Nonlinear System Identification

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Overview

The field seeks to recover how a system changes over time from measurements, so its behavior can be explained, predicted, or controlled. Approaches include sparse equation discovery such as SINDy and symbolic regression, Koopman and kernel methods, and neural state-space or differential-equation models; many incorporate physical structure such as energy conservation or dissipation. Methods are tested on settings ranging from chaotic ODEs and oscillators to robots, vehicles, and industrial processes, often with sparse, noisy, or partially observed data. A central challenge is that good predictions on observed trajectories do not guarantee that the model is identifiable, stable over long horizons, or reliable outside the training conditions.

Latest papers

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162 papers14 papers in the last four weeks, up 75% on the four weeks before. 0.1% of all new papers.

Apr 16, 2026

xFODE: An Explainable Fuzzy Additive ODE Framework for System Identification

Recent advances in Deep Learning (DL) have strengthened data-driven System Identification (SysID), with Neural and Fuzzy Ordinary Differential Equation (NODE/FODE) models achieving high accuracy in nonlinear dynamic modeling. Yet, system states in these frameworks are often reconstructed without clear physical meaning, and input contributions to the state derivatives remain difficult to interpret. To address these limitations, we propose Explainable FODE (xFODE), an interpretable SysID framework with integrated DL-based training. In xFODE, we define states in an incremental form to provide them with physical meanings. We employ fuzzy additive models to approximate the state derivative, thereby enhancing interpretability per input. To provide further interpretability, Partitioning Strategies (PSs) are developed, enabling the training of fuzzy additive models with explainability. By structuring the antecedent space during training so that only two consecutive rules are activated for any given input, PSs not only yield lower complexity for local inference but also enhance the interpretability of the antecedent space. To train xFODE, we present a DL framework with parameterized membership function learning that supports end-to-end optimization. Across benchmark SysID datasets, xFODE matches the accuracy of NODE, FODE, and NLARX models while providing interpretable insights.

Ertugrul Kececi, Tufan Kumbasar

AI and Intelligent Systems Laboratory, Istanbul Technical University, 34469, Istanbul, Türkiye

Interpretable MLNonlinear System IdentificationFuzzy LogicSystem Identification
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Apr 16, 2026

SOLIS: Physics-Informed Learning of Interpretable Neural Surrogates for Nonlinear Systems

Nonlinear system identification must balance physical interpretability with model flexibility. Classical methods yield structured, control-relevant models but rely on rigid parametric forms that often miss complex nonlinearities, whereas Neural ODEs are expressive yet largely black-box. Physics-Informed Neural Networks (PINNs) sit between these extremes, but inverse PINNs typically assume a known governing equation with fixed coefficients, leading to identifiability failures when the true dynamics are unknown or state-dependent. We propose SOLIS, which models unknown dynamics via a state-conditioned second-order surrogate model and recasts identification as learning a Quasi-Linear Parameter-Varying (Quasi-LPV) representation, recovering interpretable natural frequency, damping, and gain without presupposing a global equation. SOLIS decouples trajectory reconstruction from parameter estimation and stabilizes training with a cyclic curriculum and Local Physics Hints windowed ridge-regression anchors that mitigate optimization collapse. Experiments on benchmarks show accurate parameter-manifold recovery and coherent physical rollouts from sparse data, including regimes where standard inverse methods fail.

Murat Furkan Mansur, Tufan Kumbasar

AI and Intelligent Systems Laboratory, Istanbul Technical University, 34469, Istanbul, Türkiye

Neural Surrogate ModelingParameter EstimationParameter IdentifiabilityNonlinear System IdentificationPhysics-Informed ML
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Apr 14, 2026

Structure- and Stability-Preserving Learning of Port-Hamiltonian Systems

This paper investigates the problem of data-driven modeling of port-Hamiltonian systems while preserving their intrinsic Hamiltonian structure and stability properties. We propose a novel neural-network-based port-Hamiltonian modeling technique that relaxes the convexity constraint commonly imposed by neural network-based Hamiltonian approximations, thereby improving the expressiveness and generalization capability of the model. By removing this restriction, the proposed approach enables the use of more general non-convex Hamiltonian representations to enhance modeling flexibility and accuracy. Furthermore, the proposed method incorporates information about stable equilibria into the learning process, allowing the learned model to preserve the stability of multiple isolated equilibria rather than being restricted to a single equilibrium as in conventional methods. Two numerical experiments are conducted to validate the effectiveness of the proposed approach and demonstrate its ability to achieve more accurate structure- and stability-preserving learning of port-Hamiltonian systems compared with a baseline method.

Binh Nguyen, Nam T. Nguyen, Truong X. Nghiem

Department of Electrical and Computer Engineering, College of Engineering and Computer Science, University of Central Florida, Orlando, FL 32816, USA

Port-Hamiltonian SystemsHamiltonian Neural NetworksNonlinear System Identification
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Apr 9, 2026

Tensor-based computation of the Koopman generator via operator logarithm

Identifying governing equations of nonlinear dynamical systems from data is challenging. While sparse identification of nonlinear dynamics (SINDy) and its extensions are widely used for system identification, operator-logarithm approaches use the logarithm to avoid time differentiation, enabling larger sampling intervals. However, they still suffer from the curse of dimensionality. Then, we propose a data-driven method to compute the Koopman generator in a low-rank tensor train (TT) format by taking logarithms of Koopman eigenvalues while preserving the TT format. Experiments on 4-dimensional Lotka-Volterra and 10-dimensional Lorenz-96 systems show accurate recovery of vector field coefficients and scalability to higher-dimensional systems.

