An Introduction to Sparse Identification of Nonlinear Dynamics for Engineering Applications

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.

Weak Dominant Balance for Robust Identification of Dynamically Consistent Fluid Flow Structure

Extracting interpretable, localized physical mechanisms from complex spatiotemporal data is a foundational challenge across physics, biology, and engineering, but has remained out of reach on real measurements. The central obstacle is obtaining high-quality gradients of data via numerical differentiation, which amplifies noise, diverges for high-order equations, and falters on irregular geometries, limiting the scope of existing approaches to clean simulations of low-order systems. Here, we present weak dominant balance, a derivative-free framework that projects governing equations into a weak (integral) formulation, offloading differentiation onto smooth analytical test functions and leaving the data untouched. The method sustains accurate regime identification under severe noise where existing approaches categorically fail, delivers the first data-driven decomposition of a third-order partial differential equation applied to turbulent duct flow, and produces matching decompositions across direct numerical simulation and particle-image velocimetry measurements of a wavy channel flow, uncovering a previously uncharacterized dynamical regime. Weak dominant balance brings mechanism-level analysis out of simulation and onto measured data, and opens complex physical systems to direct, equation-grounded interpretation.

Multi-Fidelity SINDy: Sparse Discovery of Nonlinear Dynamical Systems with Fidelity-Weighted Measurements

Data from simulations and experiments are rarely noise-free and often exhibit heterogeneous levels of fidelity. Measurement uncertainty may vary across repeated observations, sensing devices, or even within a single experiment. This work addresses the problem of discovering nonlinear dynamical systems from such inhomogeneous data. We extend the Sparse Identification of Nonlinear Dynamical Systems (SINDy) framework to account for variable noise levels by combining Ensemble SINDy and Weak SINDy within a weighted regression formulation derived from generalized least squares. A statistical justification for the weighting strategy is also provided. The methodology is validated on several benchmark systems, including ordinary and partial differential equations. In addition, we show the benefit of multi-fidelity integration for forecasting the dynamics of a double pendulum system. The results confirm that the proposed approach mitigates the adverse effects of heteroscedastic noise and that repeated, low-cost, low-quality measurements can improve model recovery, in some cases matching or outperforming reconstructions obtained using only high-fidelity data.

How Low Can You Go? Active Learning for Sparse Model Discovery in the Ultra-Low-Data Limit

Identifying the governing equations of complex dynamical systems remains a fundamental challenge across science and engineering. While early approaches relied on empirical data and heuristics, modern data-driven methods offer greater flexibility and fewer assumptions. However, data acquisition in real-world settings is often expensive. This work addresses this challenge by introducing an active learning strategy for dynamics discovery in the ultra-low data limit. Rather than sampling randomly, our method iteratively prioritizes regions that are most informative for model identification. This approach builds on Sparse Identification of Nonlinear Dynamics (SINDy), and utilizes an ensemble extension, E-SINDy, to estimate epistemic uncertainty and guide the sampling for both ordinary and partial differential equations (ODEs/PDEs). For ODEs, an exhaustive analysis is conducted on the Lorenz system across varying data budgets and noise levels. For PDEs, two systems with contrasting dynamical characteristics are examined: the Burgers' equation, where a sharp shock front creates a distinction between informative and uninformative regions, and the Kuramoto-Sivashinsky equation, which presents a more spatially complex sampling landscape. Across all scenarios, the proposed method accurately identifies the governing dynamics with significantly fewer data samples than random sampling.

Uncovering Extreme Event Mechanisms for Prediction and Control with Sensitivity-Balanced Projections

Extreme events -- such as earthquakes and coronal mass ejections -- are common in many chaotic dynamical systems, yet are difficult to characterize and predict due to the subtle instability mechanisms that drive them. In this work, we develop an interpretable technique that reveals the underlying mechanisms behind extreme events and uses them to build data-driven forecasts and intuitive event suppression controllers. In particular, we utilize the covariance balancing reduction using adjoint snapshots (CoBRAS) method to identify linear oblique projections that best capture the sensitivity of a quantity of interest and reconstruct the original state. Importantly, we bypass the need for cumbersome adjoint calculations, instead using backpropagation via modern automatically differentiable numerical frameworks. To accommodate spatially localized events, we also introduce a new variant of CoBRAS to obtain local sensitivity-balanced projections. We demonstrate the utility of this approach to characterize extreme events across a diverse set of challenging systems, including turbulent bursts of energy dissipation in the 2D Kolmogorov Flow, spontaneous synchronization in networks of coupled FitzHugh-Nagumo oscillators, and the localized formation of ocean rogue waves from a modified nonlinear Schrödinger equation. For each example, we show that our simple forecast models accurately predict extreme events and that the underlying mechanisms may be used to design control laws to prevent these events. Finally, we demonstrate that by learning a neural network surrogate model of the dynamics directly from data, we may extend this approach to experimental systems and systems that are not natively written in an automatically differentiable programming language.

