Sparse Identification of Nonlinear Dynamics

Also known as SINDy

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2 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

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Latest papers 24

Sep 29, 2026cs.LG

Identifying ODEs from Unstructured Data with Causal Representation Learning

We study the problem of recovering the governing ODE of a dynamical system from unstructured, high-dimensional observations such as images. Existing methods for ODE discovery typically assume direct measurements of the variables, or do not provide theoretical guarantees on the learned variables and equations. While Causal Representation Learning (CRL) methods provide guarantees on identifying variables from high-dimensional observations up to component-wise diffeomorphisms, we show that in general these variables cannot be used directly as input to equation discovery methods, which typically assume that the variables will lead to sparse equations. So we introduce SParse Equivalent Equation Discovery AutoEncoder (SPEED-AE), a framework that combines a pretrained CRL method with a component-wise autoencoder that learns transformations of variables that are amenable to sparse ODE discovery. We show that for polynomial ODEs, this additional step allows us to restrict the identifiability of each variable from polynomial to monomial diffeomorphisms. Experiments on Lotka-Volterra, Lorenz, and a two-pendulum system show that SPEED-AE improves on the disentanglement of the CRL methods and that it recovers ODEs that are closest to the ground truth, while achieving state-of-the-art forecasting performance.

Alessandro Trenta, Riccardo Massidda, Davide Bacciu +1

Department of Computer Science University of Pisa · Saarland Informatics Campus Saarland University & University of Amsterdam

Causal Representation Learning Representation Identifiability Sparse Identification of Nonlinear Dynamics ODE Discovery

Sep 8, 2026cs.LG

Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamics

Weak Sparse Identification of Nonlinear Dynamics (WSINDy) provides a noise-robust approach for learning dynamical systems from data without requiring numerical differentiation. However, for high-dimensional systems, tensor-product libraries of candidate functions grow exponentially with the state dimension, making standard WSINDy expensive in both computation and memory. The Multidimensional Approximation of Nonlinear Dynamics (MANDy) addresses this scaling through a tensor-train (TT) representation of the candidate library, but does not provide a mechanism for sparse model selection. Here, we combine these approaches to develop TT-WSINDy, which performs the weak-form transformation, regression, and sparsification in TT format. We show that the TT formulation recovers the corresponding WSINDy regression problem and derive polynomial time and memory complexity bounds for the tensor-train sparsification procedure. Numerical experiments demonstrate robustness to measurement noise and computational savings for high-dimensional systems.

Will Houser, Vanja Dukic, David M. Bortz

Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526 USA

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification

Aug 13, 2026cs.LG

Sparse Orthogonal Regression Technique: A Spectral Framework for Equation Discovery, Approximation, and Integration

We develop the Sparse Orthogonal Regression Technique (SORT), a sparse spectral framework for learning orthonormal-basis expansions from noisy and irregularly sampled data. SORT estimates expansion coefficients directly from observations using L1-regularized regression, avoiding explicit quadrature or analytic inner-product evaluation. The central application is data-driven discovery of ordinary differential equations: vector fields are represented in chosen orthogonal bases and learned as sparse coefficient expansions. This provides a complementary route to symbolic regression, grammar-based discovery, and SINDy-style sparse identification by first recovering a compact spectral representation, which can later guide searches for simpler analytic forms. Across the dynamical-system experiments, SORT matches or improves upon library-based sparse-regression baselines when the basis is well adapted to the problem, and shows more stable degradation under sparse sampling, noisy derivative estimates, and representation mismatch. Specific examples illustrate why this representation is useful: if a finite library misses the problem-specific nonlinearity, the resulting model can fail. SORT is not immune to mismatch, but it shifts the problem away from brittle selection among generic terms to basis design adapted to the problem domain. The experiments also show that dominant low-order coefficients persist as model order increases, supporting order-consistent model growth. Beyond equation discovery, the same learned expansion supports nonlinear approximation and estimation of complex, high-dimensional integrals by coefficient readout. Overall, SORT provides a reusable intermediate representation for system identification, approximation, and integration, while making basis design an explicit part of the scientific modeling problem.

