Inverse Synthesis of Shock Time Series from Shock Response Spectra Using Ensembles of Conditional Variational Autoencoders

Adam Watts; Charles Young; Ryan Bowering; Andrew Jeon; Destry Newton

cs.LGOct 9, 2026

Inverse Synthesis of Shock Time Series from Shock Response Spectra Using Ensembles of Conditional Variational Autoencoders

Authors: Adam Watts, Charles Young, Ryan Bowering, Andrew Jeon, Destry Newton

Organizations: Los Alamos National Laboratory, P.O. Box 1663, Los Alamos, NM 87545, USA · University of Rochester, Rochester, NY, USA · 241 18th Street South, Suite 650, Arlington, VA 22202

Abstract

The shock response spectrum (SRS) characterizes the peak responses of single-degree-of-freedom systems to transient acceleration. Because the mapping from an acceleration time series (waveform) to its SRS is many-to-one, inverse synthesis seeks candidate waveforms consistent with a prescribed spectrum. We extend an earlier conditional variational autoencoder (CVAE) framework with a recurrent decoder that predicts multiscale wavelet coefficients and reconstructs waveforms through an inverse discrete wavelet transform. Individual models exhibit limited variability under latent sampling, motivating an ensemble of independently trained models spanning different mother wavelets and decomposition levels. Each member produces a separate candidate waveform for the same target SRS. Evaluation on four held-out benchmark datasets shows lower mean and median spectral errors for the illustrated wavelet model than for the earlier direct time-domain CVAE and a sum-of-decayed-sinusoids (SDS) baseline. A conditioning study examines frequency-grid and encoding choices. Comparisons with published inverse-SRS methods demonstrate superior performance across four holdout benchmark datasets (two operational shock datasets, one earthquake dataset, and one synthetic shock dataset) and competitive performance on synthetic targets reported by Brake. Single-model inference requires approximately 0.1-0.6 ms per waveform during batched inference compared with 5-8 s for SDS, excluding offline training. This low inference cost enables rapid generation of distinct SRS-compatible candidate waveforms across the ensemble, although their physical suitability requires assessment beyond spectral agreement.

Figures & tables

Figure 1: CVAE with recurrent wavelet decoder, illustrated for a six-level wavelet decomposition. BandBlocks predict coefficient sets from coarse to fine; the IDWT reconstructs the 9000-sample acceleration time series.

Figure 2: Representative reconstructions for one example from $\mathcal{D}_{\mathrm{test}}^{B}$ ( $n=890$ ): (a) target and reconstructed SRS, with the shaded region indicating the $\pm 3$ dB target band; (b) reference acceleration time series used to compute the target SRS; (c) SDS reconstruction; and (d)–(f) reconstructions from three CVAE ensemble members. The reconstructions exhibit similar SRS responses despite distinct amplitudes and temporal structures. The reference time series is plotted over $\pm 40$ g, whereas the reconstructions are plotted over $\pm 20$ g.

Holdout	Method	Mean	Median	Std. Dev.	Min	Max	0.025 Quant.	0.975 Quant.
A	SDS	0.206	0.176	0.110	0.051	0.894	0.082	0.489
	Direct CVAE	0.095	0.090	0.024	0.049	0.211	0.060	0.156
	Wavelet CVAE	0.065	0.058	0.033	0.024	0.368	0.031	0.148
B	SDS	0.231	0.200	0.112	0.088	0.925	0.110	0.546
	Direct CVAE	0.136	0.125	0.047	0.063	0.418	0.079	0.266
	Wavelet CVAE	0.094	0.086	0.029	0.045	0.376	0.051	0.161

Table 1: Descriptive statistics of per-sample RMSLE across the four held-out benchmark datasets. SDS and Direct CVAE results are reproduced from our earlier work [ 19 ] . Wavelet CVAE denotes the db10 , level 8 model. Boldface identifies the lowest mean and median within each dataset. Lower RMSLE indicates closer agreement with the target SRS.

Panel of Figure 3 — Figure 3: Distributions of per-sample RMSLE for a single CVAE ensemble member using the Daubechies 10 ( db10 ) wavelet at decomposition level 8 (blue) and classical SDS reconstructions (orange). The four held-out datasets are $\mathcal{D}_{\mathrm{test}}^{A}$ ( $n=963$ , top left), $\mathcal{D}_{\mathrm{test}}^{B}$ ( $n=890$ , top right), $\mathcal{D}_{\mathrm{test}}^{C}$ ( $n=815$ , bottom left), and $\mathcal{D}_{\mathrm{test}}^{D}$ ( $n=1000$ , bottom right). The CVAE achieves lower RMSLE than SDS for 98.1%, 98.9%, 90.8%, and 92.8% of samples, respectively. Vertical dashed lines indicate the median RMSLE for each method. Lower RMSLE indicates closer agreement with the target SRS.

Panel of Figure 4 — Figure 4: Distributions of per-frequency absolute SRS errors for a single CVAE ensemble member using the Daubechies 10 ( db10 ) wavelet at decomposition level 8 (blue) and classical SDS reconstructions (orange). The four held-out datasets are $\mathcal{D}_{\mathrm{test}}^{A}$ (top left), $\mathcal{D}_{\mathrm{test}}^{B}$ (top right), $\mathcal{D}_{\mathrm{test}}^{C}$ (bottom left), and $\mathcal{D}_{\mathrm{test}}^{D}$ (bottom right). Errors are pooled across all samples and $F=100$ evaluation natural frequencies, yielding $|\mathcal{D}_{j}^{\mathrm{test}}|\times F$ error values per method for each dataset. Vertical lines mark the 1 dB and 3 dB thresholds, with annotations reporting the corresponding empirical cumulative fractions. The CVAE achieves higher fractions of SRS points within both thresholds across all four datasets.