Tatsuya Kishimoto, Jun Ohkubo

Graduate School of Science and Engineering, Saitama University 255 Shimo-Okubo, Sakura-ku, Saitama 338–8570, Japan

Koopman Operator LearningNonlinear System IdentificationTT Decomposition
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Apr 7, 2026

A machine learning framework for uncovering stochastic nonlinear dynamics from noisy data

Modeling real-world systems requires accounting for noise - whether it arises from unpredictable fluctuations in financial markets, irregular rhythms in biological systems, or environmental variability in ecosystems. While the behavior of such systems can often be described by stochastic differential equations, a central challenge is understanding how noise influences the inference of system parameters and dynamics from data. Traditional symbolic regression methods can uncover governing equations but typically ignore uncertainty. Conversely, Gaussian processes provide principled uncertainty quantification but offer little insight into the underlying dynamics. In this work, we bridge this gap with a hybrid symbolic regression-probabilistic machine learning framework that recovers the symbolic form of the governing equations while simultaneously inferring uncertainty in the system parameters. The framework combines deep symbolic regression with Gaussian process-based maximum likelihood estimation to separately model the deterministic dynamics and the noise structure, without requiring prior assumptions about their functional forms. We verify the approach on numerical benchmarks, including harmonic, Duffing, and van der Pol oscillators, and validate it on an experimental system of coupled biological oscillators exhibiting synchronization, where the algorithm successfully identifies both the symbolic and stochastic components. The framework is data-efficient, requiring as few as 100-1000 data points, and robust to noise - demonstrating its broad potential in domains where uncertainty is intrinsic and both the structure and variability of dynamical systems must be understood.

Matteo Bosso, Giovanni Franzese, Kushal Swamy +3

Nonlinear System IdentificationSymbolic RegressionUncertainty Quantification
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Apr 7, 2026

PH-KAN: Port-Hamiltonian Kolmogorov-Arnold Network

Data-driven machine learning approaches have become increasingly attractive for nonlinear system identification, but standard models often fail to preserve the underlying physical structure and remain difficult to interpret, especially when no analytical model is available. In this context, port-Hamiltonian (pH) models provide a natural physics-informed representation. However, when these models are parameterized with standard multilayer perceptrons (MLPs), the learned constitutive components often remain poorly interpretable. In this paper, we propose a structure-preserving identification framework for nonlinear port-Hamiltonian systems based on Kolmogorov-Arnold Networks (KANs). The proposed PH-KAN model parameterizes the interconnection matrix, dissipation matrix, Hamiltonian, and input mapping using dedicated KAN blocks, while enforcing the port-Hamiltonian constraints by construction. This yields constitutive representations in which the nonlinear functions defining the identified pH components can be explicitly inspected, leading to a more interpretable model than with standard MLP-based parameterizations.

Achraf El Messaoudi, Karim Cherifi, Yann Le Gorrec +1

UMLP, ENSMM, FEMTO-ST · Université Marie et Louis Pasteur, SUPMICROTECH, CNRS, institut FEMTO-ST, F-25000 Besançon, France

Nonlinear System IdentificationPort-Hamiltonian SystemsNeural Network Interpretability
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Apr 6, 2026

A Robust SINDy Autoencoder for Noisy Dynamical System Identification

Sparse identification of nonlinear dynamics (SINDy) has been widely used to discover the governing equations of a dynamical system from data. It uses sparse regression techniques to identify parsimonious models of unknown systems from a library of candidate functions. Therefore, it relies on the assumption that the dynamics are sparsely represented in the coordinate system used. To address this limitation, one seeks a coordinate transformation that provides reduced coordinates capable of reconstructing the original system. Recently, SINDy autoencoders have extended this idea by combining sparse model discovery with autoencoder architectures to learn simplified latent coordinates together with parsimonious governing equations. A central challenge in this framework is robustness to measurement error. Inspired by noise-separating neural network structures, we incorporate a noise-separation module into the SINDy autoencoder architecture, thereby improving robustness and enabling more reliable identification of noisy dynamical systems. Numerical experiments on the Lorenz system show that the proposed method recovers interpretable latent dynamics and accurately estimates the measurement noise from noisy observations.

Kairui Ding

Department of Mathematics, Columbia University

Nonlinear System IdentificationSystem IdentificationSparse Identification of Nonlinear DynamicsLatent Dynamics ModelingAutoencoders
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Apr 3, 2026

Identification and Inference in Nonlinear Dynamic Network Models

We study identification and inference in nonlinear dynamic systems defined on unknown interaction networks. The system evolves through an unobserved dependence matrix governing cross-sectional shock propagation via a nonlinear operator. We show that the network structure is not generically identified, and that identification requires sufficient spectral heterogeneity. In particular, identification arises when the network induces non-exchangeable covariance patterns through heterogeneous amplification of eigenmodes. When the spectrum is concentrated, dependence becomes observationally equivalent to common shocks or scalar heterogeneity, leading to non-identification. We provide necessary and sufficient conditions for identification, characterize observational equivalence classes, and propose a semiparametric estimator with asymptotic theory. We also develop tests for network dependence whose power depends on spectral properties of the interaction matrix. The results apply to a broad class of economic models, including production networks, contagion models, and dynamic interaction systems.

