setting the output size of the respective last layer to 300.
3.3 Baseline Methods
We further report baseline results the following alternative methods: A k-Nearest Neighbors regressor, that finds the nearest neighbors in the latent space an autoencoder. DeepONet (ЛI. with RNI8 backbone and MLP to encode the query frequencies based on Fourier Neural Operators [15]. One employing an FNO as a replacement for the query- based decoder based RN18 features. The second directly takes the input geometry and is trained to map it to the velocity fields.
branch net. Two architectures
3.4 Training
All methods are trained in data-driven fashion for 500 epochs the training dataset of 5000 samples. 500 samples from the training dataset are excluded and employed for validation. We report evaluation results the previously unseen test set consisting of 1000 additional samples.
For methods that predict V i.e. UNet based methods and the FNO variation, the training loss is set to Ly where LV represents the MSE on the log-transformed, normalized squared velocity field (Ablation on loss function in Appendix D) For methods that directly predict F, the loss is set Lf, the MSE on the normalized frequency response. Choosing the log- eig-transfomee quantities enables the loss to be sensitive to errors outside of resonance frequencies. Otherwise, such errors would have little influence on the total loss, as their magnitude is much lower. See Appendix C for further details the architectures and training procedure.
Experiments
We train the architecture variations and baseline methods on the Vibrating Plates dataset (see Ta- ble []. To assess which architecture aspects described Section 3 are beneficial, we perform the following comparisons. Regarding Q1 (frequency variations consistently yield better predictions than equivalent grid-based methods, where responses for all frequencies are predicted at once: The ÊMSE and the ΣEDD are lower, more peaks are repro- duced, and the peak positions are more precise. Regarding Q3 (velocity field prediction), predicting the velocity ficlds and then transforming them to the frequency response leads to better results than directly predicting the frequency response. Specifically, the UNet based architectures strongly out- perform all alternatives, which we attribute the richer training data of velocity fields. Regarding Q2, the VIT encoder leads to worse results than the CNN-based encoders.
approach). the Frequency- Quurrryyy--rryyyy Operator
All evaluated baseline methods achieve comparatively worse results than our proposed methods. Despite using the same RN18 geometry encoder FQO-RN18, DeepONet [1] performs worse. We assume that this due to incorporating frequency information through a single weighted summation, which limits the model's expressivity [37]. In contrast, TQO-RNIS introduces the queried frequency

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(a) Geometry
(b) Frequency Response Prediction
(e) Reducing Size
0.7
rgo-aN1a
FNo-UNet
Ampthude
30
Reference
-10
10
roa-Unet
Frequency'
0.1
102
200
2250
4500
Requency
sample
(c) Ground Truth Velocity Field
(d) Predicted Velocity Field
(f) Using less Freqs.
Plate
1934 FFM more
0.4
1.5 Via PEN evul
27 mm
0.3
ITE modi
0.1
350 302
Frequerncias per pl geomstry
Figure 5: Results. (b) to (d) show the velocity field at frequency and prediction for the plate geometry in (a) from FQO-UNet. (e) shows the test MSE for training two methods with reduced numbers of samples from V-5000. (f) shows effects of different data generation strategies. The blue line an isoconture for a fixed compute budget of 150,000 data points, with varying number of frequencies per plate geometry. The green star represents using larger dataset at 15 frequencies per plate (half of V-5000). The red cross represents a model trained on V-5000. Training with fewer frequencies per plate is more efficient.
earlier into the model. Two Fourier Neural Operator [5] baseline methods evaluated: the first, RNI8 + FNO, which substitutes the query-based decoder with an FNO decoder, underperforms compared FQO-RNI8 on both datasets. The second FNO baseline, trained directly to predict velocity fields, yields poorer results.
Results for the G-5000 setting are slightly worse than for the V-5000 setting. The difference is surprising small considering the seven additional varied parameters in the G-5000 setting. One reason might be the average number of peaks in the frequency response: the plates in G-5000 are on average smaller and because of this stiffer, leading to fewer peaks (on average 3.9 in G-5000 and 5.9 in V-5000). This interpretation supported by the fact that the average eπтоr becomes higher with increasing frequency and thus increasing peak density (Figure t).
Looking at prediction example (Figure d) for our best model, FQO-UNet, the predicted velocity field has subtle differences to the ground truth. The prediction captures the two modes and their shape quite well, but the shape is slightly less regular than in the reference. Despite that, the resulting frequency response prediction at f 131 is close to the FEM reference. In comparison to the grid-based prediction, where peaks tend be blurry, the frequency response peaks generated by FQO-UNet are more pronounced. Additional visualizations are provided in Appendix [E.3] and in the code repository. For the best architecture in experiments, FQO-UNet, report mean and standard deviation results for multiple runs in Appendix E.2 and provide an ablation of model size for the FQO-UNet and Grid-UNe architectures in Appendix
Transfer learning. To quantify to which degree features learned on subset of the design space transfer to different subset, the V-5000 setting is split into two equally-sized parts based on the number of mesh elements that are part of a beading. The "more beadings" set contains only 5. peaks on average because the plates are stiffened by the beadings, compared to 6.7 peaks on average for the "less beadings" set. The training on plates with less beadings leads to a smaller drop in prediction quality (see Table 2). This indicates that training responses might be more efficient. In addition, we train a single model on both G-5000 and V-5000. Performance increases, indicating that training benefit from training with data based on similar mechanical models (3)
data with more complex frequency
Sample efficiency. We train the FQO-UNet and the FQO-RN18 with reduced numbers of samples (see St). notable, that the FQO-UNet with a quarter of the training data achieves nearly the same prediction quality as the FQO-RN18 with full training data. This highlights the benefit of including the velocity fields into the training process. Quantitative results are given Appendix E. for both dataset settings.

