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

TL;DR

A conditional variational autoencoder (CVAE) that learns a data-driven inverse mapping from SRS to acceleration time series, establishing deep generative modeling as a scalable and efficient approach for inverse SRS reconstruction.

Abstract

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.

Figures (13)

• Figure 1: Construction of the shock response spectrum (SRS) using a bank of single-degree-of-freedom (SDOF) oscillators. The top panel shows the SRS (blue) with selected evaluation frequencies highlighted (red markers). The middle row presents representative SDOF acceleration responses at those frequencies, illustrating frequency-dependent amplification and decay. The bottom schematic depicts the corresponding mass–spring–damper oscillators, each tuned to a natural frequency $\mathfrak{f}_i$ and subjected to a common base acceleration input (right), with each SRS ordinate obtained from the peak response of its oscillator.
• Figure 2: Base-excited SDOF oscillator with mass $m$, a particular spring stiffness $k_i$, and damping coefficient $c_i$ chosen such that the damping ratio $\zeta$ is constant across oscillators. The absolute displacement is $y(t)$, the base displacement is $x(t)$, and the relative displacement is $z(t) = y(t) - x(t)$.
• Figure 3: Forward and inverse formulations of the shock response spectrum (SRS) mapping. Top: multiple acceleration time histories $\{x^{(j)}[n]\}_{j=1}^J$ are mapped through the SRS operator to corresponding spectral vectors $\{\bm{\mathfrak{s}}^{(j)}\}_{j=1}^J$, from which an example target spectrum $\bm{\mathfrak{s}}_{\mathrm{target}}$ is constructed (here shown as the ensemble mean). Bottom: the inverse SRS problem seeks admissible time-domain signals whose spectra match the target. Because the SRS operator is not injective, the inverse mapping is set-valued, $\mathrm{SRS}^{-1}(\bm{\mathfrak{s}}_{\mathrm{target}}) \mapsto \mathcal{X}(\bm{\mathfrak{s}}_{\mathrm{target}})$, where $\mathcal{X}(\bm{\mathfrak{s}}_{\mathrm{target}})$ denotes the set of acceleration time histories satisfying $\mathrm{SRS}(\hat{x}) \approx \bm{\mathfrak{s}}_{\mathrm{target}}$. The inverse problem therefore consists of selecting one admissible reconstruction $\hat{x}[n] \in \mathcal{X}(\bm{\mathfrak{s}}_{\mathrm{target}})$.
• Figure 4: Overview of the proposed machine-learning inverse SRS framework. The network receives the target SRS and corresponding natural frequency vector as inputs and produces a reconstructed acceleration time series. During training, the loss is computed between the SRS of the reconstructed signal and the target spectrum and is backpropagated to update model parameters. Additional loss terms, such as time-domain or regularization penalties, can be combined with the SRS loss to guide learning and improve stability.
• Figure 5: Dual view of the Variational Autoencoder (VAE). Left: the inference (encoder) direction $\mathbf{x}\!\to\!\mathbf{z}$ via $q_\phi(\mathbf{z}|\mathbf{x})$ that approximates the intractable posterior $p_\theta(\mathbf{z}|\mathbf{x})$. Right: the generative (decoder) direction $\mathbf{z}\!\to\!\mathbf{x}$ via $p(\mathbf{z})$ and $p_\theta(\mathbf{x}|\mathbf{z})$. Maximizing the ELBO in Eq. \ref{['EQ:VLB']} couples these two models.