Towards a data-scale independent regulariser for robust sparse identification of non-linear dynamics

Jay Raut; Daniel N. Wilke; Stephan Schmidt

cs.LGMar 5, 2026

Towards a data-scale independent regulariser for robust sparse identification of non-linear dynamics

Authors: Jay Raut, Daniel N. Wilke, Stephan Schmidt

Organizations: Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria, South Africa · School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand, Johannesburg, South Africa

Abstract

Data normalisation, a common and often necessary preprocessing step in engineering and scientific applications, can severely distort the discovery of governing equations by magnitudebased sparse regression methods. This issue is particularly acute for the Sparse Identification of Nonlinear Dynamics (SINDy) framework, where the core assumption of sparsity is undermined by the interaction between data scaling and measurement noise. The resulting discovered models can be dense, uninterpretable, and physically incorrect. To address this critical vulnerability, we introduce the Sequential Thresholding of Coefficient of Variation (STCV), a novel, computationally efficient sparse regression algorithm that is inherently robust to data scaling. STCV replaces conventional magnitude-based thresholding with a dimensionless statistical metric, the Coefficient Presence (CP), which assesses the statistical validity and consistency of candidate terms in the model library. This shift from magnitude to statistical significance makes the discovery process invariant to arbitrary data scaling. Through comprehensive benchmarking on canonical dynamical systems and practical engineering problems, including a physical mass-spring-damper experiment, we demonstrate that STCV consistently and significantly outperforms standard Sequential Thresholding Least Squares (STLSQ) and Ensemble-SINDy (E-SINDy) on normalised, noisy datasets. The results show that STCV-based methods can successfully identify the correct, sparse physical laws even when other methods fail. By mitigating the distorting effects of normalisation, STCV makes sparse system identification a more reliable and automated tool for real-world applications, thereby enhancing model interpretability and trustworthiness.

Figures & tables

Panel of Figure 1 — Figure 1: Experiment demonstrating the degradation of SINDy sampling requirements for the Lorenz system in Equation ( 2 ) when data is unscaled and normalised. The top and bottom rows respectively show noiseless data and data with 0.1% uniformly distributed noise added. The left and right columns, respectively, show the results of SINDy on unnormalised and normalised data. White indicates a 100% success rate in identifying the correct model sparsity, while black indicates a 0% success rate.

Algorithm 1: Sequential Thresholding of Coefficient of Variation (STCV)
Inputs: Design matrix $\boldsymbol{\Theta}$ , derivative data $\dot{\boldsymbol{X}}$ , STLSQ threshold $\lambda$ , initial and final ridge penalties $\gamma_{0},\gamma_{f}$ , initial and final CP thresholds $\lambda_{CP,0},\lambda_{CP,f}$ , and number of steps $n_{steps}$ .
Output: Sparse coefficient matrix $\boldsymbol{\Xi}$ .
Procedure: 1. Precondition data for numerical stability. 2. Initialise: a. Perform an initial model fit using STLSQ with a near-zero threshold, $\lambda$ , and initial ridge penalty, $\gamma_{0}$ , to get an initial coefficient matrix, $\boldsymbol{\Xi}_{0}$ . b. Use BLR to estimate the standard deviation $\sigma_{\xi_{ij}}$ for each coefficient in $\boldsymbol{\Xi}_{0}$ . c. Calculate the CP value for each coefficient, $CP_{ij}=\mu_{\xi_{ij}}/\sigma_{\xi_{ij}}$ . d. Apply an initial threshold: eliminate terms where $\|CP_{ij}\|<\lambda_{CP,0}$ a. Create ramped schedules for the ridge penalty (from $\gamma_{1}$ to $\gamma_{n_{steps}}$ = $\gamma_{f}$ ) and the CP threshold (from $\lambda_{CP,1}$ to $\lambda_{CP,{n_{steps}}}$ = $\lambda_{CP,f}$ ). 3. Iterative Refinement: b. For each step $k$ , from 1 to $n_{steps}$ : i. Set current ridge penalty, $\gamma_{k}$ , and CP threshold, $\lambda_{CP,k}$ . ii. Inner Loop: While the model sparsity is changing: * Fit a new model, $\boldsymbol{\Xi_{k}}$ , using STLSQ on the currently active terms with ridge penalty $\gamma_{k}$ . * Estimate coefficient standard deviations using BLR. * Calculate CP values for all active terms. * Eliminate terms where $\|CP_{ij}\|<\lambda_{CP,k}$ * Check for convergence (i.e., if the set of active terms has changed). 4. Return $\boldsymbol{\Xi}=\boldsymbol{\Xi_{n_{steps}}}$ .