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Table 2: Transfer learning performance: We split V-5000 into two halves based on amount of bead- ings and evaluate transfer learning performance across these splits: training subset H test subset. The gray rows denote test results on the original subset that has been used for training.
less beadings + more beadings more beadings + less beadings
EMSE
ΣEND
EPeaKS
EMSE
ÊAND
ÊPeaKs
FQO-RNI8
0.61
16.19
0.20
9.3
0.82
15.79
0.36
8.3
(origin)
0.33
10.48
0.18
5.0
0.42
12.00
0.29
5.8
FQO-UNet
0.39
11.17
0.21
5.6
0.54
12.02
0.25
5.5
(origin)
0.18
8.68
0.19
2.6
0.17
7.83
0.13
3.0
Table 3: A FQO-UNet is trained in parallel on batches from V-5000 and G-5000 and evaluated on the G-5000 test set. Performance increases in all metrics.
EMSE
εEMD
EPEAKS
G-5000 G-5000 + V-5000
0.111
7.47
0.079
3.1
0.093
6.97
0.071
We further investigate the optimal ratio of numbers of frequencies per geometry and total num- ber of geometries, by generating an additional dataset in the V-5000 setting consisting of 50,000 plate geometries but with only 15 frequency evaluations per geometry. These frequencies uni- formly spaced with a random starting frequency. Reducing the frequencies per geometry drasti- cally increases the data efficiency of method. With tenth of data points compared original dataset, the MSE metric approaches the original value (Figure 5 quantitative results in Appendix E.
Design optimization. We investigate the potential of our FQO-UNet to be used for optimizing a beading pattern for reduced vibrations in a specified frequency range. Following the approach described in (38). generate plates with reduced vibrations, a diffusion model trained generate novel beading patterns is combined with gradient information from our FQO-UNet as follows: A gradient the pixels of the input beading pattern is obtained by passing a beading pattern through loss and then performing
the network, computing the sum of the predicted frequency response backpropagation the input beading pattern. This gradient is then used to guide the diffusion model to generate beading patterns with reduced vibrations. We optimize beading patterns to reduce vibrations between 100 and 200 Hz using the FQO-UNet trained on the V-5000 dataset (Figure ( Resulting plates have a lower mean frequency response in the targeted range than any plate in the training dataset.
5 Related Work
Acoustics. While research surrogate models for the spatio-temporal evolution of vector fields directly predicting frequency responses through neural networks is an understudied problem. general CNN architecture is applied in [22] calibrate the parameters of an analytical model for a composite column on a shake table. The data includes spectrograms representing the structural response in time-frequency domain. The frequency-domain response of acoustic metamaterials is considered in material design task by conditional generative adversarial The frequency response of multi-mass oscillator is
is fairly common 39 40 41
networks reinforcement learning 43 44
45
75
75
Amrlittdde
50
Amplltude
50
25
25
Dest generated bast: in training data
0
200
300
200
24.36
26.83
Frequency
Frequency
Figure 6: Design optimization. Exemplary generation result with lowest mean response between 100 Hz and 200 Hz out of 32 generations (left, mean response below). Plate with lowest response out of all 5000 training examples from V-5000 (middle left). Comparison of responses from left plates (middle right). Responses from 16 generated plates (right).

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predicted with transformer- based methods [46]. Within the context of aeroacousties, the propagation of a wwo-dimensional acoustic wave while considering sound- scattering obstacles predicted in time-domain by CNN 47 48 A review of machine learning in acoustics is given by [49]. Several acoustic benchmarks for numerical methods are available [50), however, these benchmarks do not systematically vary input geometries, making them not directly applicable to data-driven models.
Scientific machine learning. Data-driven machine learning techniques were successfully applied in many different disciplines within engineering and applied science; for example for alloy discovery
[51] crystal structure prediction [52), climate modeling [53) and protein folding [54]. A popular use case for dataenriven methods is to accelerate fluid dynamics, governed by the Navier-Stokes equations [39 40 [55] 56 [57].
The question of how to structure and train neural networks for predicting the solution of partial differential equations (PDE) has been the topic of intense research. Many methods investigate the inclusion of physics informed loss terms [58] 59 60 156 61]. Some methods directly solve PDEs
with neural networks surrogate model [62 63 Graph neural networks often employed, e.g.
for interaction of rigid and deformable objects as well as fluids [64] 65)
Operator learning and implicit models. A promising avenue of research for incorporating in-
ductive biases for physical models has been operator learning M 15 66 37 4i. Operator learning structures neural networks such that they implement a function that can be evaluated at real values instead of a fixed discrete grid. DeepONet [I1] implements operator learning by taking the value at which it is evaluated as an input and processes this value in a separate branch. Fourier Neural Operators 15 use a point-wise mapping to latent space which is processed through sequence of individual layers in Fourier space before being projected to the output space.
Implicit models (or coordinate- based representation) are models where location is utilized as an input to obtain a location-speciff prediction, instead of predicting the entire grid at once and thus fit in the operator learning paradigm. Such models were used to represent shapes [12 67 68 [3], later their representations were improved [69 770 and adapted for representing neural radiance fields (NeRFs) 77 [4]. Our method applies techniques from these implicit models to operator learning.
6 Conclusion
We introduced the problem of predicting structural vibrations and associated frequency response functions of mechanical systems. Unlike other benchmarks for deep learning surrogate models, this task necessitates predicting a steady-state solution that remains constant over time, but varies across different excitation frequencies. To this end, we created the Vibrating Plates dataset and bench- mark and provide reference scores for several methods. Our Frequency- -uuueyycc-Quyyyyooooee Operator method addresses the benchmark and achieves better results than the DeepONet and FNO baselines. We find that query-based approaches and the indirect prediction of a mean frequency response through predicted field quantities lead to better results. Surrogate models as shown in this work can greatly accelerate the prediction of physical quantities over the finite element method: Our models achieved a speed-up of around 4 to 6 orders of magnitude (see Appendix C.. which makes tasks such as design optimization feasible. This efficiency, however, depends on the availability of enough pre-generated training data and requires model training. We further investigated effects of changing the composi- tion of the training dataset and found that using less frequencies per plate and more different plates positively impacts prediction accuracy.
Limitations and future work. Our dataset and method serve as an initial step in the development of surrogate models for vibration prediction. The dataset focuses on plates, common geometric primitive used a great number of applications. However, many structures beyond plates exist, involving curved shells, multi-component geometries and complex material parameters. While some results from our study might transfer to these cases, more flexible architectures, able to deal with 3D data, would be needed. Different mechanical models, might also produce more complex frequency responses with c.g. more closely spaced modes, making the prediction task more challenging. As more complex geometries incur higher computational costs of FEM simulations, key questions how to enhance sample-effiiincc further, for example through transfer learning. A further limitation is the manufacturability of the considered beading patterns. The plate beadings could in principle be manufactured by deep drawing of sheet metal, but would require specifically designed stamps.
Acknowledgements. This research is funded by the Deutsche Forschungseemeinsccatt (DFG, German Research Foundation), project number 501927736, within the DFG Priority Programme
10

