Dynamical Systems

Identifying linear time-invariant (LTI) dynamical systems is challenging when trajectories are short, noisy, or high-dimensional. Traditional system identification typically treats each system independently and cannot exploit shared structure across related systems. We propose PBML-LTI, a PAC-Bayesian meta-learning framework for few-shot LTI system identification that learns a transferable prior over task-specific dynamics while preserving task heterogeneity. Each task corresponds to an unknown LTI system, and the meta-learner uses training trajectories to learn a data-dependent prior over transition matrices. For a new system with limited data, PBML-LTI performs Bayesian adaptation under this prior to obtain a task-specific posterior, providing accurate estimates and principled uncertainty quantification. A key challenge is temporal dependence, since LTI trajectories violate the i.i.d. assumptions underlying most PAC-Bayes meta-learning analyses. We address this with a martingale PAC-Bayes analysis for dependent trajectory losses and derive a support-query predictive-risk bound that motivates a fit-KL meta-training objective. The bound clarifies the roles of empirical fit, posterior complexity, and prior quality in few-shot adaptation under sequential dependence. We further derive corollaries for transition-matrix recovery and multi-step trajectory prediction, connecting uncertainty-aware meta-identification with finite-sample guarantees for dependent dynamical data.

This work proposes a physics-enhanced machine learning approach for the system identification of Linear Time-Varying (LTV) systems under time-varying operating conditions in terms of fast-varying natural frequencies and damping ratios by combining a long short-term memory network with an Extended Kalman Filter (EKF). The proposed approach uses vibration data (displacement and velocity measurements), domain knowledge of modal damping ratios, and a physics-based model that can yield an approximate natural frequencies time-dependency model. The approach is validated using synthetic data generated from a finite element model of a 2-blade offshore wind turbine under realistic environmental and operating conditions. This system displays fast time-varying frequencies due to operating conditions, whose identification is particularly challenging because of the wind and wave loading. The robustness of the proposed approach is assessed under assumed incorrect system information (e.g. damping ratio). The proposed approach is evaluated across different environmental and operating conditions to show its applicability to different operating regimes. The results show the approach can accurately identify the selected fast-varying natural frequency, 1st Fore-Aft (FA-1) mode, with a maximum root mean square error of 0.0012 Hz. The results demonstrate that the model trained on EKF estimates depends on accurate damping values, whereas the model trained on physics-based data exhibits robustness to incorrect damping assumptions. The approach is extended to damping ratio identification for the selected mode by estimating the root mean square error between models trained on EKF estimates and physics-based data. The results show that the approach can yield a good approximation of the FA-1 mode damping ratio using grid search, offering an improvement over covariance-driven stochastic subspace identification.

Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.

Many real-world processes exhibit long-range dependence, where the current state depends on a slowly decaying trace of past states rather than on the most recent state alone. This paper studies system identification for discrete-time fractional-order linear time-invariant systems from a single observed trajectory of length $t$, a setting that captures such non-Markovian dynamics through the Grünwald--Letnikov difference operator. Unlike Markovian systems, fractional-order systems couple estimation across the entire history, making both statistical analysis and practical identification more challenging. We propose \emph{Fractional-Order Ordinary-Least-Squares Grid-Search (FO-GS)}, a simple two-stage estimator that exploits the diagonal structure of the fractional-difference operator to decouple the identification problem row-wise. Under the stability assumption, we establish high-probability, non-asymptotic error bounds for estimating both the fractional order and the system matrix in the heterogeneous setting, with both estimation errors scaling as $\mathcal{O}(t^{-1/2})$. Through experiments, we show that \emph{FO-GS} outperforms existing baselines in recovering both the fractional order and the underlying system dynamics.

Generative modeling of time-dependent data is typically formulated on a discrete temporal grid, restricting supervision to the observed timestamps in the training data. We instead frame generation as continuous-time evolution on a learned data manifold. To this end, we leverage pretrained score-based models as geometric priors and learn a vector field that evolves data along score-induced interpolation paths. Because these dynamics follow transitions that respect the geometry learned by the score model, they support generation at arbitrary timestamps and temporal super-resolution beyond the discretization of the training data. Moreover, this geometric formulation allows us to train the vector field simulation-free through a regression objective. To improve long-horizon rollout robustness, we introduce an objective that promotes path-relative transverse exponential stability. While motivated by stability theory, it admits a practical interpretation as denoising score matching transverse to the interpolation path. Further, we extend the framework to a probabilistic setting that models a distribution over plausible future trajectories. We demonstrate the method on natural video and scientific dynamical data, including temporal super-resolution, PDE-based spatiotemporal fields, and molecular dynamics. Our results show that score-based priors provide a strong foundation for learning stochastic continuous-time generative dynamics.

Legged robots have demonstrated a remarkable ability to traverse various terrains, yet generating effective loco-manipulation behaviors remains challenging. A key difficulty is that object and terrain parameters are typically unknown to the robot, and mismatches between these parameters and their simulated counterparts introduce a sim-to-real gap that degrades control performance. Classical system identification (Sys-ID) methods often assume differentiable dynamics, an assumption that does not hold for contact-rich legged systems. Sampling-based Sys-ID avoids this restriction by directly matching simulated and recorded state trajectories through massively parallel rollouts, but existing approaches are typically applied offline and do not adapt as environmental conditions change. We present Adaptive-MHE an online sampling-based Sys-ID framework, based on moving horizon estimation (MHE), that estimates the physical parameters of objects and terrain in the environment (e.g., mass, friction) and couples this estimate with a sampling-based model predictive controller, enabling adaptive loco-manipulation in changing and uncertain environments. In simulation and hardware experiments, our framework consistently outperforms baselines and matches the performance of a controller with access to ground-truth parameters.

The Koopman operator has been widely used for time-series prediction in dynamical systems. However, prior work that learns latent ``Koopman spaces'' using neural networks often did not construct a valid Koopman space for forecasting, as these representations may be mathematically inconsistent with the operator-theoretic formulation and fail to capture the intrinsic low-rank structure of system dynamics. To address this issue, we introduce K$^2$SVD, a method that explicitly learns the leading singular functions of the Koopman operator by optimizing a Hilbert-Schmidt objective. This yields a well-defined low-rank approximation of the Koopman operator with an interpretable linear combination, featuring a compact latent space with less than $10\%$ of the dimensions used in previous work. In the learned Koopman space, K$^2$SVD further captures temporal evolution with a linear Gaussian state-space model and performs inference via Kalman filtering, mitigating noise accumulation during multi-step prediction. Empirical results show that K$^2$SVD outperforms state-of-the-art methods across multiple datasets, with significantly faster prediction speeds and lower computational cost than previous efficiency-focused models. This highlights the benefits of principled low-rank Koopman representations and opens up broader potential for applications.

