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

arXiv: https://arxiv.org/abs/2501.13329 · HTML: https://mlsanity.com/paper/2501.13329/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 61 of its 119 references that are in the map.
- First posted on arXiv 23 Jan 2025.

## Builds on (most cited first, 30 of 61)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Architecture Search with Reinforcement Learning](https://mlsanity.com/paper/1611.01578/) (2016)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Self-Supervised Learning from Images with a Joint-Embedding Predictive Architecture](https://mlsanity.com/paper/2301.08243/) (2023)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [What is the State of Neural Network Pruning?](https://mlsanity.com/paper/2003.03033/) (2020)
- [Revisiting Feature Prediction for Learning Visual Representations from Video](https://mlsanity.com/paper/2404.08471/) (2024)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [PhysGaussian: Physics-Integrated 3D Gaussians for Generative Dynamics](https://mlsanity.com/paper/2311.12198/) (2023)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Unsupervised Learning for Physical Interaction through Video Prediction](https://mlsanity.com/paper/1605.07157/) (2016)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [Deep learning: a statistical viewpoint](https://mlsanity.com/paper/2103.09177/) (2021)
- [Disentangling Physical Dynamics from Unknown Factors for Unsupervised Video Prediction](https://mlsanity.com/paper/2003.01460/) (2020)
- [SimVP: Simpler yet Better Video Prediction](https://mlsanity.com/paper/2206.05099/) (2022)

## Built on by (most cited first, 20 of 20)

- [Reduced Order Modeling with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2502.10930/) (2025)
- [From Models To Experiments: Shallow Recurrent Decoder Networks on the DYNASTY Experimental Facility](https://mlsanity.com/paper/2503.08907/) (2025)
- [Data assimilation and discrepancy modeling with shallow recurrent decoders](https://mlsanity.com/paper/2512.01170/) (2025)
- [T-SHRED: Symbolic Regression for Regularization and Model Discovery with Transformer Shallow Recurrent Decoders](https://mlsanity.com/paper/2506.15881/) (2025)
- [Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models](https://mlsanity.com/paper/2506.20771/) (2025)
- [Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders](https://mlsanity.com/paper/2505.13569/) (2025)
- [Disentangling Dynamical Systems: Causal Representation Learning Meets Local Sparse Attention](https://mlsanity.com/paper/2603.14483/) (2026)
- [PySHRED: A Python package for SHallow REcurrent Decoding for sparse sensing, model reduction and scientific discovery](https://mlsanity.com/paper/2507.20954/) (2025)
- [Invertible Koopman neural operator for data-driven modeling of partial differential equations](https://mlsanity.com/paper/2503.19717/) (2025)
- [Synergizing Transport-Based Generative Models and Latent Geometry for Stochastic Closure Modeling](https://mlsanity.com/paper/2602.17089/) (2026)
- [UQ-SHRED: uncertainty quantification of shallow recurrent decoder networks for sparse sensing via engression](https://mlsanity.com/paper/2604.01305/) (2026)
- [EML Trees Are Universal Approximators](https://mlsanity.com/paper/2606.23179/) (2026)
- [Discovering Latent Response Laws in Forced Physical Systems](https://mlsanity.com/paper/2607.09801/) (2026)
- [CS-SHRED: Enhancing SHRED for Robust Recovery of Spatiotemporal Dynamics](https://mlsanity.com/paper/2507.22303/) (2025)
- [Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs](https://mlsanity.com/paper/2601.11259/) (2026)
- [LAtent Phase Inference from Short time sequences using SHallow REcurrent Decoders (LAPIS-SHRED)](https://mlsanity.com/paper/2604.01216/) (2026)
- [Fast and principled equation discovery from chaos to climate](https://mlsanity.com/paper/2604.11929/) (2026)
- [Learning Control-Affine Reduced-Order Models via Autoencoders](https://mlsanity.com/paper/2606.05045/) (2026)
- [Real-time optimal control with shallow recurrent decoder networks](https://mlsanity.com/paper/2607.19302/) (2026)
- [STRIDE: State-Transition Representation via Increment Dynamics and Evolution](https://mlsanity.com/paper/2609.32367/) (2026)

## Abstract (arXiv)

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.