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2353: Daring More Intelligence Design Assistants in Mechanics and Dynamics'. The authors gratefully acknowledge the computing time made available them on the bigh-performance com- puters HIRN-IV at GWDG at the NHR Centers NHR @ Göttingen. These centers are jointly sup- ported by the German Federal Ministry Education and Research and the German state govern- ments participating in the NHR (ww..att-eeren..d//nneeee-rrrrrr
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A Dataset Construction
A.1 The Mechanical Model
In the following, we give a technical description of the mechanical model that is applied to generate the datasets. For moderately thin plates, the plate theory by Mindlin is valid differential equation
18] :
Bp [Ba++ poh h u2 + w² (50- + B -
( +
Vuu, w w =
This equation is combined with disc formulation for in-plane loads in order receive shell for- mulation for the mechanical description of arbitrarily formed moderately thin structures considering in-plane and transverse loads. The plate part is the dominating and important part for resolving bending waves. In the equation, u, denotes the normal displacement of the plate structure as degree of freedom of interest. B represents the bending stiffness, Ps the density, h the thickness, G the shear modulus and the moment of inertia. The angular frequency w is defined as w 2x The right hand-side excitation pt describes applied pressure load, which is converted to point forces through integration. As boundary conditions we apply homogencous dirichlet boundary conditions, the boundary and include a rotational stiffness the boundary model different boundary conditions, ranging from free rotating to clamped plates. The equation is transformed into weak integral formulation by weighted residuals, discretized using finite elements and integrated numerically. In particular, we use triangular shell elements with 3 nodes and linear ansatz functions. The integration delivers the sparse linear system of equations. This linear system is solved using the direct solver MUMPS [72] with a specialized FEM implementation for acoustics 24] The dis- cretization is chosen, such that the bending waves are resolved with minimum of 10 nodes. The bending wave length Xy of a plate can be calculated by
iee (x)
2T
E1
AB
12 v²)p'
where is the Young's 's modulus, t the thickness, v the Poisson ratio and the density of the plate. The final discretization is set 181 121 for G-5000 and 121 x 81 for V-5000, which is sufficient for convergence.
A.2 Datasets
The exact physical setting and variation of our mechanical model form the V-5000 and G-5000 datasets are given in Table 4 Table 6 and Table | Both dataset settings contain 6000 samples, each consisting of plate geometry with associated physical and material parameters and the computed velocity fields V(J) and frequency response F(f) for frequencies f to 300 Hz. 1000 samples from the 6000 samples are selected network training or validation. Computing a single sample out of the 6000 samples takes minutes and 19 seconds machine with a Ghz CPU (20 physical cores).
test dataset and not considered during neural
Table 4: Dataset settings. Width is the width of lines and ellipses in mm. Properties. (prop.) involves plate size, thickness, material, boundary and loading properties.
Sample space Ellipses
Sample number Train
Setting
Prop.
Lines
Width
Test
V-5000
fix
1-3
0-2
30
70
5000
1000
G-5000
vary
1-3
0-2
40
60
5000
1000
For the G-5000 dataset, the geometry, material, boundary condition and loading parameters are varied. The effect of two of the material parameters is visualized in Figure 7 Increasing the damping reduces amplitudes at resonance peaks but does not shift the overall form of the frequency response. Increasing the thickness increases the overall stiffness and shifts resonance peaks towards higher frequencies less regular manner. For boundary condition variation, we include one rotational stiffness parameter, which models the rotational stiffness along the x-axis at the lower and upper edge and along the y-axis at the left and right edge. The rotational stiffness added at the respective rotational degree of freedom at the boundaries and varied given in Table 6
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Table 5: Geometry and material parameters for V-5000 and G-5000 datasets.
Geometry
Material (Aluminum)
longth
width
thickness
deusity
Young's 's mod.
Poisson ratio
loas factor
V-5000
0.9 m
0.6 m
0.003
2700 kg/m"
7e 10 N/m
0.3
0.02
G. 8-5000
0.6 0.9 m
0.4 m
0.002 0.004 m
2700 kg/mn
7e10 N/m?
0.01 0.03
Table 6: Loading and boundary condition parameters for V-5000 and G-5000 datasets.
Loading (Point force)
Boundary condition (rot. stiffness)
x-position
y-position
GrylCra
V-5000
0.36 m 0.18 0.72
0,225 m
0.0 Nm
G-5000
0.12 0.48 m
0.0 100 Nm
0.002
0.003
60
0.004
amplitude
amplitude
50
30
30
20
0.02
853
0.03
50
100
150
200
250
300
100
150
200
250
300
froquency
frequency
Figure 7: One-at-a-time parameter variation of the thickness parameter and the damping loss factor. Increasing the damping reduces the amplitudes at the resonance peaks. Increasing the plate thickness increases the stiffness of the plate and thus shifts the resonance peaks towards higher frequencies
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B Metrics - Peak Frequency Error
We provide examples of the find peak operation which serves as the basis for the peak frequency error on ground truth (Fig- ( and predictions (Fig. 9 using kNN baseline) and visualize the matched peaks for calculating the peak frequency error (Fig. T0P. Note that find- peaks is run with prominence threshold set to 0.5 meaning that the peak must at least units higher than their surroundings.
UMA
Figure 8: Find peak results on random ground truth samples.
Figure Find peak results on predictions.
Figure 10: Visualization of the matching between ground truth (blue) and prediction (orange) peaks. Matched peaks are indicated in red.
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C Architectures
In the following, our neural network architectures and the training procedure described. The training and neural networks can be reproduced based on the publicly available code repository. To give overall impression of the employed models, Table 7 gives overview of the number of parameters and the speed for a forward pass prediction.
Table 7: Model size and speed comparison for forward pass of batch of 16 plate geometries on an A100 GPU. In comparison, solving one geometry via FEM takes 2 minutes and 19 seconds. The slowest deep learning method then around 6000 times faster.
# weights in Mio
Time (s)
RN18 + FNO
11.8
0.014
DeepONet
11.3
0.005
FNO (velocity field)
134
0.008
Grid-RNI8
12.9
0.005
FQO-RNI8
12.7
0.005
FQO-ViT
9.28
0.013
Grid-UNet
27.9
0.010
FQO-UNet
7.1
0.338
FEM (20 CPU cores)
2224.000
C.1 Frequency-Query-basse Methods
