Sparse Identification of Nonlinear Dynamics

• Sparse Identification of Nonlinear Dynamics is a data-driven framework that infers parsimonious governing equations by applying sparse regression to a curated library of nonlinear functions.
• It systematically recovers interpretable models with a minimal number of active terms, revealing the structural mechanisms behind complex dynamical systems.
• Extensions like SINDYc incorporate control inputs and feedback, ensuring robust performance and noise resilience in both autonomous and forced systems.

Sparse identification of nonlinear dynamics (SINDy) is a family of data-driven methodologies for systematically inferring parsimonious governing equations of complex dynamical systems from time-series measurements. By leveraging sparse regression over a curated library of candidate nonlinear functions, SINDy recovers interpretable models with a minimal number of active terms, often revealing the structural mechanisms underlying the observed evolution. The framework has been extended to accommodate control inputs (SINDYc), side information and physical constraints (SINDy-SI), structure-preserving formalisms, high noise robustness, optimal library selection, industrial uncertainty quantification, efficient computation for high-dimensional settings, boundary value problems, mixed-integer optimization, and more.

1. Mathematical Foundations and Core SINDy Algorithm

In the core SINDy formulation, consider a continuous-time autonomous system

$\frac{dx}{dt} = f(x),\qquad x(t)\in\mathbb{R}^n,$

where $f(x)$ is a generally unknown and potentially nonlinear map. Given $m$ time-series state measurements $X = [x(t_1),\dots,x(t_m)]\in\mathbb{R}^{n\times m}$ and corresponding time derivatives $\dot X = [\dot x(t_1),\dots,\dot x(t_m)]$, one constructs a library $\Theta(X)\in\mathbb{R}^{p\times m}$, composed of nonlinear basis functions (e.g., $x_k$, $x_k^2$, $x_k x_\ell$, $x_k^3$, $\sin x_k$, etc.), where $p$ denotes the number of candidate features.

The identification problem is then posed as a sparse regression: $\dot X \approx \Xi\,\Theta(X),$ where $\Xi\in\mathbb{R}^{n\times p}$ contains the coefficients for each basis function in each equation. Sparsity is imposed via convex regularization (LASSO): $$\xi_k = \arg\min_\xi\,\|\dot X_k - \xi\,\Theta(X)\|_2^2 + \alpha \|\xi\|_1,$$ or, more efficiently, a sequential thresholded least-squares scheme:

• Initialize $\Xi$ by standard least squares.
• Zero all entries with $|\Xi_{ij}| < \lambda$.
• Re-solve least squares on the reduced support, iterating to convergence.

This yields compact ODE models in which only a minimal subset of candidate functions are retained, providing both interpretability and predictive fidelity (Brunton et al., 2016).

2. Extensions for Control, Inputs, and Feedback: SINDYc

Generalization to systems with control or forcing requires modification of both the library and regression procedure. For dynamics

$\frac{dx}{dt} = f(x, u),$

where $u(t)\in\mathbb{R}^d$ denotes control or input variables, the data snapshots $U = [u(t_1),\cdots,u(t_m)]$ are incorporated into an extended library

$$\Theta_c(X,U) = [\, \Theta(X);\, \Upsilon(U);\, \text{cross}(x,u) ],$$

including monomials, input functions, and cross-terms between state and input. The regression becomes

$\dot X \approx \Xi_c\,\Theta_c(X,U),$

with $\Xi_c\in\mathbb{R}^{n\times p_c}$, fit by sparse regression or sequential thresholded least squares.

Feedback control $u = k(x)$ may render the identification ill-conditioned due to shared dependencies. SINDYc partitions the problem:

• Fit $U \approx \Xi_u\,\Theta(X)$ for the control law.
• Inject small noise or perturbations $d(t)$ into $u$ during identification (i.e., $u = k(x) + d(t)$) to disentangle control from intrinsic system feedback.
• Apply distinct regularization parameters for dynamics and control sparsity.

This approach enables flawless identification of known nonlinear systems with external forcing (Lotka-Volterra with $u^2$ term, Lorenz with cubic forced input) and state-feedback control laws, achieving coefficient recovery to machine precision in noise-free data, and robust predictive accuracy on unseen validation forcing functions (Brunton et al., 2016). Selection of appropriate basis functions, threshold parameters, and excitation strategies is crucial.