Diego Vallarino

Inter-American Development Bank (IDB), Washington, DC, United States

Parameter IdentifiabilityNonlinear System Identification
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Apr 3, 2026

On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations

This work establishes a rigorous bridge between infinite-dimensional delay dynamics and finite-dimensional Koopman learning, with explicit and interpretable error guarantees. While Koopman analysis is well-developed for ordinary differential equations (ODEs) and partially for partial differential equations (PDEs), its extension to delay differential equations (DDEs) remains limited due to the infinite-dimensional phase space of DDEs. We propose a finite-dimensional Koopman approximation framework based on history discretization and a suitable reconstruction operator, enabling a tractable representation of the Koopman operator via kernel-based extended dynamic mode decomposition (kEDMD). Deterministic error bounds are derived for the learned predictor, decomposing the total error into contributions from history discretization, kernel interpolation, and data-driven regression. Additionally, we develop a kernel-based reconstruction method to recover discretized states from lifted Koopman coordinates, with provable guarantees. Numerical results demonstrate reliable prediction of nonlinear delay systems, with potential relevance to future control applications.

Santosh Mohan Rajkumar, Dibyasri Barman, Kumar Vikram Singh +1

Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH, USA. · Department of Chemical, Paper, and Biomedical Engineering at Miami University, Oxford, OH, USA. · Department of Mechanical and Manufacturing Engineering at Miami University, Oxford, OH, USA.

Extended Dynamic Mode DecompositionDynamical SystemsKoopman Operator LearningLatent Dynamics ModelingNonlinear System Identification
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Apr 2, 2026

Learning interacting particle systems from unlabeled data

Learning the potentials of interacting particle systems is a fundamental task across various scientific disciplines. A major challenge is that unlabeled data collected at discrete time points lack trajectory information due to limitations in data collection methods or privacy constraints. We address this challenge by introducing a trajectory-free self-test loss function that leverages the weak-form stochastic evolution equation of the empirical distribution. The loss function is quadratic in potentials, supporting parametric and nonparametric regression algorithms for robust estimation that scale to large, high-dimensional systems with big data. Systematic numerical tests show that our method outperforms baseline methods that regress on trajectories recovered via label matching, tolerating large observation time steps. We establish the convergence of parametric estimators as the sample size increases, providing a theoretical foundation for the proposed approach.

Viska Wei, Fei Lu

Department of Applied Mathematics and Statistics and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA. · Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA.

System IdentificationNonlinear System Identification
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Apr 2, 2026

Koopman-Based Nonlinear Identification and Model Predictive Control of a Turbofan Engine

This paper investigates Koopman operator-based approaches for multivariable control of a two-spool turbofan engine. A physics-based component-level model is developed to generate training data and validate the controllers. A meta-heuristic extended dynamic mode decomposition is adapted, with a cost function designed to accurately capture both spool-speed dynamics and the engine pressure ratio (EPR), enabling the construction of a single Koopman model that can be reused across multiple control strategies. Using the identified time-varying Koopman model, an adaptive Koopman-based model predictive controller (AKMPC) with a disturbance observer is developed and compared with a Koopman-based feedback linearization controller (K-FBLC) and its integrator-augmented version (K-FBLC-I). The Koopman representation further enables nonlinear GTE output limiters, such as rotor-acceleration and turbine-inlet-temperature limits, to be expressed as linear constraints in the AKMPC. The controllers are evaluated for two control configurations of spool speeds and EPR, under both sea-level and varying flight conditions. The results demonstrate that the proposed identification approach enables accurate predictions of both spool speeds and EPR, allowing the Koopman model to be reused flexibly across different control formulations. While all strategies achieve comparable performance in sea-level conditions, the AKMPC demonstrates improved performance under varying flight conditions due to its ability to capture nonlinear dynamics, handle constraints, and compensate for model mismatch. Moreover, the EPR control strategy improves the thrust response. The study highlights the applicability of Koopman-based control and the advantages of the AKMPC framework for robust turbofan engine control.

David Grasev

Department of Aviation Technology, University of Defence, Brno, Czech Republic.

Adaptive MPCExtended Dynamic Mode DecompositionFeedback ControlModel Predictive ControlNonlinear MPCNonlinear System Identification+2
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Apr 1, 2026

Predicting Dynamics of Ultra-Large Complex Systems by Inferring Governing Equations

Predicting the behavior of ultra-large complex systems, from climate to biological and technological networks, is a central unsolved challenge. Existing approaches face a fundamental trade-off: equation discovery methods provide interpretability but fail to scale, while neural networks scale but operate as black boxes and often lose reliability over long times. Here, we introduce the Sparse Identification Graph Neural Network, a framework that overcome this divide by allowing to infer the governing equations of large networked systems from data. By defining symbolic discovery as edge-level information, SIGN decouples the scalability of sparse identification from network size, enabling efficient equation discovery even in large systems. SIGN allows to study networks with over 100,000 nodes while remaining robust to noise, sparse sampling, and missing data. Across diverse benchmark systems, including coupled chaotic oscillators, neural dynamics, and epidemic spreading, it recovers governing equations with high precision and sustains accurate long-term predictions. Applied to a data set of time series of temperature measurements in 71,987 sea surface positions, SIGN identifies a compact predictive network model and captures large-scale sea surface temperature conditions up to two years in advance. By enabling equation discovery at previously inaccessible scales, SIGN opens a path toward interpretable and reliable prediction of real-world complex systems.

