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

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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

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 Θ\boldsymbol{\Theta} , derivative data X˙\dot{\boldsymbol{X}} , STLSQ threshold λ\lambda , initial and final ridge penalties γ0,γf\gamma_{0},\gamma_{f} , initial and final CP thresholds λCP,0,λCP,f\lambda_{CP,0},\lambda_{CP,f} , and number of steps nstepsn_{steps} .
Output: Sparse coefficient matrix Ξ\boldsymbol{\Xi} .
Procedure: 1. Precondition data for numerical stability. 2. Initialise: a. Perform an initial model fit using STLSQ with a near-zero threshold, λ\lambda , and initial ridge penalty, γ0\gamma_{0} , to get an initial coefficient matrix, Ξ0\boldsymbol{\Xi}_{0} . b. Use BLR to estimate the standard deviation σξij\sigma_{\xi_{ij}} for each coefficient in Ξ0\boldsymbol{\Xi}_{0} . c. Calculate the CP value for each coefficient, CPij=μξij/σξijCP_{ij}=\mu_{\xi_{ij}}/\sigma_{\xi_{ij}} . d. Apply an initial threshold: eliminate terms where ∣CPij∣<λCP,0|CP_{ij}|<\lambda_{CP,0} a. Create ramped schedules for the ridge penalty (from γ1\gamma_{1} to γnsteps\gamma_{n_{steps}} = γf\gamma_{f} ) and the CP threshold (from λCP,1\lambda_{CP,1} to λCP,nsteps\lambda_{CP,{n_{steps}}} = λCP,f\lambda_{CP,f} ). 3. Iterative Refinement: b. For each step kk , from 1 to nstepsn_{steps} : i. Set current ridge penalty, γk\gamma_{k} , and CP threshold, λCP,k\lambda_{CP,k} . ii. Inner Loop: While the model sparsity is changing: * Fit a new model, Ξk\boldsymbol{\Xi_{k}} , using STLSQ on the currently active terms with ridge penalty γk\gamma_{k} . * Estimate coefficient standard deviations using BLR. * Calculate CP values for all active terms. * Eliminate terms where ∣CPij∣<λCP,k|CP_{ij}|<\lambda_{CP,k} * Check for convergence (i.e., if the set of active terms has changed). 4. Return Ξ=Ξnsteps\boldsymbol{\Xi}=\boldsymbol{\Xi_{n_{steps}}} .
Table 1: The Sequential Thresholding of Coefficient of Variation (STCV) Algorithm
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.
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.
Figure 6: Measured and integrated data of the physical linear (left) and nonlinear (right) oscillating system.
MethodLinear systemNonlinear system
Expecteds˙=+1.0vv˙=−1642s−0.143v\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-1642s-0.143v\end{aligned}Unknown
STLSQs˙=+1.0vv˙=−1519.43s−2.53v−363.36s2+4.75sv−218332.31s3−191.22s2v−228.92sv2+2.99v3\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}s˙=+1.0vv˙=−17.541−1653.2s−20.92v−41718.39s2+309.9sv−12.02v2−218332.31s3+12314.87s2v+949.39sv2+64.41v3\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-SINDys˙=+1.0vv˙=−1519.43s−2.53v−363.36s2−218332.31s3+191.22s2v−228.92sv2+2.99v3\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}s˙=+1.0vv˙=−17.461−1567.36s−20.69v−39480.1s2−9.45v2−429447.2s3+5187.76s2v+742.63sv2+50.02v3\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}
STCVs˙=+1.0vv˙=−1564.26s−1.93v\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-1564.26s-1.93v\end{aligned}s˙=+1.0vv˙=−17.651−1360.26s−9.16v−33094.72s2−371947.09s3\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-17.651-1360.26s-9.16v-33094.72s^{2}\\ &\quad-371947.09s^{3}\end{aligned}
STCV  
-STSLQs˙=+1.0vv˙=−1564.26s−1.93v\begin{aligned} \dot{s}&=+1.0v\\ \dot{v}&=-1564.26s-1.93v\end{aligned}s˙=+1.0vv˙=−17.651−1360.26s−9.16v−33094.72s2−371947.09s3\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 k1k_{1} and k2k_{2} , are produced. The values of k1k_{1} and k2k_{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 k1k_{1} present), a linear-nonlinear model (both k1k_{1} and k2k_{2} present), or a nonlinear model (only k2k_{2} present). k1k_{1} and k2k_{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

