Qualitative 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.

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.

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.

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.

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.

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.

Observable Matrix Dynamics (OMD) is a diagnostic framework that probes the dynamics of high-dimensional internal representations of inputs by a neural network via a fixed-size $N \times N$ distance matrix $M(t)$ on a held set of $N$ inputs. OMD uses methods of random matrix theory and particle dynamics to explore spectral reorganisations that are missed by scalar loss functions, but are informative of the training process. We read $M(t)$ against a perturbative ambient-versus-latent decomposition extending the Bogomolny--Bohigas--Schmit (BBS) theory of random distance matrices, with per-snapshot diagnostics for the top-of-spectrum band structure and ambient noise, trajectory-level observables linking snapshots, and a 3D MDS embedding (bottom-three eigenvectors) rendering training as a moving particle cloud. Across seven experiments, diffusive regimes lack stable top-of-spectrum band structure, while sharp endogenous or externally driven reorganisations produce stable fingerprints: consistent with smooth or product latent geometries in BBS-adjacent cases, and with finite-cluster or Fourier-soliton structures otherwise. OMD thus reads the geometric regime of a representation rather than reporting a single intrinsic dimension.

Machine learning (ML) algorithms are known to be opaque. We do not know the reasons for their predictions. The learning process leading to the prediction function is also opaque. We do not fully understand the time evolution of the weight values of neural nets (NN) and related dynamical phenomena. While prediction opacity is widely studied, learning opacity remains largely underexplored. This article studies learning opacity trough the lens of complex dynamical systems. We argue that NN learning is essentially a complex system and that learning opacity is due to dynamical complexity and the epistemological challenges that arise from it. We identify three key properties of training complexity -- sensitivity to weight initialization, feedback in gradient based optimization, and sensitivity to the training data -- and show how each contributes to learning opacity. As these properties are fundamental to the learning process damping or eliminating them would fundamentally alter how ML systems learn. Some sources of opacity in ML may hence be irreducible.

Policies must operate across diverse conditions, yet a single policy is often conservative while fully adaptive schemes can be complex. We study zero-shot generalization in contextual dynamical systems and introduce a performance-centric, directional task dissimilarity--the signed divergence--that upper bounds the generalization gap from a source context to a target context. The signed divergence induces $\varepsilon$-tolerance sets that certify when a source policy class generalizes, and it yields a concrete notion of task-space complexity: the minimum number of source contexts needed so that every target context incurs at most $\varepsilon$ generalization gap. Under a mild local smoothness assumption on performance, the induced tolerance sets admit certified inner/outer balls and instance-dependent volume bounds on task-space complexity. In the finite-oracle setting, source selection reduces to set cover; a greedy strategy inherits the standard $H(n)$ approximation guarantee. Using a Mass-Spring-Damper system with linear-quadratic regulator (LQR) controllers and a nonlinear CartPole system with deep reinforcement learning controllers, we show that greedy selection achieves the same $\varepsilon$-coverage with fewer policies than uniform or random baselines. Our approach delivers a performance-based task similarity measure and practical certificates for building generalizable control with simple policies.

We investigate a structure-first approach to dynamical learning in which the organization of stateful interactions is prescribed explicitly rather than left entirely to a generic recurrent parameterization. We introduce causal recurrent units built from an ordered sequence of local, state-modulated transformations. The construction is motivated by wave-based interaction models, but the units studied here do not impose scattering, passivity, or energy-balance constraints. Because fixed recurrent dynamics, designed reservoir topologies, readout-only learning, and recurrent depth are already well established, the empirical question is deliberately narrower: does the proposed interaction organization provide a useful inductive bias under controlled computational conditions? We compare a one-layer structured model, a two-layer structured model, and a generic echo-state network (ESN), all with 12 recurrent states and the same strictly linear ridge readout. Each model family receives the same random-search budget on calibration data that are disjoint from the final test data, after which the selected hyperparameters are frozen. On a custom nonlinear identification task, the one-layer structured model attains a mean validation NMSE of 2.76 x 10^{-4}, compared with 3.19 x 10^{-4} for the two-layer model and 3.94 x 10^{-4} for the ESN. On NARMA10 the ordering reverses: the ESN attains 0.312, compared with 0.348 and 0.357 for the one- and two-layer structured models. Thus, the proposed organization can be competitive and advantageous on one task, but it is not universally superior; moreover, recurrent depth does not provide a systematic benefit under matched state dimension. The results support a task-dependent interpretation of structural inductive bias and position the present architecture as a controlled precursor to stronger wave- and system-theoretic constructions.