Sabin Roman, Ljupco Todorovski, Saso Dzeroski

Department of Knowledge Technologies, Jožef Stefan Institute · Faculty of Mathematics and Physics, University of Ljubljana

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Orthogonal Representation Learning ODE Discovery

Aug 4, 2026cond-mat.soft

Data Driven Equation Discovery for Phase-Ordering Dynamics : From Allen Cahn to the Ising Model

Data-driven discovery of governing equations from spatiotemporal data offers a promising route to obtaining coarse-grained descriptions of complex dynamical systems. Here, we investigate the performance of PDE-SINDy for discovering phase-ordering dynamics using the Allen--Cahn equation as a benchmark and the Ising model with Glauber spin-flip dynamics as a microscopic system. We systematically analyze the effects of data availability, size of the candidate library, and noise on the efficiency of the equation discovery. We find that stability-selection PDE-SINDy can robustly identify the relevant terms in the governing dynamics even under limited or noisy data, while the recovered coefficient values are substantially more sensitive to these factors. We further show that enlarging the candidate library can strongly affect both term identification and coefficient recovery. Incorporating library bagging with stability selection reduces this sensitivity and improves the efficiency of equation discovery. For the Glauber spin flip Ising model dynamics, the resulting coarse-grained equation reproduces the characteristic phase-separation and coarsening dynamics of the underlying microscopic system. Overall, our results demonstrate the potential of PDE-SINDy for phase-ordering systems while highlighting the importance of carefully assessing the factors that influence the efficiency of equation discovery.

Partha Sarathi Mondal, Manav Kumar Jalan, Anish Kumar +1

Department of Physics, Indian Institute of Technology (BHU) Varanasi, India 221005

Sparse Identification of Nonlinear Dynamics PDE Discovery Phase Transitions Ising Model

Jul 31, 2026cs.LG

Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations

PDE discovery from sparse observations requires reconstructing a continuous field and selecting the correct differential terms. Our analysis of optimization paths in coupled neural PDE discovery reveals three behaviors: the exact support can persist to the end of training, appear only transiently, or fail to emerge. To decouple equation selection from neural optimization, we develop a freeze-then-select method combining a structured field adapter with Stability-Validated Weak Selection (SVWS). Trained from observations without a PDE residual, the adapter factorizes the field into learned spatial features and temporal coefficients represented by cubic splines. After freezing the field, SVWS identifies recurrent terms across independent weak-form systems, refits candidate supports, and selects the final equation on held-out weak-form systems. Beyond fixed libraries, we apply the same principle to expressions generated by genetic programming and recover the power-law form of an unknown nonlinear diffusion function from sparse, noisy observations. Across all six sparse MDBench regimes, our method attains the highest exact support recovery rate, with its clearest gains over classical and neural baselines on challenging Kuramoto-Sivashinsky dynamics.

Juncheng Zhong, Chenghuang Shen, Jianfeng Liu +5

School of Mathematical Sciences, Fudan University, Shanghai 200433, China · Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200438, China · Alibaba Group, Hangzhou, China +3

Sparse Recovery Sparse Identification of Nonlinear Dynamics PDE Discovery Symbolic Regression