SystemODEIntegratorInitial valueSampling ratePeriodTolerance
LorenzRef. \cite[cite][\@@bibrefref41]\begin{aligned} Ref.~\cite[cite]{[\@@bibref{}{ref41}{}{}]}\\ \end{aligned}RK45\begin{aligned} RK45\\ \end{aligned}\begin{aligned} \\ \end{aligned}100Hz\begin{aligned} 100Hz\\ \end{aligned}10s\begin{aligned} 10s\\ \end{aligned}10−12\begin{aligned} 10^{-12}\\ \end{aligned}
RösslerRef. \cite[cite][\@@bibrefref41]\begin{aligned} Ref.~\cite[cite]{[\@@bibref{}{ref41}{}{}]}\\ \end{aligned}RK45\begin{aligned} RK45\\ \end{aligned}\begin{aligned} \\ \end{aligned}1kHz\begin{aligned} 1kHz\\ \end{aligned}10s\begin{aligned} 10s\\ \end{aligned}10−12\begin{aligned} 10^{-12}\\ \end{aligned}
van der PolRef. \cite[cite][\@@bibrefref41]\begin{aligned} Ref.~\cite[cite]{[\@@bibref{}{ref41}{}{}]}\\ \end{aligned}LSODA\begin{aligned} LSODA\\ \end{aligned}\begin{aligned} \\ \end{aligned}100Hz\begin{aligned} 100Hz\\ \end{aligned}30s\begin{aligned} 30s\\ \end{aligned}default\begin{aligned} default\\ \end{aligned}
DuffingRef. \cite[cite][\@@bibrefref41]\begin{aligned} Ref.~\cite[cite]{[\@@bibref{}{ref41}{}{}]}\\ \end{aligned}LSODA\begin{aligned} LSODA\\ \end{aligned}\begin{aligned} \\ \end{aligned}100Hz\begin{aligned} 100Hz\\ \end{aligned}10s\begin{aligned} 10s\\ \end{aligned}default\begin{aligned} default\\ \end{aligned}
BearingEquation \refeqn:bearing\begin{aligned} Equation~\ref{eqn:bearing}\\ \end{aligned}RK45\begin{aligned} RK45\\ \end{aligned}\begin{aligned} \\ \end{aligned}1MHz\begin{aligned} 1MHz\\ \end{aligned}5μs\begin{aligned} 5\mu s\\ \end{aligned}10−12\begin{aligned} 10^{-12}\\ \end{aligned}
Linear HCRef. \cite[cite][\@@bibrefref42]\begin{aligned} Ref.~\cite[cite]{[\@@bibref{}{ref42}{}{}]}\\ \end{aligned}Radau\begin{aligned} Radau\\ \end{aligned}\begin{aligned} \\ \end{aligned}5kHz\begin{aligned} 5kHz\\ \end{aligned}2.12s\begin{aligned} 2.12s\\ \end{aligned}10−12\begin{aligned} 10^{-12}\\ \end{aligned}
Appendix
Table 3: Numerical integration parameters for dynamical systems.
SystemCP threshold rangeRidge penalty (STCV-STLSQ)
Lorenz\begin{aligned} \\ \end{aligned}10−1\begin{aligned} 10^{-1}\\ \end{aligned}
Lorenz (normalised)\begin{aligned} \\ \end{aligned}10−1\begin{aligned} 10^{-1}\\ \end{aligned}
Rössler\begin{aligned} \\ \end{aligned}10−1\begin{aligned} 10^{-1}\\ \end{aligned}
Rössler (normalised)\begin{aligned} \\ \end{aligned}10−5\begin{aligned} 10^{-5}\\ \end{aligned}
van der Pol\begin{aligned} \\ \end{aligned}10−1\begin{aligned} 10^{-1}\\ \end{aligned}
van der Pol (normalised)\begin{aligned} \\ \end{aligned}10−1\begin{aligned} 10^{-1}\\ \end{aligned}
Appendix
Table 4: STCV hyperparameters used for the benchmarking experiment.
MethodLinear systemNonlinear system
STLSQγ=10−16λ=0.9\begin{aligned} \gamma&=10^{-16}\\ \lambda&=0.9\\ \end{aligned}γ=10−16λ=0.9\begin{aligned} \gamma&=10^{-16}\\ \lambda&=0.9\\ \end{aligned}
E-SINDyγ=10−16λ=0.9Incl.thres.=(1−10−16)\begin{aligned} \gamma&=10^{-16}\\ \lambda&=0.9\\ Incl.thres.&=(1-10^{-16})\end{aligned}γ=10−16λ=0.9Incl.thres.=(1−10−16)\begin{aligned} \gamma&=10^{-16}\\ \lambda&=0.9\\ Incl.thres.&=(1-10^{-16})\end{aligned}
STCVγ0=10−16γf=10−16λ=0.01CP=0.3\begin{aligned} \gamma_{0}&=10^{-16}\\ \gamma_{f}&=10^{-16}\\ \lambda&=0.01\\ CP&=0.3\\ \end{aligned}γ0=10−16γf=10−16λ=0.01CP=0.4\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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