This paper proposes a least squares approach for the training of quadratic neural networks with regularization. The proposed methodology yields a lower bound on the solution of the training optimization problem for the case where the regularization coefficient is positive. Moreover, it yields closed-form expressions for the approximate solution and its sensitivity The lower bound is tight and the approximate solution is the optimal solution when the regularization coefficient is zero. Having a closed-form expression for the weights reduces considerably the computational time when compared with iterative numerical methods such as backpropagation that can get stuck in local minima. The proposed approach has three main contributions, namely, (i) it yields an analytical expression for the weights, (ii) an analytical expression for the sensitivity of the weights to errors in the data is also provided, (iii) it establishes a connection between the optimization to compute a lower bound and nuclear norm minimization. The proposed least squares training is successfully applied to a nonlinear system identification example where the proposed lower bound is compared with the optimal value.

In this work, we propose a structured methodology for the system identification of underwater vehicles through the design of optimal excitation trajectories. To this end, the trajectories are parameterized using Bezier curves, which ensure smooth and differentiable motion profiles while facilitating the enforcement of constraints through appropriate manipulation of the control points. An optimization problem is formulated to determine a dynamically feasible excitation trajectory that respects safety limits and maximizes the quality of the collected data, thereby enabling reliable estimation of the vehicle's dynamic parameters using least squares. The proposed methodology is experimentally validated in a laboratory water tank, where the dynamic parameters, identified from the optimized trajectory, are evaluated by predicting the vehicle's velocity through forward simulation on previously unseen trajectories.

We present the development and application of a three-dimensional Physics-Informed Neural Network (PINN) framework for the investigation of haemodynamic behaviour in the human aorta. The model incorporates a time-resolved simulation of pulsatile blood flow over a two-minute interval, enabling the extraction of pressure and velocity fields with high temporal fidelity. The mechanical stress exerted on the aortic wall was quantified through Laplace's law, with temporal averaging applied to derive representative stress distributions. This approach circumvents the computational overhead associated with conventional computational fluid dynamics (CFD) methods by eliminating mesh generation and exploiting the automatic differentiation capabilities inherent to neural networks. The proposed methodology demonstrates that PINNs can serve as an efficient and accurate alternative for modelling complex vascular flow phenomena, offering significant advantages in scalability and computational cost reduction while maintaining physical consistency.

Recent work has identified incremental learning in shallow networks trained on single-index and multi-index models. However, existing analyses often rely on simplifying settings, such as small initialization, correlation loss, or layer-wise training. These choices reduce neuron interactions and leave some feature learning dynamics under standard initialization unexplored. We study training dynamics for polynomial-width two-layer networks learning orthogonal multi-index targets under standard initialization using polynomially many samples. We first prove that incremental learning still occurs: the loss decreases sequentially according to the Hermite expansion of the target, with lower-order components learned before higher-order components recover the individual target directions. In this standard initialization regime, training also shows a competitive reallocation of parameter mass: after the total mass fits the target mean and stabilizes, mass shifts into the target subspace and then concentrates on aligned neurons. Our theoretical analysis uses slightly modified gradient flow, while vanilla gradient descent empirically exhibits the same qualitative dynamics. Technically, we introduce a symmetry-based finite-width approximation via symmetrized networks, rather than comparing directly with an infinite-width limit. This yields better control of approximation errors and may be of independent interest.

We propose a learning procedure for system identification in interacting particle systems from single-snapshot observations of collective behaviors, unlike existing approaches that rely on observations of trajectories. This setting leads to a fundamentally ill-posed inverse problem, which we solve by using a regularization strategy based on the empirical distribution of observed configurations, drawn from different, unobserved initial conditions. We test our learning procedure on a variety of representative models with steady-state and quasi-stationary patterns, where collective behaviors encode implicit information about the interaction mechanisms, demonstrating that our approach enables stable and accurate recovery of the underlying interaction laws, leading to faithful reproduction of the collective behavior, and in many cases even of the dynamics leading up to it.

In recent years, weak-form methods have made significant advances in data-driven discovery of dynamical systems. However, in high-dimensional settings, current techniques can prove expensive in both computation and memory. In this work, we introduce TT-WSINDy, which combines techniques of the Multidimensional Approximation of Nonlinear Dynamics (MANDy) and Weak Sparse Identification of Nonlinear Dynamics (WSINDy) methods, implementing requisite computations in the tensor-train (TT) format. We demonstrate that this method is able to search an exponentially-growing space of candidate functions -- performing weak-form transformation, regression, and sparsification -- without suffering from the curse of dimensionality.

Numerical simulation of dynamical systems is usually organized as a causal march through time: each state is computed from the previous one. We explore a different formulation for coupled systems. For each subsystem type we train a neural surrogate mapping a full driving trajectory and initial condition directly to a full output trajectory; following classical waveform relaxation, coupled systems are assembled by enforcing self-consistency among these trajectories: simulation becomes a fixed-point problem over complete trajectories rather than a stepwise rollout. On coupled van der Pol oscillators and Hodgkin-Huxley neuron networks, sequential depth becomes the number of solver iterations: 4-10 Newton iterations where the reference integrator takes 1500 steps. The gradient likewise loses its time recursion: it becomes a linear system solved by GMRES at memory independent of solver depth. A single scalar measured from the learned operator, the spectral radius of its Jacobian, predicts in advance where the coupled solve will converge; past that boundary, unrolled backpropagation diverges and a Neumann adjoint fails, while the implicit gradient remains correct to 0.04%. We report where the approach succeeds and where surrogate error degrades it.

Animals' body morphology shapes the gait patterns they can perform, where mechanical resonance reduces the need for active control. By tuning posture and muscle stiffness, they leverage their embodied intelligence to achieve effective gaits for different speeds. In contrast, most quadruped robots are not specifically designed to exploit mechanical resonance due to the complexity of nonlinear dynamics and require dedicated locomotion controllers. To provide an alternative, we present a proof of concept framework making the nonlinear dynamics of a robot predictable in the design process and show how this knowledge can be leveraged such that multi-gait locomotion can emerge from nonlinear resonances, shaped by gravity, inertia, and elasticity. We present the highly compliant quadruped robot eBert, on which we identify six nonlinear normal modes (NNMs) using our new theoretical tools and validate their existence in simulation and hardware. With black-box optimization to determine step length, simulations show how each NNM naturally develops into a distinct gait, manifesting different speeds, which also largely transfers to the robotic hardware. Our experiments show that eBert can exploit its mechanics to generate task-specific movements which may serve as foundation for designing a new generation of agile and efficient robots leveraging embodied intelligence.