3. Connections to Dynamic Mode Decomposition and Koopman Operator Theory

SINDy is tightly linked with operator-based data-driven modeling frameworks:

• Standard DMD emerges as a special case with a linear-only library ($\Theta(X)=X$), yielding linear maps $X' \approx A X$.
• DMDc generalizes to linear state and input libraries: $X' \approx [A\, B][X ; U]$.
• Koopman operator theory interprets SINDy as a finite-dimensional, typically sparse, approximation of the generator $L$ acting on (possibly nonlinear) observables $y(x)$, since $d y(x)/dt = L y(x)$.

These relationships provide theoretical grounding and facilitate cross-fertilization between regression-based and operator-based identification methods, including extensions toward eDMD and SINDYc (Brunton et al., 2016).

4. Practical Implementation Strategies and Performance Guidelines

Best practices for practical deployment are detailed:

• Library selection: Initiate with low-order polynomials and domain-specific candidate functions, iteratively expand to include cross-terms only if residual analysis indicates necessity.
• Derivative estimation: Employ regularized methods, specifically total variation regularization, to avoid amplification of measurement noise.
• Threshold tuning: Perform coarse-to-fine sweeps (Pareto front search) over the regularization parameter to optimize the trade-off between sparsity and validation error.
• Feedback/control identification: Guarantee sufficient excitation via additive noise or impulsive inputs to decouple control effects from internal nonlinear feedback.
• Weighted penalties: For systems in which state and input variables exhibit divergent sparsity profiles, adjust $\ell_1$ regularization weights correspondingly.

These procedural recommendations ensure robust, generalizable model discovery, especially in scenarios with limited data or nontrivial control interactions.

5. Representative Examples

Two instructive examples illustrate the method:

• Lotka-Volterra with $u^2$ forcing: SINDYc perfectly recovers $a$, $b$, $c$, $d$ coefficients, correctly predicts system attractor under validation forcing; naive SINDy (omitting $u$) catastrophically fails to reproduce attractor dynamics.
• Lorenz system with cubic forcing and state-feedback: SINDYc extracts both internal chaotic nonlinearities and external cubic forcing, and, with feedback excitation and a proper library, simultaneously disentangles nonlinear system terms and feedback law. Validation against independent forcing functions confirms high-fidelity extrapolation.

6. Limitations, Interaction with Other SINDy Extensions, and Future Directions

While SINDYc and its core framework have demonstrated reliability in controlled and noiseless regimes, several limitations persist:

• Library construction: Erroneous or incomplete basis function choices may impede correct model recovery.
• Noise sensitivity: Though methods exist for mitigating noise amplification (total variation, cross-validation), SINDy can degrade under heavy sensor noise or incomplete measurements; weak-form integral SINDy and automatic-differentiation approaches have been proposed to address these issues.
• Identification under feedback ambiguity: Poor excitation or ill-conditioned regression (overlapping basis terms) can negatively affect control law recovery.
• Scaling to high-dimensional systems: Iterative and bagged SINDy variants (library pruning, basis selection, ensemble learning) address computational and statistical limitations for large systems.

Ongoing research seeks to further enhance SINDy for constraints, side information (SINDy-SI via SOS certificates (Machado et al., 2023)), structured learning (Hamiltonian, GENERIC (Lee et al., 2021)), boundary-value and differential-operator identification (SINDY-BVP (Shea et al., 2020)), uncertainty quantification (conformal prediction (Fasel, 15 Jul 2025)), robust regression (TRIM, mixed-integer (Kiser et al., 2023, Bertsimas et al., 2022)), and high-noise environments (reweighted $\ell_1$, automatic differentiation, weak form (Cortiella et al., 2020, Kaheman et al., 2020, López et al., 23 Oct 2024)).

Table: Algorithmic Ingredients of SINDY/SINDYc

Component	Functionality	Practical Advisory
Candidate Library	Encodes possible nonlinearities	Begin with polynomials, expand as needed
Sparse Regression	Enforces parsimony of dynamics	Use STLSQ, LASSO, weighted $\ell_1$, etc.
Control Extension	Models forced or feedback systems	Augment library and fit multiple equations
Derivative Estimation	Estimates time rates from noisy data	Use total variation regularization
Threshold/Regularizer	Balances error vs. model sparsity	Sweep $\lambda$ or $\alpha$; Pareto knee selection
Feedback Decoupling	Distinguishes intrinsic/control terms	Inject input noise, parameterize control law

7. Context and Impact

SINDy and SINDYc have enabled widespread progress in interpretable system identification and data-driven modeling. Their utility in deciphering underlying physical laws, constructing digital twins, inferring control laws, and providing mechanistic insights into complex systems—while operating under realistic data constraints—continues to be expanded by ongoing methodological developments.