Panel A: Reconstructed SRS ordinates (m/s 2 )
			CVAE reconstruction
			Log-random 100-point grid	Fixed 10-point grid		Fixed 100-point grid
Frequency	Target	Brake	$\mathrm{C}$	$\mathrm{C}$	$\mathrm{H}$	$\mathrm{C}$	$\mathrm{H}$
(Hz)		(2011)
10	2.000	1.960	1.983	2.057	1.958	1.844	1.956
100	6.000	5.709	5.825	5.958	5.793	5.781	5.741

Table 2: Reconstructed SRS ordinates and corresponding error metrics for the portion of the target spectrum from Table 1 of Brake [ 2 ] spanning 10 Hz to 5 kHz. The symbols

\mathrm{C}

and

\mathrm{H}

denote concatenation and Hadamard encoding, respectively. Errors are evaluated at the 10 retained frequencies reported by Brake.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Loss term	Coefficient	Value
Waveform shape	$\lambda_{\mathrm{shape}}$	3.4601
Time series	$\lambda_{\mathrm{TS}}$	0.7658
Welch PSD	$\lambda_{\mathrm{PSD}}$	0.1885
SRS	$\lambda_{\mathrm{SRS}}$	2.9037
KL divergence	$\lambda_{\mathrm{KL}}$	4.955

Appendix

Table 3: Coefficients used in the total training objective.

Symbol	Parameter	Value or sampling distribution
$\mathcal{B}$	Number of basis components	$\operatorname{Unif}\{1,\ldots,10\}$
$\xi_{i}^{\prime}$	Independently sampled offset fraction	$\mathcal{U}(0,0.75)$
$\gamma$	Probability of inheriting the preceding offset	$0.5$
$\phi_{i}$	Phase (rad)	$\mathcal{U}(0,2\pi)$
$f_{i}$	Component frequency (Hz)	$\mathcal{U}(2,6000)$
$\lambda_{i}$	Sinusoidal decay constant ( $\mathrm{s}^{-1}$ )	$\mathcal{U}(0.004\pi f_{i},0.2\pi f_{i})$

Appendix

Table 4: Synthetic shock generation parameters.

\mathcal{U}

denotes a continuous uniform distribution and

\operatorname{Unif}

a discrete uniform distribution. The component-frequency range is 2–6000 Hz unless otherwise specified. Basis definitions and temporal placement rules are given in the accompanying equations. Here,

g

denotes standard gravitational acceleration.

Figure 5: Held-out median RMSLE by wavelet configuration for Dataset A.

Figure 6: Held-out median RMSLE by wavelet configuration for Dataset B.

Figure 7: Held-out median RMSLE by wavelet configuration for Dataset C.

Figure 8: Held-out median RMSLE by wavelet configuration for Dataset D.

Figure 9: Percentage of pooled SRS points within 1 dB and 3 dB error thresholds by wavelet configuration for Dataset A.

Figure 10: Percentage of pooled SRS points within 1 dB and 3 dB error thresholds by wavelet configuration for Dataset B.

Figure 11: Percentage of pooled SRS points within 1 dB and 3 dB error thresholds by wavelet configuration for Dataset C.

Figure 12: Percentage of pooled SRS points within 1 dB and 3 dB error thresholds by wavelet configuration for Dataset D.

		Brake (2011)	Present study
Natural frequency (Hz)	Target	Original	CVAE
10	2	1.960	2.150
100	6	5.709	6.349
400	6	5.937	6.272
500	9	9.350	9.087
600	9	7.484	8.879

Appendix

Table 5: SRS comparison for the target in Table 1 of Brake [ 2 ] .

		Brake (2011)	Present study
Natural frequency (Hz)	Target	Original	CVAE
10	10	10.030	9.927
100	11	11.050	11.115
1000	16	15.800	16.121
1500	15	15.060	15.053
10000	15	15.100	15.656

Appendix

Table 6: SRS comparison for the target in Table 2 of Brake [ 2 ] .

		Brake (2011)		Present study
Natural frequency (Hz)	Target	Original	GA-refined	CVAE
10	5	5.019	5.033	4.739
100	5	6.355	5.024	5.000
200	7	7.340	7.543	7.007
500	7	8.589	7.258	7.167
1000	9	9.929	8.192	8.912

Appendix

Table 7: SRS comparison for the target in Table 3 of Brake [ 2 ] .

		Brake (2011)		Present study
Natural frequency (Hz)	Target	Original	GA-refined	CVAE
10	8	8.089	8.076	7.532
100	10	11.240	10.010	10.560
200	12	12.370	11.790	12.012
500	13	13.020	12.910	13.219
1000	13	13.020	13.010	12.760

Appendix

Table 8: SRS comparison for the target in Table 4 of Brake [ 2 ] .

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Inverse Reconstruction of Shock Time Series from Shock Response Spectrum Curves using Machine Learning

The shock response spectrum (SRS) is widely used to characterize the response of single-degree-of-freedom (SDOF) systems to transient accelerations. Because the mapping from acceleration time history to SRS is nonlinear and many-to-one, reconstructing time-domain signals from a target spectrum is inherently ill-posed. Conventional approaches address this problem through iterative optimization, typically representing signals as sums of exponentially decayed sinusoids, but these methods are computationally expensive and constrained by predefined basis functions. We propose a conditional variational autoencoder (CVAE) that learns a data-driven inverse mapping from SRS to acceleration time series. Once trained, the model generates signals consistent with prescribed target spectra without requiring iterative optimization. Experiments demonstrate improved spectral fidelity relative to classical techniques, strong generalization to unseen spectra, and inference speeds three to six orders of magnitude faster. These results establish deep generative modeling as a scalable and efficient approach for inverse SRS reconstruction.

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