Learning a Markov transition model is not merely conditional density estimation: the learned object must be a valid transition kernel before it is iterated in downstream dynamics. This paper introduces a Doeblin-anchored contrastive chart, a statistical-to-dynamical coordinate framework for learning transition kernels from contrastive objectives. Given a restart law and an anchor strength, the chart mixes the target transition with the restart law. The resulting anchored kernel is simultaneously a Doeblin-minorized Markov kernel, the positive conditional law in a binary contrastive experiment, and an explicitly invertible coordinate for the original transition law. We prove that the anchored contrastive risk identifies the anchored transition density and calibrates excess risk to density error. Since inversion of a learned score may produce a signed or unnormalized object, we introduce a measurable Markovization operator that restores kernel validity while preserving integrated $L^1$ accuracy up to a constant factor. Oracle inequalities and Hölder--ReLU approximation bounds yield nonparametric rates for independent transition pairs. For stationary geometrically $β$-mixing trajectories, a conservative thinning-and-coupling extension yields the same reconstruction interface with an effective sample size. Occupancy-weighted perturbation bounds transfer one-step kernel error to finite-horizon marginal, path-law, and occupation-measure errors under explicit coverage.

We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.

Time-varying dependence is often modeled with dynamic correlations or Gaussian graphical models, but multivariate systems can change through tail behavior, asymmetry, or conditional structure even when correlations are nearly stable. We introduce Dynamic Vine Copulas (DVC), a temporal vine-copula framework for estimating and diagnosing sequence-wide non-Gaussian dependence. DVC fixes a chosen vine factorization for comparability; the framework applies to C-, D-, and R-vines, and our experiments use fixed-root-order C-vines. Pair-copula states evolve through smooth parameter trajectories or temporally regularized family-switching paths. The main diagnostic is a held-out comparison between a full vine and its matched 1-truncated version, which separates flexible first-tree pairwise dependence from evidence contributed by higher-tree conditional terms. At the population level, under a correct fixed vine and the simplifying assumption, this contrast equals the higher-tree component of a vine total-correlation decomposition; in finite samples, it is a predictive diagnostic. In controlled benchmarks, DVC detects Student-t degrees-of-freedom changes, Clayton-to-Gumbel switches, and recurrent conditional-interaction episodes missed or conflated by Gaussian dynamic baselines. The higher-tree score remains near zero in pairwise-only regimes and rises during conditional-interaction regimes. On Allen Visual Behavior Neuropixels data, DVC identifies a reproducible time-indexed higher-tree signal that is positive across held-out splits and vanishes under a decorrelated null, indicating simultaneous cross-area dependence. DVC therefore provides a flexible temporal copula model and an interpretable test of whether temporal dependence changes are pairwise or conditional.