Qi Shao, Duxin Chen, Jiawen Chen +5

School of Mathematics, Southeast University, Nanjing, 210096, P. R. China. · School of Electronic Engineering and Computer Science, Queen Mary University of London, London, E1 4NS, UK. · School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK. +4

Dynamical SystemsNonlinear System IdentificationSparse Identification of Nonlinear DynamicsSpatiotemporal ForecastingSpatiotemporal GNNs
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Mar 31, 2026

Hybrid Energy-Based Models for Physical AI: Provably Stable Identification of Port-Hamiltonian Dynamics

Energy-based models (EBMs) implement inference as gradient descent on a learned Lyapunov function, yielding interpretable, structure-preserving alternatives to black-box neural ODEs and aligning naturally with physical AI. Yet their use in system identification remains limited, and existing architectures lack formal stability guarantees that globally preclude unstable modes. We address this gap by introducing an EBM framework for system identification with stable, dissipative, absorbing invariant dynamics. Unlike classical global Lyapunov stability, absorbing invariance expands the class of stability-preserving architectures, enabling more flexible and expressive EBMs. We extend EBM theory to nonsmooth activations by establishing negative energy dissipation via Clarke derivatives and deriving new conditions for radial unboundedness, exposing a stability-expressivity tradeoff in standard EBMs. To overcome this, we introduce a hybrid architecture with a dynamical visible layer and static hidden layers, prove absorbing invariance under mild assumptions, and show that these guarantees extend to port-Hamiltonian EBMs. Experiments on metric-deformed multi-well and ring systems validate the approach, showcasing how our hybrid EBM architecture combines expressivity with sound and provable safety guarantees by design.

Simone Betteti, Luca Laurenti

RIAS Lab, The Italian Institute of Artificial Intelligence for Industry, Turin, 10129, IT · Delft Center for Systems and Control, TU Delft, Delft, 2600 AA, The Netherlands

Energy-Based ModelsDynamical SystemsNonlinear System IdentificationLyapunov StabilityPort-Hamiltonian Systems
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Mar 31, 2026

Learning Surrogate LPV State-Space Models with Uncertainty Quantification

The Linear Parameter-Varying (LPV) framework enables the construction of surrogate models of complex nonlinear and high-dimensional systems, facilitating efficient stability and performance analysis together with controller design. Despite significant advances in data-driven LPV modelling, existing approaches do not quantify the uncertainty of the obtained LPV models. Consequently, assessing model reliability for analysis and control or detecting operation outside the training regime requires extensive validation and user expertise. This paper proposes a Bayesian approach for the joint estimation of LPV state-space models, including their scheduling map, together with characterization of the model uncertainty and confidence bounds on the predicted model response directly from input-output data. Both aleatoric uncertainty due to measurement noise and epistemic uncertainty arising from limited training data and structural bias are considered. The resulting model preserves the LPV structure required for controller synthesis while enabling computationally efficient simulation and uncertainty propagation. The approach is demonstrated on the surrogate modelling of a two-dimensional nonlinear interconnection of mass-spring-damper systems.

E. Javier Olucha, Amritam Das, Roland Tóth

Control Systems Group, Eindhoven University of Technology, The Netherlands · Systems and Control Laboratory, HUN-REN Institute for Computer Science and Control, Hungary

Surrogate ModelingState Space ModelsUncertainty QuantificationNonlinear System Identification
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Mar 31, 2026

Lie Generator Networks for Nonlinear Partial Differential Equations

Linear dynamical systems are fully characterized by their eigenspectra, accessible directly from the generator of the dynamics. For nonlinear systems governed by partial differential equations, no equivalent theory exists. We introduce Lie Generator Network-Koopman (LGN-KM), a neural operator that lifts nonlinear dynamics into a linear latent space and learns the continuous-time Koopman generator (LkL_k) through a decomposition Lk=S−DkL_k = S - D_k, where SS is skew-symmetric representing conservative inter-modal coupling, and DkD_k is a positive-definite diagonal encoding modal dissipation. This architectural decomposition enforces stability and enables interpretability through direct spectral access to the learned dynamics. On two-dimensional Navier--Stokes turbulence, the generator recovers the known dissipation scaling and a complete multi-branch dispersion relation from trajectory data alone with no physics supervision. Independently trained models at different flow regimes recover matched gauge-invariant spectral structure, exposing a gauge freedom in the Koopman lifting. Because the generator is provably stable, it enables guaranteed long-horizon stability, continuous-time evaluation at arbitrary time, and physics-informed cross-viscosity model transfer.