Accurate dynamic models play a central role in achieving reliable control of quadcopters. Classical system identification methods remain widely used, mainly because of their interpretability. However, they often fail to capture important nonlinear effects, especially in small-scale aerial platforms where such effects become more pronounced. Data-driven approaches offer a different perspective. They can represent complex nonlinear dynamics more effectively, but this comes at the cost of reduced interpretability and the absence of well-calibrated uncertainty estimates. In this work, we propose a framework that combines physics-based modeling with data-driven learning, while explicitly accounting for uncertainty. A physics-based model is first identified using the Prediction Error Method (PEM), which captures the main structure of the system. The remaining dynamics are then modeled using a Gaussian Process (GP), allowing the residual behavior to be learned directly from data. This separation makes it possible to distinguish between known physical effects and unmodeled dynamics. The proposed framework is validated on a Duckiedrone-like experimental setup. The results show that the PEM-GP model achieves prediction accuracy comparable to that of a Long Short-Term Memory (LSTM) network, while additionally providing calibrated uncertainty estimates. This combination improves model reliability and supports uncertainty-aware decision-making.

Simulation of admittance-type models requires physically consistent dynamic models that are rarely available for off-the-shelf, everyday objects, limiting the fidelity of haptic interfaces that rely on such simulations. This paper presents the first complete workflow for producing physically consistent models of large real-world objects with various constraints and mechanisms, guaranteeing physical consistency of inertia and friction parameters. The workflow separates each object and identifies the handle and body in two stages, requiring no torque sensors at hinges, axles, or other constrained joints. Models are produced for a heavy, closer-actuated door and a wheelbarrow, representing objects of differing constraint types and model complexity. The door is modeled using four-bar linkage kinematics and a fluid dynamics-based lumped parameter model including opening, backcheck, swing, and latch zones. The wheelbarrow is modeled as a rigid body with a spherical wheel and no slip during rolling. Handle estimation RMS errors were below 0.64 N and 0.042 Nm across both objects. Door body estimation had RMS error of 2.19 Nm and wheelbarrow body estimation had RMS error of 6.77 Nm.

This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require ``anti-collapse mechanisms'' to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.

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.

Rendering haptic feedback with nonlinear virtual environments (VEs) is important in many applications that require highly accurate force feedback. This paper considers the use of the Koopman operator to represent a nonlinear VE interacting with a haptic system. Simulation and experimental results demonstrated that the proposed method provides an effective representation of the nonlinear dynamics of a Duffing-oscillator VE. A multi-user study further confirmed this conclusion. In addition, a closed-loop (CL) stability analysis is performed leveraging the Koopman representation of the nonlinear VE to access stability of the overall haptic system. This alternative way of representing nonlinear VEs enables a convenient CL stability analysis that is less conservative than traditional passivity-based methods. Since a linear combination of all lifted states is used to represent the nonlinearity, such representation is also more robust to uncertainties in the modeling of the haptic device than a traditional nonlinear model.

Forward and inverse modeling of parametric dynamical systems requires surrogate models that are not only accurate for state prediction, but also informative for parameter calibration. However, a systematic end-to-end differentiable formulation for coupling deep-learning-based reduced-order surrogates with variational parameter estimation remains underdeveloped. In this work, we introduce a physics-aware neural-network-based latent-space framework for reduced-order forward modeling and variational parameter estimation. The proposed autoencoder-based approach yields a differentiable surrogate that maps physical parameters to predicted flow fields through a latent representation. The observable supervision is used during offline training to encourage the latent variables to retain information correlated with system parameters, while the online inverse problem is solved in the parameter space through the surrogate-induced observation operator. The method is evaluated on two computational-fluid-dynamics benchmarks. The results show that reconstruction accuracy alone is insufficient for inverse modeling, owing to the lack of end-to-end differentiability or physics awareness for variational parameter calibration. Quantitative latent-space analysis further shows that observable supervision improves case-level separability and temporal organization of latent representations. Experiments with realistic measurement settings, including noisy, low-resolution, randomly masked, and block-wise partial observations, demonstrate the robustness of the proposed framework and show that it generally reduces calibration error and variability compared with the standard surrogate models.

Industrial time-series signals, such as turbine temperature and rotational speed in aero-engines, are essential for monitoring the health and operational status of complex dynamical systems. However, collecting such data is often limited by harsh environments (e.g., high temperature and high pressure) and the high cost of experimental testing. To address this challenge, we introduce PhysDGM, a stepwise physics-embedded diffusion generative model for synthesizing time-series data that are consistent with the underlying physical laws of dynamical systems. PhysDGM embeds physical laws directly into each reverse diffusion step of the generative process, ensuring trajectory-level physical consistency, rather than enforcing constraints only at the final output. A large-scale AI-synthetic dataset (4.4 million samples, 20x scale-up) constructed by PhysDGM demonstrates strong fidelity across 34 datasets spanning turbofan engines, aero-engines, batteries, and chemical processes. After incorporating the synthetic data, the downstream task performance substantially surpassed that using real data alone by 48% for remaining useful life prediction, 15% for health indicator estimation, 22% for state-of-health assessment, and 20% for fault diagnosis. Moreover, it requires 10-20x less training data than existing approaches, substantially reducing the high cost of data collection in dynamical systems. We further demonstrate PhysDGM's potential in identifying early-stage faults in aero-engines by incorporating AI-synthesized data. In summary, PhysDGM provides a solid foundation for generating physically consistent industrial time-series, paving the way for expanding physics-guided AI into diverse data-scarce environments, including both industrial machinery and complex chemical reaction dynamics.

We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strategy partitions the horizon adaptively and restarts every window from observed data: when training meets a prescribed tolerance on every window, the composite model meets it uniformly in time, with a parameter budget linear in the horizon for targets with a bounded, uniformly regular reachable tube. The Floquet strategy addresses autonomous targets with a stable limit cycle and uses no data at deployment: a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods. For the time-periodic architecture we deploy, the scalar certificate degenerates; we prove instead a uniform-in-time orbital guarantee whose hypotheses are measured on the trained model, and an obstruction showing that, for an exactly periodic learned field, small one-period error and a contracting stroboscopic map cannot hold at once. Numerical experiments on four benchmarks confirm the predicted error laws and measure the hypotheses of every guarantee.

Popularity bias in recommendation systems arises when a majority user class generates disproportionate interaction data, causing the system to increasingly favour it while degrading recommendation quality for niche users. While extensive empirical evidence of popularity bias exists, the dynamics leading to its emergence are not well understood. In this work, we study the coupled evolution of recommender model updates and user engagement through the lens of dynamical systems. We formulate a stochastic process and analyse its asymptotic behaviour through an ordinary differential equation (ODE) framework grounded in two-time-scale stochastic approximation. We characterise the equilibrium points of this dynamical system, and derive conditions under which popularity bias is provably emergent, as well as conditions under which symmetric retention of all user classes is possible. We conduct experiments on synthetic data and real-world production logs derived from a large-scale commercial music recommendation platform to validate our theoretical results.