Predicting F(f) directly: FQO-RNI8 and FQO-ViT. To directly predict the frequency response instead of predicting the velocity fields and then transforming it to the frequency response, use ResNet [31] and a vision transformer (ViT) 332 as geometry encoders. For the ResNet, we opt for the ResNet18 backbone. We replace batch normalization with layer normalization [5), this to work substantially better. In addition, we employ the vision transformer (ViT) architecture [32]. The ViT supports interactions across different image regions early layers. We use variation of the ViT-Base configuration with a reduced token size of 192, an intermediate size of 768 and three attention heads. For both the RNI8 and the VIT encoder, through average pooling from the last feature map or the encoded tokens. Scalar parameters are introduced to the encoding by a film layer. As decoder, we employ an MLP. The MLP takes both and scalar frequency value f as input to predict the response for that specific query frequency, i.e r(x, f) E TR. The frequency query introduced by a film layer to x. By querying the decoder with all frequencies individually, we obtain results for the frequency band between and 300 Hz. The MLP has six hidden layers with 512 dimensions each and ReLU activation functions.
found
obtain the d-dimensional global feature
Predicting F(f) through the velocity field V(f): FQO-UNet. To predict V(f) instead of di- rectly predicting F(f), we employ a UNet. The plate geometry is encoded by the UNet encoder and the scalar material parameters are introduced before the last contraction block of the UNet. A frequency query is introduced after the encoder again by film layer and the decoder then produces predictions of size 40 x 60, which is sufficient to capture the structure and modes of the velocity fields. Since the decoder has to be evaluated for each frequency query individually, we opt to map to predictions for five velocity fields per query.
The UNet consists of three contraction blocks, two spatial-shapor shaap-preserringg blocks and two expan- sion blocks. Additionally, two self-attention layers are included in the encoder and one self attention layer in the decoder. To ensure global features are included in the full feature volume after the en- coder, adaptive average pooling is applied to the feature volume and the result is concatenated to the feature volume.
C.2 Grid-based Methods
To provide a direct comparison to the query-based approach, two methods that predict all frequency responses at once are tested.
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Grid-RN18. The same RNI8 is used to generate a global feature as in the FQO-RN18. Given x, an MLP predicts the frequency response on a fixed Hz interval grid, with r(x) 12000 We employ six hidden layers with 512 dimensions each and ReI.U activations.
Grid-based U-Net. For the grid-based based U-Net we also employ the same architecture as for the query-variation but double the number of channels to account for the larger number of predictions that the network has to produce at once. The U-Net is trained predict all 300 velocity fields at once.
C.3 Baseline Methods
RN18 + Fourier Neural Operator (FNO). A ld FNO as constructed by [15] takes as input x processed by linear layer to size 300, the number of frequencies to be predicted. We keep 32 modes and use eight FNO blocks with 128 hidden channels.
DeepONet. We further test DeepONet with the RNI8 as branch network and as trunk network, a four layer MLP of width 128 and 512 as output width to match the size of x. The trunk network processes the frequency queries and is then combined with x to produce the prediction Note, the RN18 branch network is the same as the encoder of the FQO-RNI8.
66
FNO (velocity fields). The FNO takes as input the geometry interpolated to the resolution 40 x 60. The 2d FNO then consists of eight FNO blocks with 128 hidden channels and finally 300 output channels to map the 300 velocity fields in this resolution. In the FNO blocks, 32 modes are preserved after the Fourier transform. Scalar parameters are introduced by film layer after the first FNO block.
C.4 k-Nearest Neighbors (k-NN)
We further test k-Nearest Neighbors algorithm as baseline, which predicts the frequency response of a plate as the mean frequency response of the & closest plates in the training set. To determine the distance between different plate designs, we use the cosine distance on the 96-dimensional latent space of a convolutional autoencoder 773 trained on the beading pattern geometries. The normal- ized scalar properties are appended to the latent space to include them. To obtain a prediction, the frequency responses of the k neighbors are averaged and the optimal k in the range [1, 25] is empir- ically determined to minimize the MSE. Determining the nearest neighbors directly the geometry space was tried out, but yielded worse results.
C.5 Training
The networks are trained using the Adam W optimizer 74] with 3 [0.9, 0.99) and weight decay set to 0.00005. We further choose cosine learning rate schedule with warm-up period [7] of 50 epochs. The maximum learning rate is set to 0.001, except for the UNet and ViT architectures, for which it is set to 0.0005. In total, the networks are trained for 500 epochs. As a validation set, 500 samples from the training dataset are set and excluded from the training and the checkpoint with the lowest MSE on these samples selected. We report evaluation results on the previously unseen test set.
Compute resources. All trainings reported in this work were computed on a cluster with single A100 GPUs The most resource intense training run took roughly ld and 16h on single A100 GPU and was the ablation of the number of channels with the highest scaling factor for the FQO-UNet method detailed in Section D All other training runs with the FQO-UNet were completed in less than 24h. The trainings for the other methods were substantially shorter with i.c. the Grid-UNet finishing training in roughly 2h- 3h and likewise the FNO (velocity field) method. The roughly 20 to 30 training runs for the FQO-UNet dominate the required compute resources. We estimate that it took in total A100 for 30 days. In addition, preliminary and failed experiments required further computational resources.
D Ablations
We provide ablation results for the loss function for training methods that predict the velocity field. We consider the loss function Titota! = aTv + (1 - T.P and provide results in Table 8 This ablation was performed with training batches consisting of 300 frequencies per geometry, instead of
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subset. We find that the loss on the velocity field prediction Ly is more important than the loss on the frequency response.
Table 8: Ablation of value for weighing loss the predicted velocity field vs. predicted frequency response. Higher a value indicates more weight on velocity field. indicates only velocity field loss. The selected parameter is printed in bold.
V-5000
ÊMSE
GRND
EPEAKS
E
0.25
8.02
0.15
0.5
0.15
5.52
0.11
2.9
0.9
0.09
3.90
0.08
1.8
0.09
4.00
0.07
1.8
We provide ablation results on the number of channels in the FQO-UNet and Grid-UNet architec- tures in Table For the FQO-UNet, this ablation was performed with training batches consisting of 300 frequencies per geometry, instead of a subset. We find that increasing the number of chan- nels did not lead to perfomance improvement for the improvements for the FQO-UNet architecture.
and leads marginal further
Table 9: Ablation number of channels of FQO-UNet and Grid-UNet. The number of channels is multiplied by constant factor over the depth, named width. The selected width parameter printed in bold.
V-5000
width
EMSE
εEMD
EPEAKS
Eg.
FQO-UNet
0.12
5.00
0.09
2.2
32
0.09
3.90
0.08
1.8
64
0.08
3.82
0.07
1.8
Grid-UNet
16
0.31
11.44
0.31
3.9
32
0.24
8.56
0.25
3.3
0.19
7.57
0.24
2.7
128
0.24
8.17
0.21
3.7
21