Jul 31, 2026cs.LG

Dynamics-aware identification of governing equations from sparse and noisy data

Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data. However, sparse and noisy temporal measurements can make derivative estimates unreliable. To address this problem, we evaluate Koopman-based upsampling techniques implemented with dynamic mode decomposition (DMD), extended DMD (EDMD), and optimized DMD. These methods learn finite-dimensional approximations of Koopman evolution on selected observables and are used to interpolate and denoise snapshots inside the observed time window before derivative estimation and sparse regression. The empirical benchmark comprises two ODE systems, Lorenz-63 and Van der Pol, and three periodic PDE systems, Burgers, Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP), and linear advection-diffusion, over sparse and noisy sampling regimes. Polynomial EDMD gives the strongest ODE results, especially in coefficient accuracy. The PDE results are system-dependent: low-rank DMD-assisted reconstructions improve Burgers and advection-diffusion discovery, while the raw baseline (without upsampling) remains competitive for the Fisher-KPP data. A comparison against linear and smoothing-spline interpolation techniques shows that the selected Koopman-based preprocessors provide overall performance gains over these non-dynamical alternatives. We also demonstrate that DMD-assisted upsampling can stabilize Pareto-based non-oracle support-size selection. Overall, Koopman-based upsampling is best viewed as a dynamics-aware preprocessing step that can reduce derivative-estimation error when its observable representation and low-rank structure are appropriate for the data.

Pongpisit Thanasutives, Yoshinobu Kawahara

Center for Advanced Intelligence Project (AIP), RIKEN, Osaka, Japan · 2Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan

Koopman Operator Learning Sparse Identification of Nonlinear Dynamics PDE Discovery ODE Discovery Dynamic Mode Decomposition

Jul 20, 2026cs.LG

The Dynamics of Discoverability: How Trajectories and Priors Shape Equation Recovery

How the dynamical regime of the observed system affects equation discovery has mainly been investigated through comparisons across systems. However, such comparisons vary both the equations and the dynamics, confounding the effect of the regime with the difficulty of recovering the equations symbolically. We separate the two by varying the forcing of Lorenz-84, moving its fully observed post-transient trajectories through fixed-point, periodic, and chaotic regimes while preserving the equations' functional form. Within each regime, we separately vary the amount of data, the noise, and prior knowledge of which terms the equations contain. We then measure how well two complementary approaches, sparse regression over a fixed library of candidate functions (SINDy) and an evolutionary search over symbolic expression trees (PySR), recover the true equations' terms and coefficients. We find that recovery depends on whether the sampled states distinguish combinations of candidate functions: equations remain poorly recovered from fixed-point data even when the candidate set contains only the true terms. We link the effect of the dynamics on both algorithms to one object: the moment matrix of the candidate functions under the invariant measure of the regime. Small eigenvalues mark weakly distinguishable combinations of candidate functions: we show that more data, less noise, and more prior knowledge can mitigate the resulting recovery difficulties, while a zero eigenvalue makes distinct equations indistinguishable on the visited states. Hence, a more precise prior needs less informative data, with consequences for data collection, method design, and evaluation.

Matteo Gallo, Fabio Anselmi, Paolo Lazzari

University of Trieste, Trieste, Italy · National Institute of Oceanography and Applied Geophysics - OGS, Trieste, Italy · NBFC, National Biodiversity Future Center, Palermo, Italy

Dynamical Systems Parameter Identifiability Sparse Identification of Nonlinear Dynamics Scientific Discovery ODE Discovery Symbolic Regression

Jul 16, 2026cs.LG

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.

Yao Cheng Li, Ana Larrañaga, Steven L. Brunton +1

Department of Aeronautics, Imperial College London, SW7 2AZ, United Kingdom · Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA · NSF AI Institute in Dynamic Systems, University of Washington, Seattle, WA 98195, USA

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification

Jul 9, 2026cs.LG

Discovering Latent Response Laws in Forced Physical Systems

Governing equations provide compact descriptions of physical systems, yet the variables in which they are simple are often hidden in high-dimensional measurements. This challenge is sharper for forced systems, whose responses depend on both intrinsic dynamics and time-dependent inputs. Here we introduce FLARE, a forced latent autoencoder for response equations that learns compact response coordinates, identifies sparse input-dependent latent dynamics and decodes equation rollouts to full responses. By estimating latent dimension from data and separating state estimation from external forcing, FLARE enables forecasts to be initialized from past responses and driven by prescribed future inputs. Across known dynamical systems, application-scale forced responses and visual observations, FLARE recovers compact forced dynamics and predicts long-horizon high-dimensional responses under inputs not used for training. By turning learned coordinates into a dynamical interface, FLARE extends equation discovery to systems whose effective states are hidden within complex observations, providing a route for interpretable modelling and prediction of high-dimensional responses in forced dynamical systems.