Table 1: The Sequential Thresholding of Coefficient of Variation (STCV) Algorithm

Panel of Figure 2 — Figure 2: Success rate results from the Lorenz, Rössler, Van der Pol, and Duffing oscillator models on the Raw Data (RD) and the Scaled Data (SD). The model form and the parameters used may be found in Appendix B .

Figure 3: The success rate of various sparse regression algorithms for SINDy on noisy (scaled only, as unscaled cannot be modelled due to numerical limitations) linear oscillator data representing a damaged bearing housing during operation.

Panel of Figure 4 — Figure 4: The success rate of various sparse regression algorithms for SINDy on noisy data from the simulated linear and nonlinear half-car models

Figure 5: Experimental setup of physical linear/nonlinear mass-spring-damper system. Component list: 1) IMU wiring guide rod, 2) Linear springs, 3) Guide rod for oscillating mass, 4) Nonlinear magnetic springs simulated by stacked magnets on guide rods, 5) Rods securing end plates, 6) End plates holding springs and guide rod, 7) IMU glued to mass, 8) Oscillating mass, 9) Arduino logger.

Panel of Figure 6 — Figure 6: Measured and integrated data of the physical linear (left) and nonlinear (right) oscillating system.

Method	Linear system	Nonlinear system
Expected	$\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-1642s-0.143v\end{aligned}$	Unknown
STLSQ	$\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-1519.43s-2.53v-363.36s^{2}+4.75sv\\ &\quad-218332.31s^{3}-191.22s^{2}v-228.92sv^{2}\\ &\quad+2.99v^{3}\end{aligned}$	$\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-17.541-1653.2s-20.92v-41718.39s^{2}\\ &\quad+309.9sv-12.02v^{2}-218332.31s^{3}\\ &\quad+12314.87s^{2}v+949.39sv^{2}+64.41v^{3}\end{aligned}$
E-SINDy	$\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-1519.43s-2.53v-363.36s^{2}\\ &\quad-218332.31s^{3}+191.22s^{2}v-228.92sv^{2}\\ &\quad+2.99v^{3}\end{aligned}$	$\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-17.461-1567.36s-20.69v-39480.1s^{2}\\ &\quad-9.45v^{2}-429447.2s^{3}+5187.76s^{2}v\\ &\quad+742.63sv^{2}+50.02v^{3}\end{aligned}$
STCV	$\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-1564.26s-1.93v\end{aligned}$	$\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-17.651-1360.26s-9.16v-33094.72s^{2}\\ &\quad-371947.09s^{3}\end{aligned}$
STCV
-STSLQ	$\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-1564.26s-1.93v\end{aligned}$	$\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-17.651-1360.26s-9.16v-33094.72s^{2}\\ &\quad-371947.09s^{3}\end{aligned}$

Table 2: Comparison of resulting sparse models for the linear and nonlinear experimental data.

Figure 7: Stiffness of nonlinear setup, statically measured vs. dynamically estimated by STCV, STLSQ, and E-SINDy.

Figure 8: STCV nonlinearity bias test. 100 different models based on Equation ( 6 ), with varying values of

k_{1}

and

k_{2}

, are produced. The values of

k_{1}

and

k_{2}

are ramped inversely, so the underlying model transitions from being strongly linear to strongly nonlinear. For each model, 1000 noisy datasets are produced and modelled using STCV. The model identification rate is the success rate at which STCV fits either a linear model (only

k_{1}

present), a linear-nonlinear model (both

k_{1}

and

k_{2}

present), or a nonlinear model (only

k_{2}

present).

k_{1}

and

k_{2}

are tuned to have similar induced natural frequencies, and this experiment should help determine whether STCV is biased toward fitting linear or nonlinear models.

Figure 9: Average coefficient presence (CP) values of the linear and nonlinear terms from the STCV bias test.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