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E
Additional Results
E.1
Sample Efficiency
To provide full baseline results for the training with a reduced amount of samples, we refer to Table
10
Table 10: Test results for different training dataset sizes for both settings, V-5000 and G-5000.
V-5000
G-5000
EMSE
ΣFMD
EPEAKS
EF
EMSE
FFRD
EPEAKS
EF
FQO-UNet
10 %
0.55
15.26
0.37
6.4
0.54
22.70
0.37
11.1
25 %
0.33
12.00
0.24
4.2
0.33
16.67
0.17
7.3
50 %
0.19
8.54
0.17
2.9
0.19
11.02
0.12
4.7
75
0.13
6.95
0.13
0.14
9.14
0.10
Full dataset
0.08
4.24
0.07
1.7
0.11
7.47
0.08
3.1
FQO-RN18
10 %
0.73
19.44
0.49
7.9
0.65
27.57
0.55
13.9
25 %
0.58
14.27
0.31
7.2
0.45
21.04
0.44
8.9
50 %
0.45
12.20
0.20
6.4
0.35
17.00
0.15
7.2
75
0.37
10.96
0.18
0.29
15.06
0.15
Full dataset
0.32
10.70
0.17
5.3
0.24
13.51
0.13
5.1
Full results on training with different ratios of frequencies and geometries are provided in Table 11
Table 11: Data generation experiment.
V-5000
# Freqs
geometries
ÊMSE
EEMD
ÊPEAKS
E3
300
500
0.48
13.16
0.32
5.7
150
1,000
0.31
11.18
0.22
4.3
30
5,000
0.12
8.74
0.14
2.1
10
15,000
0.10
11.18
0.17
1.6
3
50,000
0.10
11.47
1.4
15
50,000
0.02
4.06
0.04
0.08
original 300
5,000
0.08
4.24
0.07
1.7
E.2 Multiple Trainings with Random Splits
To provide notion of the variability of the results for different training runs, we performed four trainings for the FQO-UNet with different initial seeds and different random splits training and validation set. These trainings were performed with training batches consisting of all 300 frequen- cies per geometry, instead of subset. In Table 12 we report evaluation results on the respective validation sets and on the unseen test set and observe only modest variation.
Table 12:
models were trained on random splits in training and validation sets (4500 and 500 samples respectively). The results are denoted as mean [standard deviation].
V-5000
Evaluation set
EMSE
ΣEMD
ÊPEAKS
Validation set Test set
0.096 [0.0023]
[0.072]
0.081 [0.0071]
[0.061]
0.094 [0.0031]
4.1 [0.091]
0.08 [0.003]
1.9 [0.096]
E.3 Visualizations
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Georalry
Achall frequnnyy
Armaa flil
Ataai feed fertoquancy
Preditte voleciy froquano
Trediced veladtt Rald furfuuuancy
Prodi cood velectty fel frequency
Pradiction
Gcorddry
'rauurcc
walociy frequencc
velocky fedd rrequecc
frequency
Perquency
Cconetry
Actuol vebocty Feld for Traquency 80
Actull verecity fidl for traquency 152
Aduell velocky fedd traquency
Predictd volecity hrequano
frediceed veladay fuld frezuency
valocity tie fraquency
Sahrrance
130
200
Teequency
Ceonediy
Actuol velecty for Troguancy
Actual weleott fele trog cocr
Adual velucity reki frequency
WX
Peddited veliiiy feet tortregeeeyyy so
hedimne velvity de4 tecuuney 112
Pnodimee Pnodmss weliityyyyyeeee fils tregunryy 174
25
Nefurence Wretiction
1.0
200
Figure 1I: V-5000 example predictions. The velocity fields at the three peaks with the highest amplitude are shown. The plots are scaled with respect to the maximum velocity in the prediction and reference velocity field make differences magnitude visible.
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portequancy
hrequncc
Prediction
Geomitry
vylocit by thaquency 363
Actul ve ecty frrguuunyy
0.6
0.6
0.2
6.2
32
84
0.6
0.8
14
Prediceed Feld 166
Tredieed Paquancy
1.0
0.6
Sefermnc
100
300
Freawwy
0.0
Gcanitry
Aebllveecty held hoguany 182
1.0
3.0
t
0.5
8.0
0.2
Pradiction
0.5
100
201
100
0.2
Prodine
Andimee veisity fertoruency
Prodimet webvityy
Pratiction
Figure 12: G-5000 example predictions. The velocity fields at the three peaks with the highest amplitude shown. Empty axes indicates less than three peaks in the response. The plots scaled with respect to the maximum velocity in the prediction and reference velocity field to make the differences magnitude visible.