Yi Zhu, Su Chen, Xiaojun Li +1

State Key Laboratory of Bridge Safety and Resilience, Beijing University of Technology, Beijing, 100124, China

Dynamical Systems Latent Dynamics Modeling Sparse Identification of Nonlinear Dynamics Scientific Discovery State Representation Learning

Jun 22, 2026physics.data-an

Where Is My Physics Wrong? Localized and Identifiable Discovery of Model Discrepancy

Hybrid models combine trusted physics with data-driven correction, but a physical model is rarely wrong everywhere or in the same way. The key diagnostic question is local: where does the model fail, what missing mechanism explains the failure, and is the evidence statistically real? Existing sparse-discovery and discrepancy-learning methods usually fit one global correction, which can spread a local error into clean regimes, bias trusted physical parameters, and provide no calibrated significance for selected terms. We introduce LISDD, Localized, Identifiable Sparse Discovery of Discrepancy, a framework that localizes model error to an operating regime, identifies a sparse symbolic form for the missing mechanism, and certifies the discovery with an exact finite-sample test. LISDD fits the known physics on an automatically detected clean regime, flags discrepant regions with a calibrated residual-energy statistic, selects the local missing term by exhaustive holdout over a candidate library, and confirms significance with a sample-split

F

-test. A false-discovery-rate extension handles multiple discrepant regions with different missing mechanisms. In controlled experiments, LISDD keeps physical-parameter bias at 0.002 versus 0.43 for global-discrepancy and black-box baselines, raises localization

F_1

from 0.44 to 0.80, recovers the correct symbolic form with probability one, attains exact detection, and controls the multi-region false-discovery rate while recovering every planted mechanism. The result is a calibrated diagnostic tool for grey-box building-energy models when a fixed physical law silently breaks in one operating regime.

Yifan Wang

Department of Mechanical Engineering, McGill University, QC, H3A 2T7, Canada

Sparse Identification of Nonlinear Dynamics Model Misspecification Anomaly Detection

Jun 17, 2026cs.LG

OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systems

Dynamical systems are fundamental to modeling the natural world, yet modeling them involves a persistent trade-off: manually prescribed mechanistic models are interpretable by design but often overly simplistic and misspecified; in contrast, flexible data-driven neural methods lack physical insight. Hybrid modeling aims for the best of both worlds by combining a prescribed or symbolic, physics-based component with a flexible neural network. A critical challenge, however, is that the neural component may relearn mechanistic parts, yielding redundant and uninterpretable models, especially when the symbolic structure itself is discovered from data. Existing methods based on standard

L^2

regularization rely on a projection argument that breaks when the symbolic component is learned through sparse discovery, allowing the neural augmentation to overlap with symbolic structure. We introduce \textbf{OrthoReg} (Orthogonal Regularization), which directly penalizes overlap between the symbolic and neural components, preventing symbolic structure from being absorbed by the neural residual. This yields a complementary decomposition: the symbolic part captures what the library can express, and the neural part captures what remains. On benchmark dynamical systems with partial library mismatch, OrthoReg improves symbolic recovery and out-of-distribution behavior.

Till Richter, Niki Kilbertus

Technical University of Munich · Helmholtz Munich

Sparse Identification of Nonlinear Dynamics Neural Network Interpretability

Jun 14, 2026cs.LG

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.

Filippo Zacchei, Ana Larrañaga, Attilio Frangi +2

MOX – Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy · Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195, United States · Department of Civil and Environmental Engineering, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Multifidelity Modeling Heteroscedastic Regression

Jun 10, 2026cs.LG

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.