System	ODE	Integrator	Initial value	Sampling rate	Period	Tolerance
Lorenz	$\begin{aligned} Ref.~\cite[cite]{[\@@bibref{}{ref41}{}{}]}\\ \end{aligned}$	$\begin{aligned} RK45\\ \end{aligned}$	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 100Hz\\ \end{aligned}$	$\begin{aligned} 10s\\ \end{aligned}$	$\begin{aligned} 10^{-12}\\ \end{aligned}$
Rössler	$\begin{aligned} Ref.~\cite[cite]{[\@@bibref{}{ref41}{}{}]}\\ \end{aligned}$	$\begin{aligned} RK45\\ \end{aligned}$	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 1kHz\\ \end{aligned}$	$\begin{aligned} 10s\\ \end{aligned}$	$\begin{aligned} 10^{-12}\\ \end{aligned}$
van der Pol	$\begin{aligned} Ref.~\cite[cite]{[\@@bibref{}{ref41}{}{}]}\\ \end{aligned}$	$\begin{aligned} LSODA\\ \end{aligned}$	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 100Hz\\ \end{aligned}$	$\begin{aligned} 30s\\ \end{aligned}$	$\begin{aligned} default\\ \end{aligned}$
Duffing	$\begin{aligned} Ref.~\cite[cite]{[\@@bibref{}{ref41}{}{}]}\\ \end{aligned}$	$\begin{aligned} LSODA\\ \end{aligned}$	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 100Hz\\ \end{aligned}$	$\begin{aligned} 10s\\ \end{aligned}$	$\begin{aligned} default\\ \end{aligned}$
Bearing	$\begin{aligned} Equation~\ref{eqn:bearing}\\ \end{aligned}$	$\begin{aligned} RK45\\ \end{aligned}$	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 1MHz\\ \end{aligned}$	$\begin{aligned} 5\mu s\\ \end{aligned}$	$\begin{aligned} 10^{-12}\\ \end{aligned}$
Linear HC	$\begin{aligned} Ref.~\cite[cite]{[\@@bibref{}{ref42}{}{}]}\\ \end{aligned}$	$\begin{aligned} Radau\\ \end{aligned}$	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 5kHz\\ \end{aligned}$	$\begin{aligned} 2.12s\\ \end{aligned}$	$\begin{aligned} 10^{-12}\\ \end{aligned}$

Appendix

Table 3: Numerical integration parameters for dynamical systems.

System	CP threshold range	Ridge penalty (STCV-STLSQ)
Lorenz	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 10^{-1}\\ \end{aligned}$
Lorenz (normalised)	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 10^{-1}\\ \end{aligned}$
Rössler	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 10^{-1}\\ \end{aligned}$
Rössler (normalised)	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 10^{-5}\\ \end{aligned}$
van der Pol	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 10^{-1}\\ \end{aligned}$
van der Pol (normalised)	$\begin{aligned} \\ \end{aligned}$	$\begin{aligned} 10^{-1}\\ \end{aligned}$

Appendix

Table 4: STCV hyperparameters used for the benchmarking experiment.

Method	Linear system	Nonlinear system
STLSQ	$\begin{aligned} \gamma&=10^{-16}\\ \lambda&=0.9\\ \end{aligned}$	$\begin{aligned} \gamma&=10^{-16}\\ \lambda&=0.9\\ \end{aligned}$
E-SINDy	$\begin{aligned} \gamma&=10^{-16}\\ \lambda&=0.9\\ Incl.thres.&=(1-10^{-16})\end{aligned}$	$\begin{aligned} \gamma&=10^{-16}\\ \lambda&=0.9\\ Incl.thres.&=(1-10^{-16})\end{aligned}$
STCV	$\begin{aligned} \gamma_{0}&=10^{-16}\\ \gamma_{f}&=10^{-16}\\ \lambda&=0.01\\ CP&=0.3\\ \end{aligned}$	$\begin{aligned} \gamma_{0}&=10^{-16}\\ \gamma_{f}&=10^{-16}\\ \lambda&=0.01\\ CP&=0.4\\ \end{aligned}$

Appendix

Table 5: Comparison of modelling hyperparameters used for the linear and nonlinear experimental data.

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Many engineering problems involve phenomena whose governing equations are poorly characterized or only partially known. Surrogate modeling techniques such as neural networks can capture the behavior of these systems, but they typically demand large training datasets that are difficult to obtain in engineering contexts and yield models with limited physical interpretability. The Sparse Identification of Nonlinear Dynamics (SINDy) method addresses both limitations by performing sparse regression over libraries of candidate nonlinear terms, recovering interpretable governing equations from comparatively small datasets. Although SINDy has been demonstrated extensively on canonical benchmark systems, its application to practical engineering problems is less widely documented. This tutorial introduces the SINDy method and progressively builds toward its main extensions, from noise-robust weak-form and ensembling-based variants to constrained and parametrizable formulations. The paper and the accompanying tutorial (available at https://github.com/paullililili/SINDy4Engineers) is organized in three parts: the first introduces the standard SINDy algorithm and progressively extends it, inviting readers without prior knowledge to follow each step and adapt the methods to their own problems; the remaining two parts present detailed case studies on (1) the system identification of an unmanned aerial vehicle and (2) a chaotic thermosyphon heat exchanger. Through these examples, we aim to demonstrate that SINDy is simple to implement yet flexible enough to serve as a valuable identification tool for advanced engineering applications.

Abstract

Sparse Identification of Nonlinear Dynamics

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