What should a machine learning model learn when data is missing during training? We look at the learning process from a dynamical systems perspective, cast data missingness as a structured loss of actuation that limits controllability of the parameter error dynamics, and ultimately derive adaptation mechanisms with Lyapunov stability characteristics that throttle model updates in ways that preserve learning coherence under partial, intermittent observability. Under recurrent excitation, our analysis provides ISS-type residual-to-state bounds with respect to a bounded closed-loop mismatch between the loss residual and the preconditioned update geometry. We evaluate the efficacy of our directional observability-aware adaptive learning approach on multimodal contexts, reinforcing its premise in promoting learning coherence and stability even in pathologically sparse domains and problems.

Ensemble forecasting aims to sample the conditional distribution of outcomes; whether AI forecast ensembles do this correctly in a joint sense remains largely untested. We train a diffusion model for probabilistic subseasonal coastal sea level forecasts at eight US East Coast tide gauge stations, with sea level derived from reanalysis, and find that marginal and joint forecast quality decouple: positive skill at every station and lead time marginally, while joint spatial structure is worse than climatological draws. A shuffle-based permutation decomposition reveals this failure is invisible to the energy score but detected by the variogram score. Lorenz-96 experiments across 0.7-170 equivalent years show the gap persists regardless of training volume and is reproduced by a linear baseline, indicating structural inadequacy of the learned distribution. A dynamical ensemble does not replicate the failure while a deterministic emulator does, suggesting it is specific to learned emulators rather than ensemble forecasting generally.

Koopman theory offers a linear-operator view of nonlinear sequence dynamics by lifting observations into a space where evolution is governed by a linear time-invariant Koopman operator. While the Koopman operator provides a linear representation of nonlinear dynamics, it is generally infinite dimensional and defined under time-invariant assumptions. To model non-stationary signals with frequency-dependent behavior, a frequency-varying extension is required. In recent years, deep learning has been increasingly employed to exploit its powerful function-approximation ability for learning the Koopman operator. In this study, we propose a novel approach called neural decomposition Koopman (NDKoop), an end-to-end architecture that integrates a learnable signal decomposition module with both frequency-independent and frequency-dependent Koopman based networks for sequence forecasting. To the best of our knowledge, this is the first work to jointly realize end-to end Koopman modeling and signal decomposition within a unified neural framework. We demonstrate that decomposing a signal into a frequency-independent trend component and a frequency-dependent periodic component, each governed by a corresponding Koopman operator, improves prediction accuracy when perfect linearization is unattainable. Numerical experiments across several forecasting benchmarks indicate that the proposed NDKoop provides strong performance.

Dynamical systems model trajectory data generated by fixed underlying dynamics, with applications ranging from biology to physics. Especially in scientific settings, dynamical systems are not generic but often exhibit symmetries imposed by physical laws, formalized through equivariance with respect to group actions. The identification problem concerns recovering the parameters of a system from observed trajectories. In this work, we study adaptive symmetry discovery for dynamical system identification and address how a system can be identified from a single trajectory when it is equivariant with respect to an unknown symmetry group. To this end, we first show that for known symmetries, the system can be identified from a significantly shorter single trajectory than in the generic setting, and we precisely characterize this improvement. We then consider the automatic symmetry discovery setting, proposing a method to learn the symmetry group directly from a single trajectory and incorporate it into the identification procedure, achieving the same optimal trajectory length as in the known-symmetry case. Our analysis relies on tools from group representation theory and the expander properties of Cayley graphs, and may be of independent interest for the study of symmetries in dynamical systems.

We develop a formal framework for analyzing indirect geoeconomic influence. The influencing state (sender) does not attempt to change a target nation's policy directly. Instead, the sender restructures the target's internal political economy so that its own citizens, firms, and institutions generate the compliance pressure. The framework rests on a switching dynamical system (SDS) in which a target's political economy evolves under mode-dependent rules. We analyze two modes: a permissive mode, in which a mechanism transmits pressure toward the sender's preferred policy, and a contested mode, entered naturally once the target detects and attributes the mechanism. Crucially, the sender's mechanism design shapes the transition into the contested mode rather than paying a static toll for legibility. This inverts the usual regime-switching problem: rather than estimating a latent transition kernel from data, the designer engineers the kernel to steer regime occupancy over a planning horizon. A structured switch vector decomposes any mechanism along discrete design dimensions, and a combinatorial optimizer searches this space for high-performing archetypes scored on compliance, time-to-threshold, and a durability ratio. We characterize mode-conditional equilibria and derive comparative statics on credibility and legibility, showing that the legibility penalty is scaled by the salience of the government channel and therefore interacts with the mechanism's cost incidence. We illustrate the framework with two stylized mechanisms, report a proof-of-concept simulation over a reduced switch space, and report a small blind-audit study of the pipeline's optional language-model generation stage.

Developing nonlinear models that are both expressive and computationally efficient remains a challenge in machine learning and nonlinear system identification. Tensor network kernel machines (TNKM) address this challenge by combining nonlinear feature representations with compact low-rank tensor-network parameterizations. However, practical and extensible software frameworks for developing TNKM models remain limited. In this work, we introduce "tnkm", an open-source Python library for constructing and training TNKM models using JAX. The library provides a unified interface for combining different feature maps, tensor-network architectures, and optimization strategies, including alternating least squares and gradient-based methods. We demonstrate the capabilities of "tnkm" on nonlinear benchmark problems, showing that the implemented models achieve competitive prediction accuracy while retaining compact parameterizations and efficient training. The proposed framework facilitates reproducible development and application of tensor-network-based learning methods.

Although remarkable capabilities have been demonstrated by Large Language Models (LLMs) across scientific domains, feedback controller design remains underexplored. Existing benchmarks focus mainly on linear single-Degree-of-Freedom (DoF) systems and large API-hosted models, leaving performance on complex controller-design tasks and feasibility for edge deployment unclear. To address these limitations, we introduce the Complex Dynamics-to-Control Benchmark for Large Language Models (CoDyControlBench), comprising 132 system configurations across five evaluation dimensions: number of DoF, system type, coupling level, damping regime, and controller type. Six state-of-the-art LLMs were evaluated over three independent runs, including three commercial models (GPT, Gemini, and Claude) and three open-source models (GLM, DeepSeek, and Qwen). GPT achieved the highest design success rate at 94.8%, whereas Qwen showed the lowest rate at 50.0%. Across the benchmark dimensions, DoF and controller type exhibited the largest model-averaged variations in design success, with success-rate ranges of 36.3% and 17.6%, respectively, exceeding those associated with system type, coupling level, and damping regime. Comparison of GPT and Qwen showed that their performance gap arose mainly from the control-design knowledge, particularly gain selection and the use of transient-limiting mechanisms. For edge deployment, a specialized 1.5B-parameter model was developed through reasoning distillation. The reasoning-distilled model outperformed the answer-distilled and base model on CoDyControlBench, maintained stable performance across 1-6 DoFs, and achieved successful traget tracking in all three physical trials on a pneumatic-artificial-muscle-driven robotic arm. These results establish a benchmark baseline and highlight the potential of lightweight, edge-deployable controller-design models.

