This work proposes neural network nudging, a data-driven method for learning nudging terms in nonlinear state space models and establishes a theoretical existence result based on the Kazantzis--Kravaris--Luenberger observer theory.
Abstract
Nudging is an empirical data assimilation technique that incorporates an observation-driven control term into the model dynamics. The trajectory of the nudged system approaches the true system trajectory over time, even when the initial conditions differ. For linear state space models, such control terms can be derived under mild assumptions. However, designing effective nudging terms becomes significantly more challenging in the nonlinear setting. In this work, we propose neural network nudging, a data-driven method for learning nudging terms in nonlinear state space models. We establish a theoretical existence result based on the Kazantzis--Kravaris--Luenberger observer theory. The proposed approach is evaluated on three benchmark problems that exhibit chaotic behavior: the Lorenz 96 model, the Kuramoto--Sivashinsky equation, and the Kolmogorov flow.
Accurate prediction of nonlinear dynamical systems becomes particularly challenging when the evolution of the dynamics depends on hidden, time-varying factors that are not directly observable. Although reservoir computing (RC) provides an efficient framework for modeling complex dynamics, standard approaches based on a single trained readout often experience reduced accuracy in such non-autonomous settings. We propose a multi-regime RC framework in which multiple readouts are trained under different dynamical conditions and combined through a short observation window to form a trajectory-dependent linear readout. This enables both regime identification and adaptation to unseen or intermediate dynamics. The method is evaluated on a Duffing oscillator with a time-varying forcing input and a Rössler system driven by chaotic forcing from a Chen system. The results show improved prediction accuracy compared to both regime-specific and single global models trained on data aggregated from multiple regimes.
S. Hadipour Lakmesari, H. Kantz, Francesco Sorrentino· Chaos· 0 citations
We present a machine learning framework for identifying sparse, interpretable models of dynamical systems directly from time-series data. Our approach parameterizes the underlying vector field using a neural architecture and trains it by minimizing a multi-step prediction loss over a finite horizon. To ensure numerical tractability, we optimize a mean absolute error objective averaged across prediction steps, and progressively increase the horizon during training. A key feature of this formulation is that it enforces consistency under repeated composition of the learned dynamics. As a result, the identified models exhibit significantly improved stability compared with approaches based on one-step regression of the vector field. When combined with sparsity-promoting regularization, this leads to parsimonious models that generalize beyond the training data. We demonstrate accurate recovery of systems exhibiting a wide range of behaviors, including stable and unstable fixed points, periodic orbits, and chaotic attractors. For chaotic systems, while long-term trajectory prediction is inherently limited by sensitivity to initial conditions, we show that multi-step training yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents. Moreover, we establish theoretical bounds linking trajectory error to statistical accuracy, providing a step toward a principled explanation for this behavior.
In this work, we consider the identification and control of nonlinear systems with finite action spaces. The unknown dynamics are estimated from finite samples with Koopman operator regression in a reproducing kernel Hilbert space, yielding a linear switching predictive model, the switches governed by the value of the control variable. In order to perform control in closed-loop, the learned dynamics are employed in an infinite-horizon optimal control problem with time-varying stage cost, which is solved by means of model predictive control. In a theoretical analysis, we derive learning rates for the Koopman dynamics approximation. We further quantify, under suitable assumptions, the sub-optimality of the model predictive control strategy, both in the case of exact Koopman dynamics, and in the case of learned ones. Numerical simulations on the Duffing oscillator complement our theoretical findings.
Edoardo Caldarelli, Oleksii Kachaiev, C. Molinari et al.· 0 citations
Neural autoregressive models have rapidly emerged as powerful emulators of high-dimensional chaotic systems, yet their long-term instability and error growth remain poorly understood, leading to ad-hoc solutions. Here, we develop an eigenanalysis framework that reveals the dynamical origin of this error growth. By analyzing the Jacobian of the learned one-step update map with respect to the state, we show how inference-time error growth, and thus model stability, is governed by its spectral radius. Direct-step architectures (models that predict the next state from the previous one) generically admit unstable eigenvalues with magnitudes exceeding one, explaining the rapid divergence of these widely used models. In contrast, integration-constrained models (where the time derivative is estimated and integrated with a higher-order integrator) collapse their eigenspectrum onto the unit circle, yielding neutral stability and a universal linear error-scaling law. The largest eigenvalue of this Jacobian provides an architecture-agnostic, a priori diagnostic of short-term skill, long-term stability, and spectral bias, without requiring an expensive rollout. Leveraging this theory, we introduce a stability-promoting loss that explicitly regularizes Jacobian-driven error amplification, improving both forecast accuracy and dynamical robustness. Demonstrated across $29$ models spanning two architectures, several explicit and implicit integrators, and multiple loss functions on the Kuramoto-Sivashinsky system, our results establish a theoretical foundation for the design and evaluation of neural emulators of chaotic multi-scale dynamics. More broadly, our framework is a step toward the kind of a priori stability analysis that numerical analysis provides for discretizations of differential equations and that scientific machine learning currently lacks.
C. Ainslie, P. Hassanzadeh, Michael W. Mahoney et al.· 0 citations
This work examines the nonlinear dynamics of a generalized Korteweg–de Vries–Zakharov–Kuznetsov equation, a model that appears in plasma physics, shallow water flows, and nonlinear wave propagation. By applying a solitary-wave transformation, the governing partial differential equation is reduced to an autonomous dynamical system, enabling a direct study of its phase portraits and equilibrium behavior. Stability of the fixed points is assessed through Jacobian matrices and eigenvalue classification, revealing parameter regimes that admit saddle states, centers, and oscillatory structures. The system’s richer behavior is explored by varying key parameters, with phase-space trajectories exhibiting periodic, quasiperiodic, and irregular wave patterns. To probe the onset of complexity, we employ several diagnostic tools, including time-series evolution, Lyapunov exponents, bifurcation analysis, sensitivity tests, and Poincaré sections, which together indicate transitions to chaotic motion. The resulting dynamics are further captured using a nonlinear autoregressive neural network, which accurately reproduces the observed trajectories. The combination of analytical and computational perspectives provides a clear framework for understanding this generalized equation and offers a practical approach for investigating other nonlinear systems with a similar structure.
Muhammad Ghulam Abbas Malik, Muhammad Mudassir, Zia Bashir· Mathematical and Computation...· 0 citations
This letter develops a data-driven control framework for nonlinear ensemble systems using reservoir computing (RC). We consider ensemble control problems, in which the objective is to regulate a large, potentially uncountable, population of systems with unknown dynamics. To address this challenge, we introduce a moment kernelization approach that yields a dual representation and enables a valid finite-dimensional approximation of ensemble dynamics. Building on this reduction, we cast ensemble control synthesis as the approximation of a causal operator that maps moment trajectories to control inputs. We show that continuous-time reservoir systems induce well-defined causal input-output operators with the fading-memory property, providing a principled foundation for learning these feedback operators from moment trajectory data. Based on this theory, we design an RC-based controller trained on input-output moment trajectories and deployed in a closed-loop configuration for tracking and stabilization of nonlinear ensemble systems.
Yuan-Hung Kuan, Lin Tang, Jr-Shin Li· IEEE Control Systems Letters· 0 citations
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