Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs

TL;DR

This work tackles realistic time-series generation by addressing GANs' training instability and mode-collapse risks with a robust VAE-based approach. It introduces KoVAE, which imposes a linear latent Koopman dynamic prior $z_t = A z_{t-1}$ while learning a nonlinear encoder, and supports irregular sampling via NCDE-based embeddings. The approach enables physics-informed generation through eigenvalue penalties and stability analysis, yielding strong empirical gains over state-of-the-art GAN/VAE baselines on regular and irregular data, and even demonstrating density-accurate generation in climate-like tasks. Overall, KoVAE provides a principled, flexible framework for accurate, interpretable, and domain-knowledgeable time-series generation with practical impact in science and engineering.

Abstract

Generating realistic time series data is important for many engineering and scientific applications. Existing work tackles this problem using generative adversarial networks (GANs). However, GANs are unstable during training, and they can suffer from mode collapse. While variational autoencoders (VAEs) are known to be more robust to the these issues, they are (surprisingly) less considered for time series generation. In this work, we introduce Koopman VAE (KoVAE), a new generative framework that is based on a novel design for the model prior, and that can be optimized for either regular and irregular training data. Inspired by Koopman theory, we represent the latent conditional prior dynamics using a linear map. Our approach enhances generative modeling with two desired features: (i) incorporating domain knowledge can be achieved by leveraging spectral tools that prescribe constraints on the eigenvalues of the linear map; and (ii) studying the qualitative behavior and stability of the system can be performed using tools from dynamical systems theory. Our results show that KoVAE outperforms state-of-the-art GAN and VAE methods across several challenging synthetic and real-world time series generation benchmarks. Whether trained on regular or irregular data, KoVAE generates time series that improve both discriminative and predictive metrics. We also present visual evidence suggesting that KoVAE learns probability density functions that better approximate the empirical ground truth distribution.

Figures (17)

• Figure 1: A) The posterior is composed of an embedding layer (NCDE), an encoder (GRU + BN), mean/variance computation and sampling (linear + repr. trick), and a decoder (GRU + linear + sigmoid). B) The prior consists of GRU, linear, a repr. trick layer, and our novel Koopman module.
• Figure 2: We qualitatively evaluate our approach with two-dimensional t-SNE plots of the synthetic and real data (top row). In addition, we show the probability density functions of the real data, and for KoVAE and GT-GAN synthetic distributions (bottom row).
• Figure 3: On the left, we show the spectrum of the approximate Koopman operator without constraints during the training. On the right, we show the spectrum of the approximate Koopman operator for the model that is trained with stability constraints. We can see that, indeed, the absolute value of two eigenvalues is approximately 1.
• Figure 4: The t-SNE plots of KoVAE with $\alpha=0$ (A), KoVAE (B), and constrained KoVAE (C), and their probability density functions (D) are compared to the true nonlinear pendulum data.
• Figure 5: The spectral distribution of the approximate Koopman operator of the prior for each dataset in the regular setting.