---

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31


Reviews:
[{"review_id": "TOL6uM1vlz", "reviewer": "Reviewer_1y6b", "summary": "The paper report development of a surrogate model for prediction of the structural vibrations.  The paper reports outperformance of the their method to physics informed architectures such as DeepOnet.", "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?\nCan the authors explain how the surrogate model can predict the structural vibration of structures that has not been seen in the training phase?\nHow the can one use the model for scaling predicting for higher scale geometries? Have the authors utilized the model for extrapolations? \nHow would the model behave in presence of domain shift of the input data?\nAfter 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?", "limitations": "I still doubt about the generalization and performance of of the model beyond the domain of the training data set. Can authors perform experiments to evaluate the performance of the model on a different system other than plates? such a rotary machine data or any other vibration data authors can identify.", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The authors tackles and interesting and important problem in engineering domain which is prediction of structural vibration. This problem traditionally was solved by means of numerical techniques which is computational costly. A surrogate model can enhance the time to solution.", "weaknesses": "It is not clear well if such surrogate model is robust to domain shift. This means what happen if we expose to model to the structural vibrations at different scale (lower or higher) comparing to what has been used in the training phase.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer,\n\nthank 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. \n\n\n**Questions:**\n\n> 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? \n\nOur numerical model can be considered a mapping from the input space to resulting vibrations, which is approximated by our network. \nOur 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.\n\n> Can the authors explain how the surrogate model can predict the structural vibration of structures that has not been seen in the training phase? \n\nConceptually, 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\napproximates the ground truth function (in our case numerical simulation).\nEmpirically, we find our model to generalize to unseen beading patterns within distribution (Table 1 general answer) and out-of-distribution (Table 2).\n\n> How the can one use the model for scaling predicting for higher scale geometries? \n> Have the authors utilized the model for extrapolations? How would the model behave in presence of domain shift of the input data? \n\nWe 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. \nFurther, 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.\nWithout fine-tuning, we believe large extrapolations will fail with the current setup.\n\n>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?\n\nTo 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UDMMoYpPoh", "reviewer": "Reviewer_ifbc", "summary": "This paper proposes a new surrogate deep learning model, the Frequency-Query Operator (FQO), designed to study structural vibrations in excited plates by mapping these plates as well as specific excitation frequencies to the resulting vibrations patterns. It introduces a new benchmark featuring 12000 plate geometries with varying geometric and material properties, and associated velocity field responses to excitations computed numerically using the finite element method. The FQO’s performance is then compared with numerous other architectures.", "questions": "-\tHow 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 ?\n-\tWould adding a physics loss based on equation be feasible and yield better accuracy ?", "limitations": "-\tNew encoder/decoders would need to be designed to deal with more complex geometries\n-\tCostly FEM simulations would need to be run on complex geometries to extend the dataset in this case", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "-\tThe tackling of a new time-independent problem in solid mechanics using deep learning \n-\tThe introduction of a new FQO architecture to infer the structure vibrations, showing better performance than other classic neural operator models and a high speedup when compared against classical FEM.\n-\tA new 12,000 element benchmark data-set of plate responses to various frequency excitations, for different plate geometries and material properties\n-\tThe paper is well written and organized", "weaknesses": "-\tOnly very simple plate geometries (rectangular shaped) are considered here.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer, \n\nthanks 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! \n\nPlease 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.\n\n\n**Questions:**\n\n>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 ?\n\nPredicting 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.\n\n>Would adding a physics loss based on the equation be feasible and yield better accuracy ?\n\nAdding 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.\n\nWe will add this discussion to the paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nx4mD06pUE", "reviewer": "Reviewer_YeHT", "summary": "This work presents a benchmark dataset of 12,000 rectangular plate geometries with different beading patterns and their corresponding vibrational responses. The authors suggest that the dataset can be used to construct surrogate models to aid in the design and optimization of plate structures for noise reduction purposes. In addition to the dataset, the authors propose evaluation metrics to measure prediction accuracy: mean squared error, earth mover's distance, and peak frequency error. Lastly, they introduce a new neural network architecture that can map plate geometry and excitation frequencies to vibration patterns.", "questions": "Can the authors please justify their comparisons to the baselines?\nWhy are these specific architectures chosen? \nHow do they differ? \nWhy do these networks have gaps in performance?", "limitations": "Although the authors outline many limitations of their approach, they do not address the challenge of extending their neural network architecture to different geometries. The current design relies on an image-like grid, which is not easily adaptable to other shapes or forms.", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The paper introduces a novel dataset and neural network architecture.", "weaknesses": "The study is limited in scope regarding the forcing terms, material properties, and boundary conditions. It focuses exclusively on rectangular plates with elliptical and linear beading patterns.\nIt is unclear how the beading patterns are integrated into the Mindlin differential equation.\nExtending the methodology to more complex systems is not straightforward, and the assumption of simply supported boundary conditions is restrictive.\nThe 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.\nThe motivation for the study is weak and lacks clear justification.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer,\n\nthank 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. \n\n\n**General Comments:** \n\n>The study is limited in scope regarding the forcing terms, material properties, and boundary conditions. \n\nPlease 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).\n\n**Possible misunderstandings and lack of clarity:**\n\n> It is unclear how the beading patterns are integrated into the Mindlin differential equation.\n\nTechnically, 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. \n\n>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.\n\nWe 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.\nFurthermore, 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. \n\n> The motivation for the study is weak and lacks clear justification.\n\nAs 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. \n\nThank you for drawing our attention to these points. This will enable us to improve the relevant sections of our paper.