# Discovering Sparse Interpretable Dynamics from Partial Observations

arXiv: https://arxiv.org/abs/2107.10879 · HTML: https://mlsanity.com/paper/2107.10879/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 47 references that are in the map.
- First posted on arXiv 22 Jul 2021.

## Builds on (most cited first, 22 of 22)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting](https://mlsanity.com/paper/2010.04456/) (2020)
- [Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery](https://mlsanity.com/paper/1912.04825/) (2019)
- [Don't Unroll Adjoint: Differentiating SSA-Form Programs](https://mlsanity.com/paper/1810.07951/) (2018)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data](https://mlsanity.com/paper/2009.08810/) (2020)
- [Extracting Interpretable Physical Parameters from Spatiotemporal Systems using Unsupervised Learning](https://mlsanity.com/paper/1907.06011/) (2019)
- [Learning Latent Dynamics for Partially-Observed Chaotic Systems](https://mlsanity.com/paper/1907.02452/) (2019)
- [Extended Dynamic Mode Decomposition with Learned Koopman Eigenfunctions for Prediction and Control](https://mlsanity.com/paper/1911.08751/) (2019)
- [OccamNet: A Fast Neural Model for Symbolic Regression at Scale](https://mlsanity.com/paper/2007.10784/) (2020)
- [Physics-Incorporated Convolutional Recurrent Neural Networks for Source Identification and Forecasting of Dynamical Systems](https://mlsanity.com/paper/2004.06243/) (2020)

## Built on by (most cited first, 25 of 25)

- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Deep Generative Symbolic Regression](https://mlsanity.com/paper/2401.00282/) (2023)
- [Deep Learning and Symbolic Regression for Discovering Parametric Equations](https://mlsanity.com/paper/2207.00529/) (2022)
- [Marrying Causal Representation Learning with Dynamical Systems for Science](https://mlsanity.com/paper/2405.13888/) (2024)
- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [Discovering Conservation Laws using Optimal Transport and Manifold Learning](https://mlsanity.com/paper/2208.14995/) (2022)
- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [Scalable Mechanistic Neural Networks for Differential Equations and Machine Learning](https://mlsanity.com/paper/2410.06074/) (2024)
- [On the definition and importance of interpretability in scientific machine learning](https://mlsanity.com/paper/2505.13510/) (2025)
- [Learning Macroscopic Dynamics from Partial Microscopic Observations](https://mlsanity.com/paper/2410.23938/) (2024)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [Characterizing control between interacting subsystems with deep Jacobian estimation](https://mlsanity.com/paper/2507.01946/) (2025)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [Decomposing heterogeneous dynamical systems with graph neural networks](https://mlsanity.com/paper/2407.19160/) (2024)
- [SODAs: Sparse Optimization for the Discovery of Differential and Algebraic Equations](https://mlsanity.com/paper/2503.05993/) (2025)
- [CTSR: Cartesian tensor-based sparse regression for data-driven discovery of high-dimensional invariant governing equations](https://mlsanity.com/paper/2504.07618/) (2025)
- [Variational Rank Reduction Autoencoders for Generative Thermal Design](https://mlsanity.com/paper/2509.08515/) (2025)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [Discovering Latent Response Laws in Forced Physical Systems](https://mlsanity.com/paper/2607.09801/) (2026)
- [Automating the Discovery of Partial Differential Equations in Dynamical Systems](https://mlsanity.com/paper/2404.16444/) (2024)
- [Self-induced stochastic resonance: A physics-informed machine learning approach](https://mlsanity.com/paper/2510.22848/) (2025)
- [Knowledge-Informed Kernel State Reconstruction from Heterogeneous Partial Observations](https://mlsanity.com/paper/2601.22328/) (2026)
- [Data Enrichment for Symbolic Regression Using Diffusion Models](https://mlsanity.com/paper/2606.00988/) (2026)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)
- [Retracing Hodgkin and Huxley: State Recovery Does Not Certify Mechanism](https://mlsanity.com/paper/2609.34566/) (2026)

## Abstract (arXiv)

Identifying the governing equations of a nonlinear dynamical system is key to both understanding the physical features of the system and constructing an accurate model of the dynamics that generalizes well beyond the available data. We propose a machine learning framework for discovering these governing equations using only partial observations, combining an encoder for state reconstruction with a sparse symbolic model. Our tests show that this method can successfully reconstruct the full system state and identify the underlying dynamics for a variety of ODE and PDE systems.