Can nonlinear dynamical systems be learned through a compact linear state-space representation, without directly solving a non-convex system-identification problem? We give a provable pipeline for doing so. Starting from observations of an unknown nonlinear dynamical system, we first learn an implicit spectral predictor using Observation Spectral Filtering (OSF), a convex method that competes with the best linear observer for the system. We then apply spectral-to-LDS distillation to convert this predictor into an explicit recurrent linear dynamical system. Our main theorem shows that the average prediction error of the distilled LDS decomposes into an exponentially-small distillation term and the OSF learning term governed by the Luenberger complexity of the best observer. The guarantee is dimension-free: it depends on observer complexity rather than on the latent dimension needed to represent the nonlinear system. To our knowledge, this yields the first end-to-end provable method for extracting a best-in-hindsight LDS representation of nonlinear dynamics through convex learning followed by provable distillation. Experiments on linear LDS benchmarks and MuJoCo behavior cloning show that the train-then-distill pipeline produces compact LDS predictors that match or outperform directly trained baselines.

Echo State Networks (ESNs) offer an efficient framework for temporal prediction, but their randomly initialized reservoirs are often over-parameterized and dynamically redundant. Existing pruning methods largely rely on static connectivity or activation statistics, which may overlook neurons that shape input-driven state transitions. We propose Dynamical Mode Pruning (DMP), a reservoir pruning method that ranks neurons by their contribution to dominant transition modes obtained from a trajectory-averaged Jacobian Gramian. DMP removes low-impact units and retrains only the readout. Experiments on chaotic and real-world time-series benchmarks show that DMP improves or preserves forecasting accuracy while reducing redundant reservoir components. Our results suggest that dynamical influence is a useful criterion for reservoir refinement beyond static structural importance alone.

Time series in real-world applications are often generated by nonlinear dynamical systems, making accurate forecasting challenging. Existing approaches that explicitly model system dynamics typically rely on linear assumptions or Koopman-based linearizations, which may inadequately capture complex nonlinear behaviors and lead to error accumulation in long-horizon prediction. To address this limitation, we propose the Neural Bilinear Dynamical Model (NBDM), which models nonlinear system dynamics through a bilinear latent dynamical formulation. Specifically, NBDM leverages Koopman theory to lift the original nonlinear dynamics into a higher-dimensional latent space, where a bilinear dynamical model is constructed to characterize state evolution. To mitigate the approximation error introduced by bilinear representations, we further incorporate a parameterized error compensation term. Within this formulation, control inputs are explicitly integrated into the dynamics, using auxiliary variables when available and learned feedback signals otherwise. To handle scenarios with missing control inputs, we design a memory-enhanced controller that infers latent controls through multiplicative interactions between historical states and control signals. Experiments on five real-world datasets demonstrate that NBDM consistently outperforms competitive baselines in both given-control and missing-control settings, particularly for multi-step and long-horizon forecasting.

Pretrained large language models (LLMs) have demonstrated in-context learning (ICL) capabilities for numerical inference over sequences serialized as text. Prior work has identified and characterized this form of numerical inference primarily through output-level evaluations such as prediction error. However, how numerical information is organized within LLM representations remains much less understood. To study this internal organization, we adopt a graph signal processing perspective in which attention induces a weighted graph over tokens, while token hidden states define signals on its nodes. Quantitative graph-spectral diagnostics and qualitative token-graph visualizations reveal that representations become more clearly differentiated by input dynamical complexity as context length increases. Simpler inputs produce attention-induced token graphs with stronger global connectivity and smoother, spectrally concentrated hidden-state signals, whereas more complex inputs produce more localized graphs and hidden-state signals with broader spectral support and greater high-frequency energy. Together, these findings point to systematic, context-dependent internal signatures associated with numerical ICL that are conserved across model families.

Reinforcement learning and control theory are two adjacent scientific fields that focus on optimizing the controller of unknown dynamical systems using feedback. While both fields have common roots in dynamic programming, they have evolved with distinct methodologies, goals, and cultures. Despite decades of mutual influence, a significant gap persists between the two communities. This tutorial introduces adaptive control, actor-critic reinforcement algorithms, and a new way to combine these two paradigms for data-driven decision making on a classical locomotion control problem. Our aim is to provide a foundation for understanding the core differences between the two approaches and insights to help experts in each field better understand and engage with the tools and approaches of the other.

Physics-informed neural networks (PINNs) combine sparse observations with physical equations, providing an important approach for modeling complex plasma processes and inferring unknown physical quantities. The steep-gradient pedestal of high-confinement-mode tokamaks is closely linked to plasma confinement and edge transport. Analyzing ion-temperature-gradient (ITG) drift waves in this region requires jointly identifying complex eigenfrequencies and reconstructing two-dimensional complex-valued mode fields. Localized high-frequency oscillations, strong real-imaginary coupling, and nonlinear coupling between the mode field and eigenfrequency challenge PINN representation and joint optimization. To address these challenges, we propose a physics-informed neural framework combining Fourier feature encoding, complex-valued feature propagation, and three-stage training. Under sparse observations and physical constraints, it jointly solves for the complex eigenfrequency and mode field of a representative ground-state ITG branch. Experiments show that the framework accurately recovers the target complex eigenfrequency and two-dimensional complex-valued mode field and outperforms representative PINN baselines. It also provides a basis for analyzing higher-order and multiple-branch drift-wave modes.

This paper studies feedback Nash equilibrium (FBNE) seeking for multi-agent trajectory planning in nonlinear dynamical systems with unknown agents' objectives and state-dependent inter-agent coupling. While dynamic game theory provides a principled framework for such problems, existing approaches typically assume fully rational agents with known objectives or rely on fixed regularization, limiting their ability to capture bounded rationality and spatially varying interaction intensity in safety-critical settings. To this end, we propose a KL-regularized dynamic game with a state-dependent weight that adaptively balances optimality and behavioral priors. To infer unknown cost parameters from demonstrated behaviors, we develop a context-aware inverse game module based on maximum-entropy inverse reinforcement learning with physics-informed regularization, ensuring structural consistency with the forward game. We establish per-iteration well-posedness of the regularized local game and show that the adaptive weighting function remains Lipschitz continuous under bounded nominal-trajectory updates. Numerical simulations and multi-robot experiments on cooperative navigation and merging scenarios validate the effectiveness of the proposed framework.

Increasing the temperature of an ordinary many-state system increases access to a wider range of states and hence increases its entropy. We find the opposite in ChatGPT-like AIs, even though raising the decoder temperature likewise increases access to a wider range of states (next-token choices). Across 12,000 continuations from 11 AIs, autoregressive feedback drives the long-time output population through an entropy maximum and into population inversion. The transition features frozen states, cycles, intermittency and noise-induced ordering. We present evidence of a hidden coordinate that acts as the state variable of an effective nonlinear map. Its trajectory average strongly predicts output repetition in separate test trajectories. ChatGPT-like AIs therefore behave not as `stochastic parrots', but as a new class of controllable nonlinear physical systems whose internal dynamics can be measured and perturbed.

