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

TL;DR

This paper introduces SINDy-SHRED, a novel method combining sparse identification and shallow recurrent networks to model complex spatio-temporal dynamics efficiently and interpretably, with applications to physical systems and video prediction.

Contribution

The paper presents a new approach that jointly solves sensing and model identification using simple, robust, and efficient shallow recurrent networks with SINDy regularization, enabling interpretable models and discovering new physics.

Findings

Achieves superior accuracy and data efficiency in modeling PDE data and real-world measurements.

Outperforms baseline deep learning models in long-term video prediction.

Provides stable, interpretable models of complex spatio-temporal dynamics.

Abstract

Modeling real-world spatio-temporal 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 spatio-temporal 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 spatio-temporal dynamics, (ii) discovers new physics models even for well-known physical systems, (iii) achieves provably robust convergence with an observed globally convex loss landscape, and (iv) achieves superior accuracy, data efficiency, and training time, all with fewer model parameters. We conduct systematic experimental studies on PDE 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 time, and data requirements, including Convolutional LSTM, PredRNN, ResNet, and SimVP.