\n\n\n**Questions:**\n\n>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?\n\nThere 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.\nThe 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. \nFeel free to ask if you have further questions regarding our choice of baselines.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kYcy2XHowM", "reviewer": "Reviewer_75jo", "summary": "The paper suggests a method of using operator network theory in order to form predictions of the resultant steady state vibrational patterns which appear on a beaded flat plate under several levels of external excitation. They firstly use numerical structural engineering software (rooted in FEMs) to generate a diverse set of input-output relationships over these flat plates and the corresponding (i) Velocity fields, and (ii) The frequency response functions (FRF). The flat plate model chosen for the simulations is based on Mindlin plate theory. This forms the basis of a benchmark dataset. They then they use a bespoke deep operator network in order to derive the input-output relationships with respect to the functional operator framework.", "questions": "- Are there any plans to extend the nature of the dataset? Perhaps to different plate geometries (I see you have referenced this point in your limitations section -- but do you have intentions to move in this direction)? Perhaps also to include the effect of damage in the plate? A simple model could be that of punched holes in the plate represent a highly discrete form of damage, which could then also extend the range of tasks available to your benchmark to being that of binary classification.", "limitations": "- Since it would feel that the main contribution is that of the benchmark dataset generation I believe a good step forward would be to generate it even comprehensively with respect to various machine learning tasks. For example, right now the notion of FRF and velocity field prediction is performed. Perhaps also the transfer learning section can be enhanced. A small study is performed with respect subset design spaces, but in structural engineering a major point would be something akin to taking a pre-trained model on bridge A, and zero/one shot applying it to bridge B -- a different structure. This may be akin to demonstrating that you can take pre-trained models on a square plate and transfer onto an entirely different geometry such as a triangular or circular plate. \n\n - It is understood that training took place with a lot of available data. In engineering, oftentimes in real life we do not have that much data on the real world structures. It would be good to show that some pre-traind model on the suggested computational dataset, can be extended to a real world model with an absolute minimal amount of real world data (low data cardinality)\n\n - Moreover in terms of pre-training, it is understood that for example even the available computational training data for fluid flow simulations within Navier Stokes (Large Eddy Simulations) is dearth. It would be good to analyze what is the absolute minimal amount of computational training data necessary in order to produce a maximal amount usability from the proposed operator training models, especially as it pertains to model transferability. \n\nI understand these are all very difficult problems within the engineering field, but some commentary and addressing of these problems would be good.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- I think the largest strength of this paper is that of the development of the benchmark framework based upon 2D flat plate theory. It is unfortunate that many structural / mechanical / aeronautical datasets are not as publicly available, and that the problems faced in these fields are no where near as well explored when compared against those seen in CV and NLP fields, means that  the addition of an engineering benchmark 2D flat plate dataset is very welcome. \n\n - The fact that several other methodologies were used on the benchmark dataset, and that the (anonymously shared) code is neatly written is great.\n\n- Additional commentary in the appendix is welcome.", "weaknesses": "- It appears that the proposed numerical plate simulations are still somewhat simplistic, as for example there is no high frequency noise present in FRFs (which would result from experimental limitations of sensors). Moreover there is no investigation of difficult FRF phenomenon which may arise in real life such as the appearance of closely spaced modes (which are sometimes to within 0.1 Hz) of one another. Whilst such a setting may not arise according to geometry + Midlin theory, the current wide spacing of the FRF modes makes the problem somewhat removed from the real-life difficulties that would be seen as incredibly valuable.\n\n - 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. \n\n - The necessity of the FQO-UNet model having to interpolate (predict within a neighbourhood) for computational reasons when demonstrating the FRF plots at varying frequencies doesn't seem like the most adaptable solution for real life settings. It could miss closely spaced modes, and in the presence of noise perhaps introduce spurious modes.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer,\n\nthanks 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. \n\n**General Comments:**\n\n> The necessity of the FQO-UNet model having to interpolate [...] doesn't seem like the most adaptable solution for real life settings.\n\nWe are able to drop this constraint. Please see Section 3 of our general answer for details.\n\n> It appears that the proposed numerical plate simulations are still somewhat simplistic, as for example there is no high frequency noise present in FRFs\n\nWe 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.\n\n> Moreover there is no investigation of difficult FRF phenomenon which may arise in real life such as the appearance of closely spaced modes [...]\n\nConceptually, 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. \nFor modeling frequency responses with dense modes, our frequency query approach is suitable since it allows us to sample the FRF at arbitrary resolution.\n\n> 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.\n\nWe 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:\n\n* 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.\n* Balance compute between the shared embedding of the plate geometry and the frequency-specific decoder.\n* 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.\n* Use of self-attention: The global information exchange through self-attention support the prediction, since vibration patterns depend on the global plate geometry.\n\nOur 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.\n\nIf you disagree with this perspective, we gladly elaborate on this.\n\n\n**Questions:**\n\n> 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? [...]\n\nWe 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.\n\nDamage 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.\nThus, we mainly consider the design optimization task (general answer Section 2) which is directly applicable to our existing dataset. \n\n**Limitations:**\n\nWe identified two main points in the answer: transfer learning/domain shift and sample efficiency.\n\nTransfer Learning / Domain Shift:\n> 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 [...]\n\nWe 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).\nWe hope you understand that addressing transfer to entirely new geometries and real world data is not possible within the scope of this paper. \n\nSample Efficiency:\n> [...] It would be good to analyze what is the absolute minimal amount of computational training data [...]\n\nWe 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. \n\nReducing 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.\n\n**Table 4**\n|# freqs.|# geometries|MSE|EMD|E_Peaks|E_F|\n|---|---|---|---|---|---|\n|300|500|0.48|13.16|0.32|5.7|\n|150|1,000|0.31|11.18|0.22|4.3|\n|30|5,000|0.12|8.74|0.14|2.1|\n|10|15,000|0.10|11.18|0.17|1.6|\n|3|50,000|0.10|11.47|0.20|1.4|\n|15|50,000|0.02|3.61|0.04|0.08|\n|original||0.08|4.24|0.07|1.7|", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Dear reviewers,