Learning Individual Dynamics from Sparse Cross-Sectional Snapshots

Predicting how a dynamical unit evolves over time - how an individual ages, an epidemic spreads, or a physical system degrades - typically requires dense longitudinal tracking. When only extremely sparse or entirely cross-sectional data is available, inferring individualized, continuous-time trajectories is fundamentally ill-posed. Existing methods force a strict compromise: sequence models (e.g. latent ODEs) require dense longitudinal data, while cross-sectional methods (e.g. optimal transport, flow matching-based) map aggregate populations, losing individual dynamics. In this paper, we demonstrate that this dichotomy can be broken. We introduce CADENCE, a principled probabilistic framework that recovers continuous individual trajectories from isolated snapshots by anchoring latent dynamics to static, individual-level contexts. We provide novel identifiability guarantees for single-timepoint trajectory inference. By combining a score-based spatial encoder (bijective Probability Flow ODE) to eliminate diffeomorphic ambiguities with a Soft Mixture-of-Experts (SMoE) router, we show that individual dynamical parameters and routing function are jointly identifiable. Across a suite of benchmarks spanning physical systems to real-world biological data, CADENCE, trained strictly on extremely sparse snapshots with context structure, matches or exceeds the performance of state-of-the-art sequential models trained on dense, full-trajectory data.

AeroJEPA: Learning Semantic Latent Representations for Scalable 3D Aerodynamic Field Modeling

Aerodynamic surrogate models are increasingly used to replace repeated high-fidelity CFD evaluations in many-query design settings, but current approaches still face two important limitations: they often scale poorly to the very large fields arising in realistic 3D aerodynamics, and they rarely produce latent representations that are directly useful for analysis and design. We introduce AeroJEPA, a Joint-Embedding Predictive Architecture for aerodynamic field modeling that addresses both issues. Rather than predicting the full flow field directly from geometry, AeroJEPA predicts a target latent representation of the flow from a context latent representation of the geometry and operating conditions, and optionally reconstructs the field through a continuous implicit decoder. This formulation decouples latent prediction from field resolution while encouraging the latent space to organize semantically. We evaluate AeroJEPA on two complementary datasets: HiLiftAeroML, which stresses the method in a high-fidelity regime with extremely large boundary-layer fields, and SuperWing, which tests large-scale generalization and latent-space optimization over a broad family of transonic wings. Across these benchmarks, AeroJEPA is competitive as a continuous surrogate for aerodynamic fields, scales naturally to high-resolution outputs, and learns context and predicted latents that encode geometry and aerodynamic quantities not used directly as supervision. We further show that the resulting latent space supports controlled interpolation, linear probing, concept-vector arithmetic, and a constrained design latent-optimization experiment. These results suggest that predictive latent learning is a promising direction for scalable and design-meaningful aerodynamic surrogate modeling.

SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networks

Kolmogorov-Arnold networks (KANs) have arisen as a potential way to enhance the interpretability of machine learning. However, solutions learned by KANs are not necessarily interpretable, in the sense of being sparse or parsimonious. Sparse identification of nonlinear dynamics (SINDy) is a complementary approach that allows for learning sparse equations for dynamical systems from data; however, learned equations are limited by the library. In this work, we present SINDy-KANs, which simultaneously train a KAN and a SINDy-like representation to increase interpretability of KAN representations with SINDy applied at the level of each activation function, while maintaining the function compositions possible through deep KANs. We apply our method to a number of symbolic regression tasks, including dynamical systems, to show accurate equation discovery across a range of systems.

OpenReservoirComputing: GPU-Accelerated Reservoir Computing in JAX

OpenReservoirComputing (ORC) is a Python library for reservoir computing (RC) written in JAX (Bradbury et al. 2018) and Equinox (Kidger and Garcia 2021). JAX is a Python library for high-performance numerical computing that enables automatic differentiation, just-in-time (JIT) compilation, and GPU/TPU acceleration, while Equinox is a neural network framework for JAX. RC is a form of machine learning that functions by lifting a low-dimensional sequence or signal into a high-dimensional dynamical system and training a simple, linear readout layer from the high-dimensional dynamics back to a lower-dimensional quantity of interest. The most common application of RC is time-series forecasting, where the goal is to predict a signal's future evolution. RC has achieved state-of-the-art performance on this task, particularly when applied to chaotic dynamical systems. In addition, RC approaches can be adapted to perform classification and control tasks. ORC provides both modular components for building custom RC models and built-in models for forecasting, classification, and control. By building on JAX and Equinox, ORC offers GPU acceleration, JIT compilation, and automatic vectorization. These capabilities make prototyping new models faster and enable larger and more powerful reservoir architectures. End-to-end differentiability also enables seamless integration with other deep learning models built with Equinox.

Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds

Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant, so closure failures and projection errors lead to spurious eigenvalues, misleading Koopman modes, and overconfident forecasts. This paper addresses a central validation problem in data-driven Koopman methods: how to quantify invariance and projection errors for an arbitrary feature space using only snapshot data, and how to use these diagnostics to produce actionable guarantees and guide dictionary refinement? A unified a posteriori methodology is developed for certifying when a Koopman approximation is trustworthy and improving it when it is not. Koopman invariance is quantified using principal angles between a subspace and its Koopman image, yielding principal observables and a principal angle decomposition (PAD), a dynamics-informed alternative to SVD truncation with significantly improved performance. Multi-step error bounds are derived for Koopman and Perron--Frobenius mode decompositions, including RKHS-based pointwise guarantees, and are complemented by Gaussian process expected error surrogates. The resulting toolbox enables validated spectral analysis, certified forecasting, and principled dictionary and kernel learning, demonstrated on chaotic and high-dimensional benchmarks and real-world datasets, including cavity flow and the Pluto--Charon system.