Shafayeth Jamil, Rehan Kapadia

Ming Hsieh Department of Electrical and Computer Engineering University of Southern California

Koopman Operator LearningPDE Operator LearningNonlinear System IdentificationComputational Fluid Dynamics
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Mar 27, 2026

Physics-Informed Neural Networks and Sequence Encoder: Application to heating and early cooling of thermo-stamping process

In a previous work (Elaarabi et al., 2025b), the Sequence Encoder for online dynamical system identification (Elaarabi et al., 2025a) and its combination with PINN (PINN-SE) were introduced and tested on both synthetic and real data case scenarios. The sequence encoder is able to effectively encode time series into feature vectors, which the PINN then uses to map to dynamical behavior, predicting system response under changes in parameters, ICs and BCs. Previously (Elaarabi et al., 2025b), the tests on real data were limited to simple 1D problems and only 1D time series inputs of the Sequence Encoder. In this work, the possibility of applying PINN-SE to a more realistic case is investigated: heating and early cooling of the thermo-stamping process, which is a critical stage in the forming process of continuous fiber reinforced composite materials with thermoplastic polymer. The possibility of extending the PINN-SE inputs to multimodal data, such as sequences of temporal 2D images and to scenarios involving variable geometries, is also explored. The results show that combining multiple encoders with the previously proposed method (Elaarabi et al., 2025b) is feasible, we also show that training the model on synthetic data generated based on experimental data can help the model to generalize well for real experimental data, unseen during the training phase.

Mouad Elaarabi, Domenico Borzacchiello, Philippe Le Bot +2

Nantes Université, Ecole Centrale Nantes, CNRS, GeM, UMR 6183, Nantes, 44321, France · Nantes Université, IRT Jules Verne, Bouguenais, 44340, France

Physics-Informed MLNeural Surrogate ModelingNonlinear System IdentificationManufacturingSequence Modeling
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Mar 23, 2026

Identifiability and amortized inference limitations in Kuramoto models

Bayesian inference is a powerful tool for parameter estimation and uncertainty quantification in dynamical systems. However, for nonlinear oscillator networks such as Kuramoto models, widely used to study synchronization phenomena in physics, biology, and engineering, inference is often computationally prohibitive due to high-dimensional state spaces and intractable likelihood functions. We present an amortized Bayesian inference approach that learns a neural approximation of the posterior from simulated phase dynamics, enabling fast, scalable inference without repeated sampling or optimization. Applied to synthetic Kuramoto networks, the method shows promising results in approximating posterior distributions and capturing uncertainty, with computational savings compared to traditional Bayesian techniques. These findings suggest that amortized inference is a practical and flexible framework for uncertainty-aware analysis of oscillator networks.

Emma Hannula, Jana de Wiljes, Matthew T. Moores +2

LUT University, Yliopistonkatu 34, Lappeenranta, 53850, Finland · TU Ilmenau, Ehrenbergstraße 29, Ilmenau, 98693, Germany · University of Cambridge, Herschel Road, Cambridge, CB3 9AL, United Kingdom

Kuramoto ModelAmortized InferenceDynamical SystemsNeural Posterior EstimationSimulation-Based InferenceParameter Identifiability+1
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Mar 19, 2026

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.

Amanda A. Howard, Nicholas Zolman, Bruno Jacob +2

Pacific Northwest National Laboratory, Richland, WA 99354 · University of Washington, Seattle, WA 98195

Nonlinear System IdentificationSparse Identification of Nonlinear DynamicsSymbolic RegressionNeural Network InterpretabilityKolmogorov-Arnold Networks
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Mar 18, 2026

Physics-informed Deep Mixture-of-Koopmans Vehicle Dynamics Model with Dual-branch Encoder for Distributed Electric-drive Trucks

Advanced autonomous driving systems require accurate vehicle dynamics modeling. However, identifying a precise dynamics model remains challenging due to strong nonlinearities and the coupled longitudinal and lateral dynamic characteristics. Previous research has employed physics-based analytical models or neural networks to construct vehicle dynamics representations. Nevertheless, these approaches often struggle to simultaneously achieve satisfactory performance in terms of system identification efficiency, modeling accuracy, and compatibility with linear control strategies. In this paper, we propose a fully data-driven dynamics modeling method tailored for complex distributed electric-drive trucks (DETs), leveraging Koopman operator theory to represent highly nonlinear dynamics in a lifted linear embedding space. To achieve high-precision modeling, we first propose a novel dual-branch encoder which encodes dynamic states and provides a powerful basis for the proposed Koopman-based methods entitled KODE. A physics-informed supervision mechanism, grounded in the geometric consistency of temporal vehicle motion, is incorporated into the training process to facilitate effective learning of both the encoder and the Koopman operator. Furthermore, to accommodate the diverse driving patterns of DETs, we extend the vanilla Koopman operator to a mixture-of-Koopman operator framework, enhancing modeling capability. Simulations conducted in a high-fidelity TruckSim environment and real-world experiments demonstrate that the proposed approach achieves state-of-the-art performance in long-term dynamics state estimation.

