We will prove by the induction. Let's suppose that the formula holds for k up to n. We will prove that this formula also holds for k = n+1. By the definition in Eq. 4 and the chain rule, we can get that: Ns,n+1(t) = t τs A.2 Spline representation In this section, we give error bounds for spline representation. For simplicity, we consider 1D scenario and assume the target function u: [0,1] R is periodic and defined on the unit interval Ω = [0,1].

While impressive, most of these tasks can be easily performed by an intelligent human.

In this work, we revisit dictionary-based sparse regression, in particular, Sequential Threshold Least Squares (STLS), and propose a score-guided library selection to provide practical guidance for data-driven modeling, with emphasis on SINDy-type algorithms. STLS is an algorithm to solve the $\ell_0$ sparse least-squares problem, which relies on splitting to efficiently solve the least-squares portion while handling the sparse term via proximal methods. It produces coefficient vectors whose components depend on both the projected reconstruction errors, here referred to as the scores, and the mutual coherence of dictionary terms. The first contribution of this work is a theoretical analysis of the score and dictionary-selection strategy. This could be understood in both the original and weak SINDy regime. Second, numerical experiments on ordinary and partial differential equations highlight the effectiveness of score-based screening, improving both accuracy and interpretability in dynamical system identification. These results suggest that integrating score-guided methods to refine the dictionary more accurately may help SINDy users in some cases to enhance their robustness for data-driven discovery of governing equations.

Learning governing equations from data is central to understanding the behavior of physical systems across diverse scientific disciplines, including physics, biology, and engineering. The Sindy algorithm has proven effective in leveraging sparsity to identify concise models of nonlinear dynamical systems. In this paper, we extend sparsity-driven approaches to real-time learning by integrating a cornerstone algorithm from control theory -- the Kalman filter (KF). The resulting Sindy Kalman Filter (SKF) unifies both frameworks by treating unknown system parameters as state variables, enabling real-time inference of complex, time-varying nonlinear models unattainable by either method alone. Furthermore, SKF enhances KF parameter identification strategies, particularly via look-ahead error, significantly simplifying the estimation of sparsity levels, variance parameters, and switching instants. We validate SKF on a chaotic Lorenz system with drifting or switching parameters and demonstrate its effectiveness in the real-time identification of a sparse nonlinear aircraft model built from real flight data.

In this paper, we propose a novel data-driven framework for discovering probabilistic laws underlying the Feynman-Kac formula. Specifically, we introduce the first stochastic SINDy method formulated under the risk-neutral probability measure to recover the backward stochastic differential equation (BSDE) from a single pair of stock and option trajectories. Unlike existing approaches to identifying stochastic differential equations-which typically require ergodicity-our framework leverages the risk-neutral measure, thereby eliminating the ergodicity assumption and enabling BSDE recovery from limited financial time series data. Using this algorithm, we are able not only to make forward-looking predictions but also to generate new synthetic data paths consistent with the underlying probabilistic law.

Data-driven modeling of nonlinear dynamical systems is often hampered by measurement noise. We propose a denoising framework, called Runge-Kutta and Total Variation Based Implicit Neural Representation (RKTV-INR), that represents the state trajectory with an implicit neural representation (INR) fitted directly to noisy observations. Runge-Kutta integration and total variation are imposed as constraints to ensure that the reconstructed state is a trajectory of a dynamical system that remains close to the original data. The trained INR yields a clean, continuous trajectory and provides accurate first-order derivatives via automatic differentiation. These denoised states and derivatives are then supplied to Sparse Identification of Nonlinear Dynamics (SINDy) to recover the governing equations. Experiments demonstrate effective noise suppression, precise derivative estimation, and reliable system identification.

The Sparse Identification of Nonlinear Dynamics (SINDy) is a method for discovering nonlinear dynamical system models from data. Quantifying uncertainty in SINDy models is essential for assessing their reliability, particularly in safety-critical applications. While various uncertainty quantification methods exist for SINDy, including Bayesian and ensemble approaches, this work explores the integration of Conformal Prediction, a framework that can provide valid prediction intervals with coverage guarantees based on minimal assumptions like data exchangeability. We introduce three applications of conformal prediction with Ensemble-SINDy (E-SINDy): (1) quantifying uncertainty in time series prediction, (2) model selection based on library feature importance, and (3) quantifying the uncertainty of identified model coefficients using feature conformal prediction. We demonstrate the three applications on stochastic predator-prey dynamics and several chaotic dynamical systems. We show that conformal prediction methods integrated with E-SINDy can reliably achieve desired target coverage for time series forecasting, effectively quantify feature importance, and produce more robust uncertainty intervals for model coefficients, even under non-Gaussian noise, compared to standard E-SINDy coefficient estimates.

Discovering symbolic differential equations from data uncovers fundamental dynamical laws underlying complex systems. However, existing methods often struggle with the vast search space of equations and may produce equations that violate known physical laws. In this work, we address these problems by introducing the concept of \textit{symmetry invariants} in equation discovery. We leverage the fact that differential equations admitting a symmetry group can be expressed in terms of differential invariants of symmetry transformations. Thus, we propose to use these invariants as atomic entities in equation discovery, ensuring the discovered equations satisfy the specified symmetry. Our approach integrates seamlessly with existing equation discovery methods such as sparse regression and genetic programming, improving their accuracy and efficiency. We validate the proposed method through applications to various physical systems, such as fluid and reaction-diffusion, demonstrating its ability to recover parsimonious and interpretable equations that respect the laws of physics.

Many model selection algorithms rely on sparse dictionary learning to provide interpretable and physics-based governing equations. The optimization algorithms typically use a hard thresholding process to enforce sparse activations in the model coefficients by removing library elements from consideration. By introducing an annealing scheme that reactivates a fraction of the removed terms with a cooling schedule, we are able to improve the performance of these sparse learning algorithms. We concentrate on two approaches to the optimization, SINDy, and an alternative using hard thresholding pursuit. We see in both cases that annealing can improve model accuracy. The effectiveness of annealing is demonstrated through comparisons on several nonlinear systems pulled from convective flows, excitable systems, and population dynamics. Finally we apply these algorithms to experimental data for projectile motion.