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Learning to Predict Structural Vibrations
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Introduces the Vibrating Plates benchmark (12,000 FEM-simulated beaded plates) and the Frequency-Query Operator, a frequency-conditioned UNet that predicts velocity fields and derives frequency response curves, outperforming DeepONet and FNO baselines.
Problem
Surrogate prediction of steady-state, frequency-domain vibration patterns and frequency response functions of harmonically excited plates, which vary strongly with excitation frequency and geometry (beadings), while FEM is expensive.
Approach
A frequency-query operator: geometry and scalar parameters are encoded (UNet, ResNet18 or ViT), a frequency query is injected via FiLM into the decoder, and the predicted velocity field is aggregated into a frequency response; training on velocity fields provides richer supervision than the scalar response alone.
Key results
- FQO-UNet achieves MSE 0.08 and peak error 0.07 on V-5000 test, vs 0.44 for Grid-RN18 and 0.49 for DeepONet (MSE).
- FQO-UNet on G-5000 reaches MSE 0.11 vs 0.28 for RN18 + FNO and 0.49 for FNO (velocity field).
- FQO-UNet with a quarter of training data nearly matches FQO-RN18 with full data, highlighting value of velocity-field supervision; surrogate runs about 6000x faster than FEM for the slowest DL model.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use frequency queries instead of predicting all frequencies on a fixed grid (ablation) | FQO-UNet versus Grid-UNet · V-5000 | MSE: 0.19 -> 0.08 | Helped |
| Predict velocity fields and derive the frequency response instead of predicting the response directly (ablation) | FQO-UNet versus FQO-RN18 · V-5000 | MSE: 0.32 -> 0.08 | Helped |
| Replace the RN18 geometry encoder with a ViT encoder (ablation) | FQO-RN18 · V-5000 | MSE: 0.32 -> 0.68 | Hurt |
| Use velocity-field loss rather than frequency-response loss (ablation) | FQO-UNet · V-5000 | MSE: 0.25 -> 0.09 | Helped |
| Increase FQO-UNet width from 32 to 64 (ablation) | FQO-UNet · V-5000 | MSE: 0.09 -> 0.08 | Helped |
| Increase Grid-UNet width from 64 to 128 (ablation) | Grid-UNet · V-5000 | MSE: 0.19 -> 0.24 | Hurt |
| Train on both G-5000 and V-5000 instead of G-5000 alone | FQO-UNet · G-5000 | MSE: 0.111 -> 0.093 | Helped |
| Increase the training set from 25% to the full dataset | FQO-UNet · V-5000 | MSE: 0.33 -> 0.08 | Helped |
| Increase the training set from 25% to the full dataset | FQO-UNet · G-5000 | MSE: 0.33 -> 0.11 | Helped |
| Use 50,000 plates with 15 frequencies per plate instead of the original dataset | FQO-UNet · V-5000 | MSE: 0.08 -> 0.02 | Helped |
| Train on less-beaded plates and transfer to more-beaded plates | FQO-UNet · V-5000, more-beadings test subset | MSE: 0.18 -> 0.39 | Hurt |
| Train on more-beaded plates and transfer to less-beaded plates | FQO-UNet · V-5000, less-beadings test subset | MSE: 0.17 -> 0.54 | Hurt |
Surprising findings
- Predicting velocity fields and deriving the frequency response outperforms directly predicting the frequency response, even though the target is a scalar curve.
- Vision transformer encoder performed worse than the CNN-based encoders on this task.
Limitations
- Limited to simply supported plates with beadings; not validated on curved shells, 3D or multi-component structures.
- Only one mechanical model (Mindlin shell FEM) with aluminum material; more closely spaced modes could make prediction harder.
- Beading patterns may be hard to manufacture.
Experimental setup
Datasets: V-5000 G-5000
Benchmarks: Vibrating Plates V-5000 Vibrating Plates G-5000
Metrics: MSE on normalized frequency response earth mover distance (Wasserstein-1) peak frequency error peak count error
Baselines: k-NN DeepONet FNO (velocity field) RN18 + FNO Grid-RN18 Grid-UNet FQO-RN18 FQO-ViT
Builds on: DeepONet Fourier Neural Operator implicit shape models (DeepSDF-style occupancy/coordinate models) FiLM
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