# Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data

arXiv: https://arxiv.org/abs/2009.08810 · HTML: https://mlsanity.com/paper/2009.08810/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Sep 2020.

## Built on by (most cited first, 30 of 37)

- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Out-of-Domain Generalization in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2402.18377/) (2024)
- [Discovering Sparse Interpretable Dynamics from Partial Observations](https://mlsanity.com/paper/2107.10879/) (2021)
- [DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimization](https://mlsanity.com/paper/2108.00069/) (2021)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Almost-Linear RNNs Yield Highly Interpretable Symbolic Codes in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2410.14240/) (2024)
- [Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects](https://mlsanity.com/paper/2203.05164/) (2022)
- [Rapid Bayesian identification of sparse nonlinear dynamics from scarce and noisy data](https://mlsanity.com/paper/2402.15357/) (2024)
- [Noise-aware Physics-informed Machine Learning for Robust PDE Discovery](https://mlsanity.com/paper/2206.12901/) (2022)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [On latent dynamics learning in nonlinear reduced order modeling](https://mlsanity.com/paper/2408.15183/) (2024)
- [Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case](https://mlsanity.com/paper/2110.00546/) (2021)
- [A Robust SINDy Approach by Combining Neural Networks and an Integral Form](https://mlsanity.com/paper/2309.07193/) (2023)
- [A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study](https://mlsanity.com/paper/2201.12683/) (2022)
- [Learning noise-induced transitions by multi-scaling reservoir computing](https://mlsanity.com/paper/2309.05413/) (2023)
- [Sparse discovery of differential equations based on multi-fidelity Gaussian process](https://mlsanity.com/paper/2401.11825/) (2024)
- [Symbolic Regression on Sparse and Noisy Data with Gaussian Processes](https://mlsanity.com/paper/2309.11076/) (2023)
- [Machine Learning Applications to Computational Plasma Physics and Reduced-Order Plasma Modeling: A Perspective](https://mlsanity.com/paper/2409.02349/) (2024)
- [Multiphysics discovery with moving boundaries using Ensemble SINDy and Peridynamic Differential Operator](https://mlsanity.com/paper/2303.15631/) (2023)
- [Sparse Identification of Nonlinear Dynamics with Conformal Prediction](https://mlsanity.com/paper/2507.11739/) (2025)
- [Investigating Sindy As a Tool For Causal Discovery In Time Series Signals](https://mlsanity.com/paper/2212.14133/) (2022)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [Interpretable neural network system identification method for two families of second-order systems based on characteristic curves](https://mlsanity.com/paper/2509.10632/) (2025)
- [Uncovering Closed-form Governing Equations of Nonlinear Dynamics from Videos](https://mlsanity.com/paper/2106.04776/) (2021)
- [Digital Twin Data Modelling by Randomized Orthogonal Decomposition and Deep Learning](https://mlsanity.com/paper/2206.08659/) (2022)
- [Data-driven discovery of mechanical models directly from MRI spectral data](https://mlsanity.com/paper/2411.06958/) (2024)
- [Data Denoising and Derivative Estimation for Data-Driven Modeling of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2509.14219/) (2025)
- [Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identification](https://mlsanity.com/paper/2604.20141/) (2026)
- [LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generation](https://mlsanity.com/paper/2607.10546/) (2026)

## Abstract (arXiv)

The sparse identification of nonlinear dynamics (SINDy) is a regression framework for the discovery of parsimonious dynamic models and governing equations from time-series data. As with all system identification methods, noisy measurements compromise the accuracy and robustness of the model discovery procedure. In this work, we develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al. for simultaneously (i) denoising the data, (ii) learning and parametrizing the noise probability distribution, and (iii) identifying the underlying parsimonious dynamical system responsible for generating the time-series data. Thus within an integrated optimization framework, noise can be separated from signal, resulting in an architecture that is approximately twice as robust to noise as state-of-the-art methods, handling as much as 40% noise on a given time-series signal and explicitly parametrizing the noise probability distribution. We demonstrate this approach on several numerical examples, from Lotka-Volterra models to the spatio-temporal Lorenz 96 model. Further, we show the method can identify a diversity of probability distributions including Gaussian, uniform, Gamma, and Rayleigh.