Jinyu Miao, Pu Zhang, Rujun Yan +8

School of Vehicle and Mobility, and State Key Laboratory of Intelligent Green Vehicle and Mobility, Tsinghua University, Beijing, China · KargoBot Inc., Beijing, China

Vehicle DynamicsKoopman Operator LearningNonlinear System IdentificationPhysics-Informed ML
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Mar 17, 2026

Data-driven forced response analysis with min-max representations of nonlinear restoring forces

This paper discusses a novel data-driven nonlinearity identification method for mechanical systems with nonlinear restoring forces such as polynomial, piecewise-linear, and general displacement-dependent nonlinearities. The proposed method is built upon the universal approximation theorem that states that a nonlinear function can be approximated by a linear combination of activation functions in artificial neural network framework. The proposed approach utilizes piecewise linear springs with initial gaps to act as the activation functions of the neurons of artificial neural networks. A library of piecewise linear springs with initial gaps are constructed, and the contributions of the springs on the nonlinear restoring force are determined by solving the linear regression problems. The piecewise linear springs are realized by combinations of min and max functions with biases. The proposed method is applied to a Duffing oscillator with cubic stiffness, and a piecewise linear oscillator with a gap and their nonlinearities are successfully determined from their free responses. The obtained models are then used for conducting forced response analysis and the results match well with those of the original system. The method is then applied to experimentally-obtained free response data of a cantilevered plate that is subjected to magnetic restoring force, and successfully finds the piecewise linear representation of the magnetic force. It is also shown that the obtained model is capable of accurately capturing the steady-state response of the system subject to harmonic base excitation.

Akira Saito, Hiromu Fujita

Department of Mechanical Engineering, Meiji University, Kawasaki, Kanagawa 214-8571, Japan.

Nonlinear System IdentificationDynamical Systems
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Mar 17, 2026

Deep Adaptive Model-Based Design of Experiments

Model-based design of experiments (MBDOE) is essential for efficient parameter estimation in nonlinear dynamical systems. However, conventional adaptive MBDOE requires costly posterior inference and design optimization between each experimental step, precluding real-time applications. We address this by combining Deep Adaptive Design (DAD), which amortizes sequential design into a neural network policy trained offline, with differentiable mechanistic models. For dynamical systems with known governing equations but uncertain parameters, we extend sequential contrastive training objectives to handle nuisance parameters and propose a transformer-based policy architecture that respects the temporal structure of dynamical systems. We demonstrate the approach on four systems of increasing complexity: a fed-batch bioreactor with Monod kinetics, a Haldane bioreactor with uncertain substrate inhibition, a two-compartment pharmacokinetic model with nuisance clearance parameters, and a DC motor for real-time deployment.

Arno Strouwen, Sebastian Micluţa-Câmpeanu

Strouwen Statistics, Hasselt, Belgium · Biosystems Department, KU Leuven, Belgium · JuliaHub, Massachusetts, USA +1

Bayesian Experimental DesignNonlinear System Identification
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Mar 16, 2026

Minimum-Action Learning: Energy-Constrained Symbolic Model Selection for Physical Law Identification from Noisy Data

Identifying physical laws from noisy observational data is a central challenge in scientific machine learning. We present Minimum-Action Learning (MAL), a framework that selects symbolic force laws from a pre-specified basis library by minimizing a Triple-Action functional combining trajectory reconstruction, architectural sparsity, and energy-conservation enforcement. A wide-stencil acceleration-matching technique reduces noise variance by 10,000x, transforming an intractable problem (SNR ~0.02) into a learnable one (SNR ~1.6); this preprocessing is the critical enabler shared by all methods tested, including SINDy variants. On two benchmarks -- Kepler gravity and Hooke's law -- MAL recovers the correct force law with Kepler exponent p = 3.01 +/- 0.01 at ~0.07 kWh (40% reduction vs. prediction-error-only baselines). The raw correct-basis rate is 40% for Kepler and 90% for Hooke; an energy-conservation-based criterion discriminates the true force law in all cases, yielding 100% pipeline-level identification. Basis library sensitivity experiments show that near-confounders degrade selection (20% with added r^{-2.5} and r^{-1.5}), while distant additions are harmless, and the conservation diagnostic remains informative even when the correct basis is absent. Direct comparison with noise-robust SINDy variants, Hamiltonian Neural Networks, and Lagrangian Neural Networks confirms MAL's distinct niche: interpretable, energy-constrained model selection that combines symbolic basis identification with dynamical rollout validation.

Martin G. Frasch

Institute on Human Development and Disability, University of Washington, Seattle, WA 98195, USA · Health Stream Analytics, LLC, Seattle, WA, USA

Scientific DiscoveryModel SelectionPhysics-Informed MLNonlinear System IdentificationSymbolic Regression
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Mar 10, 2026

NanoBench: A Multi-Task Benchmark Dataset for Nano-Quadrotor System Identification, Control, and State Estimation

Existing aerial-robotics benchmarks target vehicles from hundreds of grams to several kilograms and typically expose only high-level state data. They omit the actuator-level signals required to study nano-scale quadrotors, where low-Reynoldsnumber aerodynamics, coreless DC motor nonlinearities, and severe computational constraints invalidate models and controllers developed for larger vehicles. We introduce NanoBench, an open-source multi-task benchmark collected on the commercially available Crazyflie 2.1 nano-quadrotor (takeoff weight 27 g) in a Vicon motion capture arena. The dataset comprises 172 flight recordings spanning multi-frequency excitation, geometric trajectory tracking at three speed regimes, and long-duration battery-drain hover. Each recording provides synchronized Vicon ground truth, raw IMU data, onboard extended Kalman filter (EKF) estimates, PID controller internals, and motor PWM commands at 100 Hz, alongside battery telemetry at 10 Hz, crossaligned by gyroscope-based correlation on a 1 ms search grid. NanoBench defines standardized evaluation protocols, train/test splits, and open-source baselines for three tasks: nonlinear system identification, closed-loop controller benchmarking, and onboard state estimation assessment. To our knowledge, it is the first public dataset to jointly provide actuator commands, controller internals, and estimator outputs with millimeter-accurate ground truth on a commercially available nano-scale aerial platform.

