Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks

Modeling real-world spatiotemporal data is exceptionally difficult due to inherent high dimensionality, measurement noise, partial observations, and often expensive data collection procedures. In this paper, we present Sparse Identification of Nonlinear Dynamics with SHallow REcurrent Decoder networks (SINDy-SHRED), a method to jointly solve the sensing and model identification problems with simple implementation, efficient computation, and robust performance. SINDy-SHRED uses Gated Recurrent Units to model the temporal sequence of sparse sensor measurements along with a shallow decoder network to reconstruct the full spatiotemporal field from the latent state space. Our algorithm introduces a SINDy-based regularization for which the latent space progressively converges to a SINDy-class functional, provided the projection remains within the set. In restricting SINDy to a linear model, a Koopman-SHRED model is generated. SINDy-SHRED i) learns a symbolic and interpretable generative model of a parsimonious and low-dimensional latent space for the complex spatiotemporal dynamics, ii) discovers governing equations even for well-known physical systems, iii) achieves an observed convex loss landscape, and iv) achieves superior accuracy, data efficiency, and training cost, all with fewer model parameters, than competing state-of-the-art methods. We conduct systematic experimental studies on Partial differential equation data such as turbulent flows, real-world sensor measurements for sea surface temperature, and direct video data. The interpretable SINDy and Koopman models of latent state dynamics enable stable and accurate long-term video predictions, outperforming all current baseline deep learning models in accuracy, training cost, and data requirements, including Convolutional LSTM, PredRNN, ResNet, and SimVP.