Ana Larrañaga, Urban Fasel, Steven L. Brunton

Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, United States · NSF AI Institute in Dynamic Systems, University of Washington, Seattle, WA 98195, United States · Department of Aeronautics, Imperial College London, SW7 2AZ, United Kingdom

Sparse Identification of Nonlinear Dynamics PDE Discovery Nonlinear System Identification Active Learning System Identification Data-Efficient Learning

Jun 10, 2026math.DS

Integral Formulation of QENDy for Robust Nonlinear System Identification

This manuscript proposes an integral formulation of the newly defined quadratic embedding method for identifying nonlinear systems (QENDy). In the original algorithm, trajectory data points along with their time derivatives are used. Methods for calculating time derivatives make the algorithm sensitive to noise. Our integral formulation does not use the time derivatives. This results in a more robust method to learn the dynamics.

Nikhil Saran, Sushant Pokhriyal, Stefan Klus +2

Department of Mathematics and Statistics at the University of South Florida · Institute of Engineering and Technology, JK Lakshmipat University, Jaipur, India · School of Mathematical & Computer Sciences at the Heriot–Watt University, Edinburgh +1

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification

May 28, 2026physics.ao-ph

Learning effective Sargassum transport dynamics from limited drifter observations

Floating-material transport is influenced by unresolved processes that are often absent from available circulation products. We develop a data-driven transport-learning framework for learning effective transport corrections from limited Lagrangian observations using physically motivated ocean--atmosphere diagnostics and finite-memory representations motivated in part by inertial-particle memory effects. The diagnostic representation is analyzed through predictive and sparse symbolic-discovery approaches under leave-one-trajectory-out validation. Applications to Sargassum-following drifters in the Puerto Rico region and the Gulf Stream show that the diagnostics contain transport-relevant information beyond the baseline circulation products. Multilayer perceptron (MLP) ensembles provide flexible predictive trajectory corrections, while Sparse Identification of Nonlinear Dynamics (SINDy) tests whether instantaneous or delayed sparse symbolic transport structure can be extracted from the diagnostics. The results differ across flow regimes: (i) in Puerto Rico, delayed sparse symbolic corrections provide modest but systematic improvement; (ii) in the Gulf Stream application, dynamically useful sparse symbolic corrections remain primarily instantaneous even though delayed predictive information persists. These results support finite-memory transport effects in coarse-grained floating-material transport while also illustrating the difficulty of obtaining stable delayed sparse symbolic closures.

F. J. Beron-VEra, M. J. Olascoaga, J. Morell +1

Department of Atmospheric Sciences Rosenstiel School of Marine, Atmospheric, and Earth Science University of Miami Miami, Florida, USA · Department of Ocean Sciences Rosenstiel School of Marine, Atmospheric, and Earth Science University of Miami Miami, Florida, USA · Department of Marine Sciences University of Puerto Rico Mayagüez, Puerto Rico

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Trajectory Prediction

May 28, 2026math.DS

Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy

Social systems consist of networks of individuals who influence one another through social interactions. Studying how processes evolve on these networks can help us better understand patterns of social behavior. We study a system that couples online and offline social activity and investigate how to learn effective models directly from data using Weak Form Sparse Identification of Nonlinear Dynamics (WSINDy), a method for discovering governing equations. We assess learning performance using data generated by a mean-field approximation model of a stochastic interaction process on networks and test how accurately the system can be recovered under different noise levels. Our results show that using more trajectories improves accuracy when noise is high, but only a small number of additional trajectories is needed to gain most of the benefit, with little improvement beyond that. We also learn effective ODE models from averaged stochastic data on networks. When traditional mean-field approximations fail, identifying continuum ODEs directly from stochastic processes yields efficient models that better match the data and provide deeper insight into the underlying dynamics.