Syed Izzat Ullah, Jose Baca

Department of Engineering Texas A&M University-Corpus Christi, Corpus Christi, TX, USA

Benchmark DesignNonlinear System IdentificationQuadrotor ControlRobot State Estimation
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Mar 10, 2026

Vision-Augmented On-Track System Identification for Autonomous Racing via Attention-Based Priors and Iterative Neural Correction

Operating autonomous vehicles at the absolute limits of handling requires precise, real-time identification of highly non-linear tire dynamics. However, traditional online optimization methods suffer from "cold-start" initialization failures and struggle to model high-frequency transient dynamics. To address these bottlenecks, this paper proposes a novel vision-augmented, iterative system identification framework. First, a lightweight CNN (MobileNetV3) translates visual road textures into a continuous heuristic friction prior, providing a robust "warm-start" for parameter optimization. Next, a S4 model captures complex temporal dynamic residuals, circumventing the memory and latency limitations of traditional MLPs and RNNs. Finally, a derivative-free Nelder-Mead algorithm iteratively extracts physically interpretable Pacejka tire parameters via a hybrid virtual simulation. Co-simulation in CarSim demonstrates that the lightweight vision backbone reduces friction estimation error by 76.1 using 85 fewer FLOPs, accelerating cold-start convergence by 71.4. Furthermore, the S4-augmented framework improves parameter extraction accuracy and decreases lateral force RMSE by over 60 by effectively capturing complex vehicle dynamics, demonstrating superior performance compared to conventional neural architectures.

Zhiping Wu, Cheng Hu, Yiqin Wang +2

State Key Laboratory of Industrial, Zhejiang University, Hangzhou 310027, China

Nonlinear System IdentificationVehicle DynamicsAutonomous RacingResidual Dynamics Learning
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Mar 5, 2026

Towards a data-scale independent regulariser for robust sparse identification of non-linear dynamics

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.

Jay Raut, Daniel N. Wilke, Stephan Schmidt

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

Sparse Identification of Nonlinear DynamicsNonlinear System IdentificationRobust Regression
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Feb 27, 2026

Physics-Embedded Neural ODEs for Learning Antagonistic Pneumatic Artificial Muscle Dynamics

Pneumatic artificial muscles (PAMs) enable compliant actuation for soft wearable, assistive, and interactive robots. When arranged antagonistically, PAMs can provide variable impedance through co-contraction but exhibit coupled, nonlinear, and hysteretic dynamics that challenge modeling and control. This paper presents a hybrid neural ordinary differential equation (Neural ODE) framework that embeds physical structure into a learned model of antagonistic PAM dynamics. The formulation combines parametric joint mechanics and pneumatic state dynamics with a neural network force component that captures antagonistic coupling and rate-dependent hysteresis. \rev{The forward model was trained on 29 selected co-contraction conditions and predicted joint motion and chamber pressures over 196 held-out conditions with a mean R2^2 of 0.88.} An inverse formulation, derived from the learned dynamics, computes pressure commands offline for desired motion and stiffness profiles, tracked in closed loop during execution. Experimental validation demonstrates reliable stiffness control across 126-176 N/mm and consistent impedance behavior across operating velocities, in contrast to a static model, which shows degraded stiffness consistency at higher velocities.

Xinyao Wang, Jonathan Realmuto

Dept. of Mechanical Engineering, University of California, Riverside, CA, USA

Pneumatic ActuationSoft Pneumatic ActuatorsRobot Dynamics LearningRobotic ControlNonlinear System IdentificationNeural ODEs
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Feb 20, 2026

PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting

Real physical systems are dissipative -- a pendulum slows, a circuit loses charge to heat -- and forecasting their dynamics from partial observations is a central challenge in scientific machine learning. We address the position-only (q-only) problem: given only generalized positions~qtq_t at discrete times (momenta~ptp_t latent), learn a structured model that (a)produces stable long-horizon forecasts and (b)recovers physically meaningful parameters when sufficient structure is provided. The port-Hamiltonian framework makes the conservative-dissipative split explicit via x˙=(J−R)∇H(x)\dot{x}=(J-R)\nabla H(x), guaranteeing dH/dt≤0dH/dt\le 0 when R⪰0R\succeq 0. We introduce PHAST (Port-Hamiltonian Architecture for Structured Temporal dynamics), which decomposes the Hamiltonian into potentialV(q)V(q), massM(q)M(q), and damping~D(q)D(q) across three knowledge regimes (KNOWN, PARTIAL, UNKNOWN), uses efficient low-rank PSD/SPD parameterizations, and advances dynamics with Strang splitting. Across thirteen q-only benchmarks spanning mechanical, electrical, molecular, thermal, gravitational, and ecological systems, PHAST achieves the best long-horizon forecasting among competitive baselines and enables physically meaningful parameter recovery when the regime provides sufficient anchors. We show that identification is fundamentally ill-posed without such anchors (gauge freedom), motivating a two-axis evaluation that separates forecasting stability from identifiability.