In biomechanical systems, observable performance is often used as a proxy for underlying organization, although similar outputs may arise from different adaptive configurations. This study considers the vertical dimension of occlusion (VDO) as a constraint applied to an adaptive neuromechanical system. A single-case design in a patient with Parkinson's disease enabled repeated intra-individual gait observations under six occlusal probes. Three complementary analytical levels were examined: (i) an aggregated scalar score of observable performance, (ii) a conceptual dynamical systems framework, and (iii) an exploratory UMAP representation of 55 standardized biomechanical variables from 270 M1 observations. The revised Level 1 analysis showed that the relative ranking of OC2.5 and OC3 depended on score construction, while their scalar distributions remained close. The Level 3 embedding showed substantial overlap among all six probes and did not identify independently separated condition-specific clusters. OC2.5 and OC3 displayed limited centroid displacement but broad observation-level overlap. The principal result is therefore representational non-identifiability: neither the aggregated score nor the selected low-dimensional embedding uniquely identifies an occlusal-condition-specific system state. VDO is interpreted as a constraint parameter rather than a causal determinant. The findings are exploratory, model dependent, and non causal. They do not establish distinct physiological states, an optimal VDO, clinical thresholds, or diagnostic, predictive, mechanistic, or prescriptive validity.

High-capacity associative memories based on Kernel Logistic Regression (KLR) exhibit strong storage capabilities, but the dynamical and geometric mechanisms underlying their stability remain poorly understood. This paper investigates the global geometry of attractor basins and the mechanisms governing the storage limit in KLR-trained Hopfield networks. We combine empirical evaluations using random sequences and real-world image embeddings (CIFAR-10) with morphing experiments and statistical Signal-to-Noise Ratio (SNR) analysis. Our experiments show that the network achieves a storage capacity for random sequences up to $P/N \approx 16$, while maintaining stable retrieval for structured data at effective loads near $P/N \approx 20$. Morphing analysis indicates that attractors on the "Ridge of Optimization" are separated by sharp, phase-transition-like boundaries, characterized by steep effective potential barriers and critical slowing down. Furthermore, by comparing an SNR analysis with a geometric reference point inspired by Cover's theorem, we show that the practical storage limit is governed primarily not by a lack of geometric separability in the feature space, but by the loss of dynamical stability against crosstalk noise. These findings suggest that KLR networks function as highly localized exemplar-based memories that operate near the onset of dynamical collapse, providing a useful perspective on the design of robust, large-scale retrieval systems.

What do recurrent neural networks, polynomial ODEs, and discrete polynomial maps each bring to computation, and what do they lack? All three operate over the continuum--real-valued states evolved by real-valued dynamics--even when the target functions are discrete. We study them through primitive recursion. We prove that primitive recursion admits equivalent characterizations in all three frameworks: bounded iteration of a fixed recurrent ReLU network, robust computation by a fixed polynomial ODE, and iteration of a fixed polynomial map with an externally supplied step-size parameter. In each, the time bound is itself primitive recursive, composition emerges from the dynamics rather than as a closure rule, and inputs are raw integer vectors. Every primitive recursive function is first compiled into bounded iteration of a single threshold-affine normal form, then interpreted as a ReLU computation and as a polynomial ODE. The equivalences expose a structural asymmetry: no fixed polynomial map can round uniformly to the nearest integer or realize exact phase selection--operations polynomial ODEs perform robustly via continuous-time flow. Each formalism compensates for a limitation the others lack: the ReLU gate provides exact branching, continuous time provides autonomous rounding and control, and the step-size parameter recovers both at the cost of discretization precision. This opens dynamical characterizations of subrecursive hierarchies and complexity classes by restricting time bounds, polynomial degrees, or discretization resources within one framework. More broadly, these models do not compute by composing subroutines: they shape the trajectory of a dynamical system through clocks, phase selectors, and error correction built into the dynamics. This differs structurally from symbolic programming, and our theorem gives a precise framework to study the difference.