Moyi Tian, Daniel A. Messenger, Vanja Dukic +2

Department of Applied Mathematics, University of Colorado, Boulder, CO 80309 United States · Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 United States

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Computational Social Science

May 10, 2026cs.LG

Discovery of Nonlinear Dynamics with Automated Basis Function Generation

Discovering governing equations from observational data remains a fundamental challenge in scientific modeling, particularly when the underlying mathematical structure is unknown. Traditional sparse identification methods like SINDy excel at discovering parsimonious models but require researchers to specify candidate basis functions a priori, a limitation that often leads to model failure when critical terms are omitted or when systems exhibit unconventional dynamics. Purely symbolic regression approaches offer unlimited flexibility but struggle with noise sensitivity and frequently produce overly complex, unstable equations. We present AutoSINDy, a hybrid Discovery-then-Solve framework that combines the exploratory power of symbolic regression with the robust sparsity-promoting capabilities of SINDy. Our method operates in three stages: (1) PySR-based symbolic regression discovers candidate functional forms from bootstrapped data chunks; (2) a curation pipeline decomposes, expands, and filters these expressions using collinearity analysis to construct a minimal yet comprehensive library; and (3) SINDy identifies sparse governing equations from this custom-tailored library. Extensive experiments across canonical nonlinear systems demonstrate that AutoSINDy consistently recovers ground-truth equations even under high observational noise, achieving a ground-truth recovery rate of 92.8% across all trials. Compared with standard SINDy using enriched libraries and standalone symbolic regression, AutoSINDy achieves higher predictive accuracy, superior generalization to unseen trajectories, and substantially lower symbolic complexity.

Mohammad Amin Basiri, Charles Nicholson

Data Science and Analytics Institute, University of Oklahoma, Norman, OK, USA · School of Industrial and Systems Engineering, University of Oklahoma, Norman, OK, USA

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Autonomous Scientific Discovery ODE Discovery Symbolic Regression

May 7, 2026cs.LG

Physics-based Digital Twins for Integrated Thermal Energy Systems Using Active Learning

Real-time supervisory control of thermal energy distribution systems requires digital twins that are accurate, interpretable, and uncertainty-aware, yet remain data and computationally efficient. High-fidelity simulations alone are costly, while purely data-driven surrogates often lack robustness. To address these challenges, this work proposes an active learning (AL) framework that couples system-level Modelica simulations with four simpler physics-informed and data-driven surrogate modeling approaches: deterministic Sparse Identification of Nonlinear Dynamics with Control (SINDyC), its probabilistic multivariate-Gaussian extension (MvG-SINDyC), feedforward neural network (FNN), and gated recurrent unit (GRU) network. Tailored to each surrogate, model-specific AL query strategies are employed, including Mahalanobis-distance sampling in coefficient space for MvG-SINDyC and error-based sampling in prediction space for SINDyC, FNN, and GRU, allowing the learning process to prioritize dynamically informative trajectories. The proposed approach is demonstrated on the glycol heat exchanger (GHX) subsystem of the Thermal Energy Distribution System (TEDS) at Idaho National Laboratory. Across key GHX outputs--the bypass mass flow rate

\dot{m}_{\mathrm{GHX}}

and heat transfer rate

Q_{\mathrm{GHX}}

-the AL framework achieves comparable predictive accuracy using as few as one-fifth of the simulation trajectories required by random sampling. Among the evaluated surrogates, the GRU achieves the highest predictive fidelity, while SINDyC remains the most computationally efficient and interpretable. The probabilistic MvG-SINDyC surrogate further enables uncertainty quantification and exhibits the largest computational gains under AL.

Umme Mahbuba Nabila, Paul Seurin, Linyu Lin +1

Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, United States · Nuclear Science & Technology Division, Idaho National Laboratory, Idaho Falls, ID 83415, United States · Department of Computer Science and Engineering, University of Michigan, Ann Arbor, MI 48109, United States

Surrogate Modeling Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Active Learning Physics-Informed ML Digital Twins

May 6, 2026cs.LG

From Video-to-PDE: Data-Driven Discovery of Nonlinear Dye Plume Dynamics

Inferring continuum models directly from video is hampered by two facts: the recorded field is uncalibrated image intensity rather than a physical state, and direct numerical differentiation of noisy frames is unstable. We develop a video-to-PDE pipeline that converts grayscale recordings of an ink plume into a normalised scalar field

u(x,y,t)