Understanding and controlling complex dynamical systems often requires executing thousands of numerical simulations across vast parametric landscapes, which is time-consuming. Machine learning surrogates significantly accelerate simulation by predicting state trajectories across different initializations and parameter values. However, surrogate models are specialized to one simulation model. Modifying the underlying differential equations - e.g., adding a physiological state or altering an epidemiological contact network - renders trained models obsolete and forces computationally expensive retraining from scratch. We introduce HyperODE, a surrogate capable of operating across an entire class of approximately mass-conserving compartmental models without retraining. By mapping the structure of ordinary differential equations (ODEs) into directed hypergraphs, HyperODE decouples the functional form of system interactions from the neural network architecture. HyperODE takes a compartmental model in the form of an ODE with an arbitrary parameter distribution defined through quantiles and transforms it into a hypergraph. It outputs the distribution of the trajectories for all the states in the original ODE in the form of quantiles. We then use this surrogate to build an encoder that takes a noisy trajectory and outputs a distribution over the parameters of the original ODE, thus calibrating the model in a single pass. On families and system sizes never seen in training, HyperODE produces calibrated quantile bands in a single forward pass, with weighted-interval score and coverage on par with specialized surrogates for each structure. For inverse inference, HyperODE produces calibration from noisy state trajectories in a few milliseconds with a single shared encoder, competitive with existing methods. HyperODE extends zero-shot to ODEs that break mass conservation and to external forcing.

Reliable forecasting of nonlinear physical systems underpins scientific discovery and engineering decision-making. Yet high-fidelity simulations are prohibitively costly, and machine-learning surrogates can be opaque and encode assumptions about system dynamics, limiting generalizability. Pretrained transformers mapping synthetic ODE trajectories to equations offer interpretable alternatives, promising transfer without system-specific equation knowledge. Transferring them reliably to high-dimensional physical data, however, remains an open challenge. We develop a verifier-guided (VG) workflow around ODEFormer as a symbolic backbone, using dynamical and physical-admissibility criteria to select from a multi-trajectory candidate equation pool, enabling transfer. On canonical Van der Pol oscillators, VG outperforms the original ODEFormer workflow across held-out initial conditions. We then address vortex shedding, a phenomenon occurring in atmospheric and plasma systems of societal relevance, through coordinate reduction and symbolic discovery at fixed and varying Reynolds numbers. VG discovers fixed-parameter reduced-order equations that recover the fundamental shedding oscillator and higher harmonics without a wake-specific candidate library or prescribed Navier-Stokes structure, while the cross-parameter model generalizes to withheld regimes. Reconstruction fidelity alone did not determine symbolic discoverability, highlighting the importance of compatibility between latent dynamics and the backbone's pretraining distribution. This work establishes a verifier-guided neural-to-symbolic methodology for interpretable and physically auditable forecasting in the natural sciences.

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.

We present a novel approach to regression tasks using classification which is motivated by the mechanism used by fruitflies to sense their environment. Specifically, we formulate a general framework for learning nonlinear input-output relationships by replacing complex global surrogate models with a finite library of representative local patterns. Since scientific data often occupy limited and recurring regions of the input space, we generate predictions by measuring similarities between a query and stored patterns, then combining their associated responses through weighted reconstruction. We apply this approach to nonlinear dynamical systems, data-driven regression, and physics-informed learning using suitable embeddings and similarity measures. For dynamical systems, our offline-online workflow extracts patterns from data or governing equations during the offline phase, while online prediction requires only similarity evaluation and response aggregation. This structure helps us reduce computational and memory demands while providing explicit control over the trade-off among accuracy, storage, and inference cost.

Modeling and forecasting nonlinear dynamics under distribution shifts is essential for robust decision-making in real-world systems. In this work, we propose MetaKoopman, a Bayesian meta-learning framework for modeling nonlinear dynamics through linear latent representations. MetaKoopman learns a Matrix Normal-Inverse Wishart (MNIW) prior over the Koopman operator, enabling closed-form Bayesian updates conditioned on recent trajectory segments. Moreover, it provides a closed-form posterior predictive distribution over future state trajectories, capturing both epistemic and aleatoric uncertainty in the learned dynamics. We evaluate MetaKoopman on a full-scale autonomous truck and trailer system across a wide range of adverse winter scenarios, including snow, ice, and mixed-friction conditions, as well as in simulated control tasks with diverse distribution shifts. MetaKoopman consistently outperforms prior approaches in multi-step prediction accuracy, uncertainty calibration, and robustness to distributional shifts. Field experiments further demonstrate its effectiveness in dynamically feasible motion planning, particularly during evasive maneuvers and operation at the limits of traction. Project website: https://mahmoud-selim.github.io/MetaKoopman/

Positive quadratic networks admit the low-rank representation f_U(x)=x^top UU^top x, where Uinmathbb{R}^{dtimes r} is identifiable only up to right orthogonal multiplication, representing a rank-r PSD matrix Q=UU^top. We study how this quotient structure governs training dynamics, curvature, recovery, and interpolation bias. On the full-column-rank stratum, we identify mathbb{R}^{dtimes r}_*/O(r) with the rank-r PSD manifold. For smooth objectives L(U)=ell(UU^top), the Euclidean factor gradient is horizontal. Thus, factor gradient flow projects exactly to quotient Riemannian gradient flow, while finite-step gradient descent induces an exact congruence recursion for the predictor. For quadratic regression, we derive the effective Hessian at interpolators as the empirical measurement Gram form restricted to the tangent space relative to the quotient metric. Under Gaussian rank-one measurements, we compute population curvature, prove uniform deviation bounds for the empirical normal operator, construct a spectral initializer, and establish local exponential convergence for gradient flow and linear convergence for small-step descent. Recovery guarantees are explicit but conservative due to reliance on full-space second-moment control. In underdetermined commuting regimes, factor gradient flow becomes an exact entropy mirror flow in joint spectral coordinates. Strictly positive initializations converge to Bregman projections onto the interpolation set. With isotropic initialization q(0)=varepsilon^2mathbf{1}, predictors approach the minimum-trace solution set as varepsilondownarrow0, resolving nonuniqueness via weighted entropy within the invariant joint spectral algebra. Finite-step descent selects interpolants differing from continuous-time Bregman projections by O(eta). Numerical experiments verify these quotient identities, curvature predictions, recovery behaviors, and selection laws.

We propose a score-based stabilization framework for numerical simulation of partial differential equations, in which a learned score model defines a stabilization operator applied to provisional numerical updates. This operator augments standard time-stepping schemes by enforcing structure and physical consistency through a correction that drives iterates toward the manifold of admissible states. We show that the stabilization operator acts as a contraction toward this manifold, yielding a correction mechanism with basin-conditional stability. Numerical experiments on Advection, Korteweg-de Vries (KdV), Nonlinear Schrodinger (NLS), and Burgers' equations demonstrate improved robustness, suppression of nonphysical instabilities, and preservation of qualitative dynamics.