Understanding how complex systems respond to perturbations, such as whether they will remain stable or what their most sensitive patterns are, is a fundamental challenge across science and engineering. Traditional stability and receptivity (resolvent) analyses are powerful but rely on known equations and linearization, limiting their use in nonlinear or poorly modeled systems. Here, we introduce a data-driven framework that automatically identifies stability properties and optimal forcing responses from observation data alone, without requiring governing equations. By training a neural network as a dynamics emulator and using automatic differentiation to extract its Jacobian, we can compute eigenmodes and resolvent modes directly from data. We demonstrate the method on both canonical chaotic models and high-dimensional fluid flows, successfully identifying dominant instability modes and input-output structures even in strongly nonlinear regimes. By leveraging a neural network-based emulator, we readily obtain a nonlinear representation of system dynamics while additionally retrieving intricate dynamical patterns that were previously difficult to resolve. This equation-free methodology establishes a broadly applicable tool for analyzing complex, high-dimensional datasets, with immediate relevance to grand challenges in fields such as climate science, neuroscience, and fluid engineering.

Nature suggests that exploiting the elasticities and natural dynamics of robotic systems could increase their locomotion efficiency. Prior work on elastic snake robots supports this hypothesis, but has not fully exploited the nonlinear dynamic behavior of the systems. Recent advances in eigenmanifold theory enable a better characterization of the natural dynamics in complex nonlinear systems. This letter investigates if and how the nonlinear natural dynamics of a kinematic elastic snake robot can be used to design efficient gaits. Two types of gaits based on natural dynamics are presented and compared to a state-of-the-art approach using dynamics simulations. The results reveal that a gait generated by switching between two nonlinear normal modes does not improve the locomotion efficiency of the robot. In contrast, gaits based on non-brake periodic trajectories (non-brake orbits) are perfectly efficient in the energy-conservative case. Further simulations with friction reveal that, in a more realistic scenario, non-brake orbit gaits achieve higher efficiency compared to the baseline gait on the rigid system. Overall, the investigation offers promising insights into the design of gaits based on natural dynamics, fostering further research.

We address the problem of classifying trajectories or sequences generated by nonlinear dynamical systems, where each class corresponds to a distinct dynamical system. We propose Dynafit, a kernel-based method that learns a distance metric between training data and the underlying dynamics. New observations are assigned to the class whose dynamics best fit the observations according to the learned metric.The learning algorithm approximates the Koopman operator, which globally linearizes the dynamics in a (potentially infinite-dimensional) feature space associated with a kernel function. The distance metric is computed in the feature space independently of its dimensionality by exploiting the kernel trick commonly used in machine learning. The kernel function can be tailored to incorporate prior knowledge of the dynamics when available. We consider a classical test example, the logistic map as a discrete dynamical system, and derive analytically the kernel function from the polynomial Koopman basis that exactly linearizes the dynamics. Dynafit is applicable to a wide range of classification tasks involving nonlinear dynamical systems and sensors. We illustrate its effectiveness through three examples: chaos detection in the logistic map, recognition of handwritten dynamical patterns, and classification of visual dynamic textures.

Models of complex systems often have many parameters, yet are constrained by far fewer experimentally accessible observables; consequently, similar activity can emerge from coordinated parameter changes. We formalize these compatible parameter sets as \emph{viable parameter manifolds}: the inverse images of target dynamical features under a parameter-to-feature map. The relevant codimension is not the number of reported features, but the effective rank of that map at the target scale. Locally redundant features lower the effective codimension, while poor conditioning, high curvature, or regime mixing degrade learnability. We train conditional score-based diffusion models on simulated parameter--feature pairs and use them as amortized samplers of prior-weighted viable sets. In the Lorenz system, scalar trajectory statistics generate thin viable sheets, and a finite-tolerance conditioning localizes a transition-adjacent corridor. In the Izhikevich neuron model, four firing descriptors lie close to a nearly two-dimensional family of features, and the learned inverse images reveal distinct regular and irregular compensation geometries. In a deterministic ODE reduction of finite spiking networks, the same framework reveals excitatory--inhibitory compensation, timescale--coupling tradeoffs, and viable manifolds across 4--12 parameter dimensions. In this view, robustness, compensation, and hidden parameter dependencies are organized as inverse geometry, with diffusion models providing practical tools for sampling, visualizing, and interrogating that geometry.