# Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systems

arXiv: https://arxiv.org/abs/2410.10023 · HTML: https://mlsanity.com/paper/2410.10023/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 38 references that are in the map.
- First posted on arXiv 13 Oct 2024.

## Builds on (most cited first, 14 of 14)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ1-regularized Least Squares](https://mlsanity.com/paper/2005.13232/) (2020)
- [Robust discovery of partial differential equations in complex situations](https://mlsanity.com/paper/2106.00008/) (2021)
- [Nonlinear System Identification of Soft Robot Dynamics Using Koopman Operator Theory](https://mlsanity.com/paper/1810.06637/) (2018)
- [A Hybrid Science-Guided Machine Learning Approach for Modeling and Optimizing Chemical Processes](https://mlsanity.com/paper/2112.01475/) (2021)
- [Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equations](https://mlsanity.com/paper/2107.07040/) (2021)
- [Data-Driven Theory-guided Learning of Partial Differential Equations using SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE)](https://mlsanity.com/paper/2109.07471/) (2021)

## Built on by (most cited first, 3 of 3)

- [Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics](https://mlsanity.com/paper/2506.18339/) (2025)
- [KAN/MultKAN with Physics-Informed Spline fitting (KAN-PISF) for ordinary/partial differential equation discovery of nonlinear dynamic systems](https://mlsanity.com/paper/2411.11801/) (2024)
- [Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data](https://mlsanity.com/paper/2603.22380/) (2026)

## Abstract (arXiv)

Sparse system identification of nonlinear dynamic systems is still challenging, especially for stiff and high-order differential equations for noisy measurement data. The use of highly correlated functions makes distinguishing between true and false functions difficult, which limits the choice of functions. In this study, an equation discovery method has been proposed to tackle these problems. The key elements include a) use of B-splines for data fitting to get analytical derivatives superior to numerical derivatives, b) sequentially regularized derivatives for denoising (SRDD) algorithm, highly effective in removing noise from signal without system information loss, c) uncorrelated component analysis (UCA) algorithm that identifies and eliminates highly correlated functions while retaining the true functions, and d) physics-informed spline fitting (PISF) where the spline fitting is updated gradually while satisfying the governing equation with a dictionary of candidate functions to converge to the correct equation sequentially. The complete framework is built on a unified deep-learning architecture that eases the optimization process. The proposed method is demonstrated to discover various differential equations at various noise levels, including three-dimensional, fourth-order, and stiff equations. The parameter estimation converges accurately to the true values with a small coefficient of variation, suggesting robustness to the noise.