We study empirical risk minimization for learning non-linear dynamical systems whose transition dynamics may switch over time. Under stability assumptions, and i.i.d switching over a set of $K$ modes, we derive non-asymptotic bounds on the prediction risk expressed in terms of the metric entropy of the underlying function class. We instantiate our general result for Hölder and linear function classes, obtaining explicit convergence rates that depend on the effective sample size $Tp_i$, where $T$ is the trajectory length and $p_i$ is the probability of observing mode $i$. Numerical simulations support our theoretical findings. To the best of our knowledge, these results are the first non-asymptotic guarantees for learning switched nonlinear dynamical systems from a single trajectory.

Elastic object manipulation (EOM) involves highdimensional, nonlinear, and elastic deformations. The diverse deformation properties of elastic objects substantially expand the relevant state space, requiring extensive exploration to learn accurate manipulation policies for tasks such as slingshot manipulation. While simulation enables large-scale and safe exploration compared to costly and potentially destructive real-world trials (e.g., repeated projectile launches), accurately calibrating elastic behavior between the real world and simulation remains challenging since elastic properties are largely indistinguishable from visual observations alone. To address these challenges, we propose Sling2Sim2Real, a one-shot Real2Sim2Real framework that identifies elastic parameters from a single non-destructive interaction and enables policy learning in simulation. The framework consists of two stages: 1) a multi-start Real2Sim system identification method that exploits parameter covariance to estimate elastic properties, and 2) simulation-based policy learning followed by zero-shot Sim2Real transfer using the calibrated simulator. We evaluate Sling2Sim2Real on a slingshot manipulation task using a Franka Emika Panda arm and elastic bands with diverse physical properties across varying target distances. Experimental results demonstrate that Sling2Sim2Real achieves accurate policy learning and robust generalization while significantly reducing the amount of required real-world interaction.

Recent work has shown that classifying large language models (LLMs)' responses can be distinguished by modeling token embeddings as trajectories of a black-box dynamical system (DS) and comparing prediction residuals of two DSs. Despite the empirical success of this dynamical approach, a theoretical understanding of why it works, how well it scales as a function of the token sequence, and when it transfers across embedding models remains lacking. We address these questions by formalizing the classification task as a binary hypothesis test between two stochastic linear DSs. We show that the total variation distance between the stationary marginal distributions of the two DSs can be arbitrarily small even when the dynamics differ substantially, which provides a fundamental accuracy floor for any classifier that ignores token dynamics. We then show that the misclassification probability of DS-based classification decays exponentially in the sequence length $L$, with the decay governed by a dynamical discriminability quantity $δ^2$ that captures the spectral distance between the two DSs. We also characterize cross-embedding generalization by introducing an approximate intertwining condition between embedding models and establishing a lower bound on the transferable discriminability in terms of the intertwining map's smallest singular value. Together, these results explain the empirical performance of DS-based classification and motivate further investigation into using DS theory to analyze AI systems, in contrast to the more common approach of using AI to model dynamical systems.

Rare-regime discovery in parameterized dynamical systems is an active-search problem: find one verified parameter at which a scientifically defined qualitative threshold is crossed, even when acceptable candidates are rare, nonconvex, or fragmented. We introduce Quantum-Classical Phase-space and Stability-Threshold Search (QC-PHAST), an evidence-gated decision protocol and query-accounting framework for finite candidate libraries. A candidate induces a dynamical object, simulator-derived criticality score, and verified first-hit predicate. Scientific metadata and charged pilot evidence are used to assess whether equation-aware search, scalar-score active search, predicate-only search, or only a query-model comparison is admissible. The quantum row is the inherited Grover/Boyer--Brassard--Hoyer--Tapp (BBHT) unknown-$M$ marked-set query reference; it is not a new quantum-search theorem, materialized circuit, or hardware-speedup claim. The result is a regime map. Direct boundary constructions, geometry controls, online simulator loops, and learned-label accounting further identify when classical structure, false positives, calibration cost, or state preparation erases the query-model margin. QC-PHAST is therefore an auditable protocol for deciding when a finite-pool marked-set reference is informative and when classical or resource-aware search should dominate.

We study the long-term behavior of the Multiplicative Weights Update (MWU) algorithm in game settings where learning dynamics frequently fail to converge to Nash equilibria and instead exhibit Li-Yorke chaos. While such chaos precludes the prediction of specific long-term strategy profiles, it does not imply a lack of statistical structure. We demonstrate that natural invariant measures - a fundamental concept from ergodic theory - provide the rigorous framework necessary to find order within this chaos. Focusing on a two-strategy congestion game, we prove that these measures allow for a comprehensive statistical characterization of the dynamics. Crucially, we show that this framework extends beyond simple strategy frequencies to \emph{general observables}, enabling the precise calculation of long-term time averages for broad classes of economic metrics - including payoffs, social cost, and regret - despite chaos. Our results reveal that this simple learning algorithm captures the full spectrum of behaviors found in one-dimensional dynamical systems, from unique or multiple absolutely continuous invariant measures to complex periodic attractors as well as coexisting chaotic and stable (periodic) behaviors. By bridging game theory and dynamical systems, we show that statistical predictability is attainable even in the absence of pointwise convergence.

Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability. Kolmogorov Arnold Networks (KANs) mitigate these limitations because their learnable spline activations are structurally aligned with the piecewise-polynomial bases of classical discretizations. However, the way a PDE is cast into a loss functional is as decisive as the choice of approximator: strong-form residual minimization requires high-order derivatives and heavily weighted losses, the energy (Bubnov-Galerkin) form is restricted to self-adjoint operators and, as we show, collapses to a trivial solution for parameter-identification problems, and boundary integral forms require a known fundamental solution. We propose PG-KINN, a physics-informed KAN built on a Petrov-Galerkin formulation in which the trial space is a KAN and the test space is an independent, compactly supported, piecewise-polynomial space evaluated with Gauss-Legendre quadrature. Integration by parts lowers the differentiation order while retaining applicability to general non-self-adjoint, nonlinear, and inverse problems; the localized test functions turn the global residual into a set of element-wise weak residuals with favorable conditioning. On a suite of benchmarks spanning crack singularities, stress concentration, Neo-Hookean hyperelasticity, inverse parameter identification in heterogeneous media, and complex geometries, PG-KINN consistently outperforms legacy MLP baselines and state-of-the-art KAN-based strong/energy/inverse formulations (PIKAN). These results position the Petrov-Galerkin coupling of KAN trial spaces and polynomial test spaces as a robust and accurate route for AI-based computational mechanics.