, isolates a bulk drift

\mathbf{v}(t)

from intrinsic spreading via the intensity-weighted centroid, and identifies an effective transport law by weak-form sparse regression. Conditioning, threshold-sweep and random-centre diagnostics show that overcomplete libraries are strongly collinear; the search is therefore restricted to compact gradient-based libraries. Coefficients are refined by an inverse physics-informed network and recalibrated against forward rollouts, with a chronological block bootstrap quantifying uncertainty. The selected reduced model

u_t+\mathbf v(t)\!\cdot\!\nabla u = 9.005\,|\nabla u|^{2}+0.666\,Δu

outperforms advection--diffusion baselines on held-out frames, retains a positive Laplacian coefficient, and admits a Cole--Hopf reduction to a linear advection--diffusion equation. The framework demonstrates that uncalibrated visual data can yield compact, predictive and structurally interpretable continuum models when discovery, calibration and uncertainty are treated as distinct stages.

Cesar Acosta-Minoli, Sayantan Sarkar

aGEDES, Universidad del Quindio, Cra 15 Cl 12-00, Armenia, 630004, Quindio, Colombia · bDepartment of Mathematics, State University of New York at Buffalo, Mathematics Building, Buffalo, 14260, New York, USA

PDE Solving Sparse Identification of Nonlinear Dynamics PDE Discovery

Apr 22, 2026cs.LG

Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identification

We introduce Fourier Weak SINDy, a minimal noise-robust and interpretable derivative-free equation learning method that combines weak-form sparse equation learning with spectral density estimation for data-driven test function selection. By using orthogonal sinusoidal test functions inspired by their prevalence in Modulating Function-based system identification, the weak-form sparse regression problem reduces to a regression over Fourier coefficients. Dominant frequencies are then selected via multitaper estimation of the frequency spectrum of the data. This formulation unifies weak-form learning and spectral estimation within a compact and flexible framework. We illustrate the effectiveness of this approach in numerical experiments across multiple chaotic and hyperchaotic ODE benchmarks.

Zhiheng Chen, Urban Fasel, Anastasia Bizyaeva

Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York, USA, 14850 · Department of Aeronautics, Imperial College London, London, United Kingdom

Sparse Identification of Nonlinear Dynamics Nonlinear System Identification System Identification ODE Discovery

Apr 20, 2026cs.LG

AC-SINDy: Compositional Sparse Identification of Nonlinear Dynamics

We present AC-SINDy, a compositional extension of the Sparse Identification of Nonlinear Dynamics (SINDy) framework that replaces explicit feature libraries with a structured representation based on arithmetic circuits. Rather than enumerating candidate basis functions, the proposed approach constructs nonlinear features through compositions of linear functions and multiplicative interactions, yielding a compact and scalable parameterization and enabling sparsity to be enforced directly over the computational graph. We also introduce a formulation that separates state estimation from dynamics identification by combining latent state inference with shared dynamics and multi-step supervision, improving robustness to noise while preserving interpretability. Experiments on nonlinear and chaotic systems demonstrate that the method recovers accurate and interpretable governing equations while scaling more favorably than standard SINDy.

Peter Racioppo

Independent Researcher Los Angeles, CA 90731

Latent Dynamics Modeling Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Symbolic Regression

Apr 20, 2026cs.LG

Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms

Data-driven discovery of governing equations has advanced significantly in recent years; however, existing methods often struggle in multiscale systems where dynamically significant terms may have small coefficients. Therefore, we propose Balance-Guided SINDy (BG-SINDy) inspired by the principle of dominant balance, which reformulates

\ell_0

-constrained sparse regression as a term-level

\ell_{2,0}

-regularized problem and solves it using a progressive pruning strategy. Terms are ranked according to their relative contributions to the governing equation balance rather than their absolute coefficient magnitudes. Based on this criterion, BG-SINDy alternates between least-squares regression and elimination of negligible terms, thereby preserving dynamically significant terms even when their coefficients are small. Numerical experiments on the Korteweg--de Vries equation with a small dispersion coefficient, a modified Burgers equation with vanishing hyperviscosity, a modified Kuramoto--Sivashinsky equation with multiple small-coefficient terms, and a two-dimensional reaction--diffusion system demonstrate the validity of BG-SINDy in discovering small-coefficient terms. The proposed method thus provides an efficient approach for discovering governing equations that contain small-coefficient terms.