Shubham Bhardwaj, Chandrajit Bajaj

Department of Computer Science, Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, USA

Time Series ForecastingPort-Hamiltonian SystemsHamiltonian Neural NetworksNonlinear System IdentificationSystem IdentificationParameter Identifiability
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Feb 17, 2026

Wideband Power Amplifier Behavioral Modeling Using an Amplitude Conditioned LSTM

Wideband power amplifiers exhibit complex nonlinear and memory effects that challenge traditional behavioral modeling approaches. This paper proposes a novel amplitude conditioned long short-term memory (AC-LSTM) network that introduces explicit amplitude-dependent gating to enhance the modeling of wideband PA dynamics. The architecture incorporates a Feature-wise Linear Modulation (FiLM) layer that conditions the LSTM's forget gate on the instantaneous input amplitude, providing a physics-aware inductive bias for capturing amplitude-dependent memory effects. Experimental validation using a 100 MHz 5G NR signal and a GaN PA demonstrates that the proposed AC-LSTM achieves a normalized mean square error (NMSE) of -41.25 dB, representing a 1.15 dB improvement over standard LSTM and 7.45 dB improvement over augmented real-valued time-delay neural network (ARVTDNN) baselines. The model also closely matches the measured PA's spectral characteristics with an adjacent channel power ratio (ACPR) of -28.58 dB. These results shows the effectiveness of amplitude conditioning for improving both time-domain accuracy and spectral fidelity in wide-band PA behavioral modeling.

Abdelrahman Abdelsalam, You Fei

School of Electronic Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China

Nonlinear System IdentificationLong Short-Term Memory NetworksRecurrent Neural Networks
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Feb 17, 2026

Hybrid Model Predictive Control with Physics-Informed Neural Network for Satellite Attitude Control

Reliable spacecraft attitude control depends on accurate prediction of attitude dynamics, particularly when model-based strategies such as Model Predictive Control (MPC) are employed, where performance is limited by the quality of the internal system model. For spacecraft with complex dynamics, obtaining accurate physics-based models can be difficult, time-consuming, or computationally heavy. Learning-based system identification presents a compelling alternative; however, models trained exclusively on data frequently exhibit fragile stability properties and limited extrapolation capability. This work explores Physics-Informed Neural Networks (PINNs) for modeling spacecraft attitude dynamics and contrasts it with a conventional data-driven approach. A comprehensive dataset is generated using high-fidelity numerical simulations, and two learning methodologies are investigated: a purely data-driven pipeline and a physics-regularized approach that incorporates prior knowledge into the optimization process. The results indicate that embedding physical constraints during training leads to substantial improvements in predictive reliability, achieving a 68.17% decrease in mean relative error relative. When deployed within an MPC architecture, the physics-informed models yield superior closed-loop tracking performance and improved robustness to uncertainty. Furthermore, a hybrid control formulation that merges the learned nonlinear dynamics with a nominal linear model enables consistent steady-state convergence and significantly faster response, reducing settling times by 61.52%-76.42% under measurement noise and reaction wheel friction.

Carlo Cena, Mauro Martini, Marcello Chiaberge

Department of Electronics and Telecommunications, Politecnico di Torino, Torino, Italy

Spacecraft Attitude DeterminationModel Predictive ControlFeedback ControlNonlinear System IdentificationPhysics-Informed ML
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Feb 17, 2026

Controlled oscillation modeling using port-Hamiltonian neural networks

Learning dynamical systems through purely data-driven methods is challenging as they do not learn the underlying conservation laws that enable them to correctly generalize. Existing port-Hamiltonian neural network methods have recently been successfully applied for modeling mechanical systems. However, even though these methods are designed on power-balance principles, they usually do not consider power-preserving discretizations and often rely on Runge-Kutta numerical methods. In this work, we propose to use a second-order discrete gradient method embedded in the learning of dynamical systems with port-Hamiltonian neural networks. Numerical results are provided for three systems deliberately selected to span different ranges of dynamical behavior under control: a baseline harmonic oscillator with quadratic energy storage; a Duffing oscillator, with a non-quadratic Hamiltonian offering amplitude-dependent effects; and a self-sustained oscillator, which can stabilize in a controlled limit cycle through the incorporation of a nonlinear dissipation. We show how the use of this discrete gradient method outperforms the performance of a Runge-Kutta method of the same order. Experiments are also carried out to compare two theoretically equivalent port-Hamiltonian systems formulations and to analyze the impact of regularizing the Jacobian of port-Hamiltonian neural networks during training.

Maximino Linares, Guillaume Doras, Thomas Hélie

IRCAM, 1, place Igor-Stravinsky, 75004, Paris, France

Port-Hamiltonian SystemsHamiltonian DynamicsNonlinear System Identification
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