Thank you for many valuable and thoughtful comments. We are pleased that the reviewers recognize the value of our novel benchmark (**ifbc, 1y6b, YeHT**)  and method (**YeHT, ifbc**) for the important problem (**1y6b, ifbc**) of structural vibration prediction. as well as rating the paper (**ifbc**) and code (**75jo**) well-written. 

First, we would like to comment on three key points, mentioned by multiple reviewers:


## 1. Dataset complexity: Extension to more complex geometries, boundary conditions, or forcing terms (**ifbc, YeHT, 75jo**)

More complex geometries are an exciting research avenue. We deliberately constrained ourselves to rectangular plates to focus on methodological issues, as an investigation of data scarcity, design optimization and transfer learning (also suggested by the reviewers) is already challenging in this setting. This enables us to benefit from the comparatively fast numerical simulations for rectangular plates and obtain insights which we expect to generalize to more complex and less constrained geometries.

Therefore, in the scope of this paper, we opted for retaining the current plate geometries which have the direct engineering application of metal beading but extend the G5000 dataset (G5000 new) by two new aspects: (a) variable rotational stiffness at the boundary, allowing us to predict simply supported as well as clamped boundary conditions, and (b) a variable point force position. This extension makes the prediction task more challenging:

**Table 1**
| |MSE |EMD |E_Peaks|E_F|
|----|----|----|-------|---|
|G5000 - original|0.09|4.94|0.07|2.5|
|G5000 - new |0.11|7.47|0.08|3.1|


## 2. Deeper investigation of generalization, data efficiency  and transfer learning (**1y6b, 75jo**)

Generalization and robustness to domain shift are critical for surrogate models to be applicable when the mechanical model changes. We reported results on transfer between different subsets of our dataset in Table 2 in the paper. To further investigate generalization we show new results when using our FQO-UNet for (1) design optimization in conjunction with a guided diffusion method (see Figure 1 in PDF), (2) an experiment on minimizing the amount of training data for a given prediction quality, and (3) transfer learning between V5000 and G5000:
1. For design optimization, the goal is to find beading patterns with the lowest possible mean frequency response in predefined frequency ranges. Based on the gradient information from the surrogate model, our guided diffusion method is able to generate beading patterns with a mean frequency response well below any plate in the training data (verified by numerical simulation). This highlights the potential for generalization beyond the training data distribution, as the resulting beading patterns look different from the training data, with more variety in thickness, form and number of beadings.
2. We generate a new dataset consisting of 50,000 plate geometries but with only 15 frequency evaluations per geometry. Reducing the frequencies per geometry drastically increases the data efficiency of our method. With a tenth of data points compared to our original dataset, the MSE metric approaches the original value (Figure 2 in PDF, and answer to reviewer **75jo**).
3. For the transfer learning experiment, we took a model trained on V5000 as an initialization for the G5000 setting. When initializing with the pretrained model, the performance improves in all metrics (Table 2).

**Table 2**
|           |                   |MSE  |EMD  |E_Peaks    |E_F|
|----|-----|-----|-----|-----------|---|
|G5000 | from scratch |0.086|4.94 |0.068      |2.5|
|G5000 |V5000 fine-tuned|0.061|4.01 |0.053      |1.9|
|G5000 - new | from scratch |0.111|7.47 |0.079      |3.1|
|G5000 - new | V5000 fine-tuned|0.095|4.63 |0.078      |1.9|


## 3. Limitations of FQO-UNet: Grids and Frequency (**YeHT, 75jo**)

Indeed, our architecture requires a regular grid structure with respect to the input plate, constraining the input geometries (see a discussion of this setting in general answer 1).
However, this limited setting enables us to focus on understanding which aspects are relevant for frequency response prediction. For example, some of our findings are:

* The frequency query approach leads to better predictions. They also enable evaluating the FRF at arbitrary locations (e.g. in 0.1 Hz steps or beyond the training frequency range).
* Predicting a field quantity (in our case velocity) first and then directly calculating a frequency response from it is more data efficient than directly predicting the frequency response.
* Convolutions pose a beneficial inductive bias for predicting the field quantity.

These insights can now inform the development of more flexible neural network architectures. 
As mentioned by **75jo**, we originally mapped to five velocity fields per query. This constraint is technically not necessary and was only introduced to speed up training.
This goal can also be achieved by training on a subset of frequencies per geometry per batch, without frequency bundling. The results (Table 3) are close to identical, while frequency subset training is around 2 times faster.

**Table 3**
|                              |MSE |EMD |E_Peaks|E_F|
|-----|-----|-----|-----------|---|
|V5000 - original|0.09|3.90|0.08   |1.8|
|V5000 - one prediction per query      |0.08|4.24|0.07   |1.7|

# Conclusion 
Concluding, we consider this work a solid foundation for the vibroacoustics and ML community to make progress in applying ML methods. Interdisciplinary work like ours first needs to establish common problems and a common language, which we are grateful to the reviewers to acknowledge. With the design optimization task and the exploration of data-efficiency we display two promising future research directions of our work.

Meta-review: The reviewers all recognized the value of proposed the application of surrogate models to prediction of structural vibrations, and the rich experimental setting. While they found the geometries a little simplistic, they appreciated the new proposed dataset and quality of the code released. They found the manuscript well written, and the authors addressed the reviewers questions in their rebuttal.
Decision: Accept (poster)