Zhenhua Dang, Lei Zhang, Long Wang +1

Sparse Identification of Nonlinear Dynamics PDE Discovery Nonlinear System Identification

Apr 18, 2026quant-ph

Q-SINDy: Quantum-Kernel Sparse Identification of Nonlinear Dynamics with Provable Coefficient Debiasing

Quantum feature maps offer expressive embeddings for classical learning tasks, and augmenting sparse identification of nonlinear dynamics (SINDy) with such features is a natural but unexplored direction. We introduce \textbf{Q-SINDy}, a quantum-kernel-augmented SINDy framework, and identify a specific failure mode that arises: \emph{coefficient cannibalization}, in which quantum features absorb coefficient mass that rightfully belongs to the polynomial basis, corrupting equation recovery. We derive the exact cannibalization-bias formula

Δξ_P = (P^\top P)^{-1}P^\top Q\,\hatξ_Q

and prove that orthogonalizing quantum features against the polynomial column space at fit time eliminates this bias exactly. The claim is verified numerically to machine precision (

<10^{-12}

) on multiple systems. Empirically, across six canonical dynamical systems (Duffing, Van der Pol, Lorenz, Lotka-Volterra, cubic oscillator, Rössler) and three quantum feature map architectures (ZZ-angle encoding, IQP, data re-uploading), orthogonalized Q-SINDy consistently matches vanilla SINDy's structural recovery while uncorrected augmentation degrades true-positive rates by up to 100%. A refined dynamics-aware diagnostic,

R^2_Q

for

\dot X

, predicts cannibalization severity with statistical significance (Pearson

r=0.70

,

p=0.023

). An RBF classical-kernel control across 20 hyperparameter configurations fails more severely than any quantum variant, ruling out feature count as the cause. Orthogonalization remains robust under depolarizing hardware noise up to 2% per gate, and the framework extends without modification to Burgers' equation.

Samrendra Roy, Syed Bahauddin Alam

Department of Nuclear, Plasma, and Radiological Engineering University of Illinois at Urbana-Champaign Urbana, IL, USA · Department of Nuclear, Plasma, and Radiological Engineering University of Illinois at Urbana-Champaign National Center for Supercomputing Applications Urbana, IL, USA

Quantum Feature Maps Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Quantum Kernel Methods

Apr 16, 2026cs.LG

One-shot learning for the complex dynamical behaviors of weakly nonlinear forced oscillators

Extrapolative prediction of complex nonlinear dynamics remains a central challenge in engineering. This study proposes a one-shot learning method to identify global frequency-response curves from a single excitation time history by learning governing equations. We introduce MEv-SINDy (Multi-frequency Evolutionary Sparse Identification of Nonlinear Dynamics) to infer the governing equations of non-autonomous and multi-frequency systems. The methodology leverages the Generalized Harmonic Balance (GHB) method to decompose complex forced responses into a set of slow-varying evolution equations. We validated the capabilities of MEv-SINDy on two critical Micro-Electro-Mechanical Systems (MEMS). These applications include a nonlinear beam resonator and a MEMS micromirror. Our results show that the model trained on a single point accurately predicts softening/hardening effects and jump phenomena across a wide range of excitation levels. This approach significantly reduces the data acquisition burden for the characterization and design of nonlinear microsystems.

Teng Ma, Luca Rosafalco, Wei Cui +2

Dynamical Systems Sparse Identification of Nonlinear Dynamics Nonlinear System Identification Few-Shot Learning