In this work we investigate reinforcement learning (RL) as a framework for the robust control of parametrized dynamical systems in presence of measurements and model uncertainties. High-dimensional state spaces, expensive numerical solvers, the partial knowledge of the governing equations, and the dependence on physical parameters that may be uncertain or difficult to estimate accurately, make the use of standard RL approaches computationally unfeasible. Indeed, lack of robustness and poor generalization across parameter variations are further amplified in presence of noisy or incomplete measurements, ultimately hampering control performance. To address these challenges, we introduce HypEMBER, a novel RL framework based on the combination of hypernetworks and ensemble learning. In the proposed approach, both the policy and value functions are represented through hypernetworks that generate the weights of the underlying models conditioned on the physical parameters of the system, thereby enabling parametric generalization across different dynamical regimes. In addition, an ensemble of policy and value approximators is employed to quantify epistemic uncertainty, leading to improved exploration strategies and enhanced robustness during and after training. The performance of the proposed framework is assessed on two representative parametrized control problems: (i) the one-dimensional Kuramoto-Sivashinsky equation and (ii) a particle-navigation task in a two-dimensional time-dependent gyre flow, focusing on robustness with respect to measurement noise and parameter misspecification. Numerical results demonstrate that HypEMBER consistently improves training stability and sample efficiency, while achieving superior robustness to uncertainties affecting both the system dynamics and the available observations, in comparison with state-of-the-art RL methods.

Chemistry describes aspects of the universe in terms of molecules and their reactions. In this exploratory work we present a way to describe aspects of any dynamical system in similar terms. To describe a dynamical system in this way three decisions have to be made. The first is how many different "places" there are at which molecules or chemical species can occur; the second is how to determine the species present (or not) at each place; and the third is the set of transitions and reactions that can occur between the species in the various places. For these choices to be compatible with the state update of the dynamical system each state must be able to determine transitions that take the currently occurring molecules to those occurring in the updated state. We also propose an additional requirement that there is always a unique way to choose the least amount of transitions occurring during state updates. We discuss gliders in the game of life cellular and argue that when following their definition of according to Randall Beer they satisfy the additional criterion as well. We also point out some issues with the approach.

Deep learning has proven highly effective for nonlinear system identification, but heavily parameterized neural networks are prone to overfitting in low-data regimes and lack reliable uncertainty quantification. The recently developed manifold meta-learning framework addresses the data efficiency problem by restricting the model parameters to a meta-learned low-dimensional manifold. However, that method is purely deterministic. We propose a fully probabilistic extension of the manifold meta-learning framework, based on amortized Variational Inference, where a generative prior over the low-dimensional parameter manifold is learned. During task-specific adaptation, we combine Maximum A Posteriori estimation with the Laplace approximation to yield a mathematically grounded posterior approximation. Evaluated on a static regression task and the Bouc--Wen dynamical system benchmark, the proposed approach achieves predictive accuracy comparable to its deterministic counterpart while successfully providing calibrated uncertainty bounds in severely low-data regimes.

Recent advances in deep-learning-based nonlinear system identification have led to encoder-based estimation of neural state-space (ANN-SS) models that achieve state-of-the-art performance in offline settings by estimating initial model states from past input-output data. These methods are typically used in multiple-shooting-based offline identification, and online learning of these models remains largely unexplored. This paper presents a batch-wise learning pipeline and a direct recursive identification algorithm for subspace encoder-based ANN-SS models. We provide convergence analysis of the recursive formulation and validate its performance through extensive simulation studies. The results demonstrate that the proposed approach enables computationally efficient online adaptation with high model accuracy.

Reinforcement learning (RL) has recently emerged as a promising feedback control strategy for nonlinear and complex dynamical systems. However, RL algorithms are sample inefficient and require a large number of interaction with the environment to synthesize optimal control strategies. Consequently, applications of RL are typically limited to sparse sensors and actuators due to the curse of dimensionality entailed by the exploration-exploitation dilemma in high-dimensional spaces. In this work, we bridge RL and traditional optimal control for dynamical system with a novel Physics-EnhAnced Reinforcement Learning (PEARL) paradigm tailored to the control of high-dimensional and parametric dynamical systems, exploiting the differentibility of their dynamics. Specifically, PEARL employs an actor-adjoint algorithm that leverages automatic differentiation to compute policy gradients over short horizons and adjoint-based sensitivities of future returns approximated via neural networks, significantly reducing the number of environment interactions, while mitigating long-term gradient instabilities. Through two challenging parametric navigation problems in unsteady flows, we show that PEARL (i) effectively exploits differentiable environments to outperform state-of-the-art RL algorithms, (ii) is sample efficient, thanks to the physics-guided policy learning, (iii) generalizes across multiple scenarios, which is crucial when dealing with parametric systems, and (iv) enables scaling RL to high-dimensional state and action spaces, without requiring low-dimensional state representations or multi-agent strategies.

Can a model be identified from one experiment? We study analytic systems that are linearly parameterized by a combination of prescribed dictionary terms, such as partial differential operators and dynamical systems. For a single input-response pair, recovery is possible exactly when the evaluated dictionary terms are linearly independent. We prove a sharp zero-one law: either no input uniquely determines the coefficients, or almost every random input sampled from a nondegenerate Gaussian measure does. This dichotomy reduces one-shot system identification to a question about degenerate inputs and provides an a posteriori certificate for any recovered model. Numerical examples recover dynamical systems, nonlinear partial differential equations, and structured matrix families from single trajectory data, while also detecting when an extra probe is necessary.

Higher-order couplings enhance the expressive power of hypergraph neural networks (HGNNs), but they also intensify representation collapse in deep propagation due to strong multi-way feature mixing. This work investigates hypergraph oversmoothing from a dynamical-systems perspective and develops a reaction--diffusion framework for depth-resistant hypergraph learning. By defining hypergraph gradient and divergence operators, we interpret message passing as an incidence-level diffusion process. The analysis of pure diffusion shows that its continuous semiflow exponentially contracts the null-mode-free component of node representations and drives the Dirichlet energy to zero, revealing hypergraph oversmoothing as an intrinsic transverse-energy dissipation phenomenon. Motivated by this analysis, we propose Hypergraph Neural Reaction--Diffusion (HNRD), which introduces a reaction mechanism acting on the transverse component to compensate diffusion-induced dissipation and stabilize discriminative variations. We establish global well-posedness of the proposed dynamics and prove that the null-mode-free Dirichlet energy remains bounded away from zero. A forward-Euler discretization provides a practical HNRD layer with a stability condition for deep propagation. Experiments on benchmark and synthetic heterophilic hypergraphs demonstrate that HNRD consistently improves over representative hypergraph baselines. Depth, robustness, and efficiency analyses further show that HNRD preserves stable performance and nonzero Dirichlet energy under deep propagation and perturbations. These results provide a principled dynamical framework for designing deep hypergraph architectures that maintain higher-order expressiveness without representation collapse.

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.