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.
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.
Neil F. Johnson, Frank Yingjie Huo, Bella Xinrui Liphysics.soc-ph cond-mat.dis-nn cs.AI nlin.AO physics.app-ph
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.
This study identifies new depression biomarkers based on the dynamical properties of tract variables, which represent geometric features describing the configuration of the speech articulators. A key advantage of this approach lies in its ability to quantify aspects of the articulatory process that have not been previously explored in the context of depression, namely predictability, complexity, and randomness. These properties are respectively characterised using the Largest Lyapunov Exponent, the Correlation Dimension, and the Sample Entropy. Thorough experiments were conducted on the Androids Corpus, a publicly available dataset comprising 64 speakers diagnosed with depression by clinicians and 54 control speakers with no reported history of mental health conditions. The results indicate that the proposed biomarkers effectively discriminate between the depressed and control speakers, as evidenced by the high Cliffs delta values across both read and spontaneous speech.
Baoyang Zhang, Dong An, Zhaoyuan Meng +4quant-ph cs.AI physics.flu-dyn
Nonlinear dynamics is ubiquitous in nature, ranging from chemical pattern formation to ocean circulation, yet its simulation on quantum computers is fundamentally limited by the unitary nature of quantum evolution. We propose the quantum Koopman method, a data-driven framework that embeds nonlinear dynamics into a learned linear representation and implements the resulting evolution using shallow quantum circuits. This method learns Koopman observables from trajectory data, projects the lifted dynamics onto a finite-dimensional subspace, and decomposes the corresponding non-unitary propagator into parallel spectral channels. We utilize the Koopman method on a superconducting processor to simulate three distinct nonlinear systems, comprising reaction-diffusion dynamics, fluid motion on a sphere, and satellite-derived observations of Gulf Stream currents, employing up to 32 parallel circuits of 10 qubits. These quantum simulations capture the dominant multiscale patterns and statistical signatures of the underlying dynamics, and reveal a transition from performance limited by hardware noise in weakly nonlinear systems to performance limited by finite-dimensional Koopman representations as nonlinear scale interactions increase. This transition identifies a practical boundary for quantum-amenable nonlinear dynamics, establishing a hardware-validated route for simulating moderately nonlinear dynamics on near-term quantum hardware.
Emmanuel E. Oguadimma, Victory C. Obieke, Xueying Yucs.LG math.AP math.NA
We consider the cubic nonlinear Schrödinger (NLS) equation on two-dimensional flat tori with varying aspect ratios. In this formulation, the choice of aspect ratio governs the Fourier resonance structure, so rational and irrational geometries can exhibit different high-frequency cascade behaviors. We present a geometry-conditioned Fourier neural operator (FNO) for the cubic defocusing NLS equation, where the input consists of the real and imaginary parts of the solution together with the aspect-ratio parameter \(ω^2\). The model is trained to approximate the one-step solution operator and is evaluated on unseen trajectories generated from random-phase initial data using Fourier pseudospectral method. Our numerical experiments show that the learned operator captures the main solution dynamics on both tori and reproduces the distinct Sobolev norm behavior of the two geometries, with stronger \(H^2\)-growth on the rational torus and more constrained behavior on the irrational torus, consistent with the findings of \cite{hrabski2021energy}. We perform ablation studies to examine the roles of retained Fourier modes, activation functions, Fourier-layer depth, and explicit geometry conditioning. The results indicate that including $ω^2$ improves long-time predictive accuracy, especially for the rational geometry, and supports the use of geometry-aware neural operators for learning spectral-transfer phenomena in nonlinear dispersive partial differential equations.
The cardiovascular system evolves along a bounded trajectory in physiological state space that converges to a compact geometric object: the cardiac attractor. A wearable photoplethysmograph (PPG) or electrocardiograph (ECG) observes a one-dimensional projection of this attractor; by Takens' embedding theorem, delay coordinates reconstruct its full geometry. Three decades of nonlinear cardiac dynamics have extracted Lyapunov exponents, recurrence statistics, and sample entropy from reconstructed attractors, yet no principled account exists of which attractor properties capture which cardiovascular quantities, or why, leaving feature selection as a search problem and negative results uninterpretable. We introduce Attractor Domain Theory (ADT), which proves that the reconstructed attractor's information partitions into three mutually non-redundant domains: the Geometry Domain G (delay embedding; native capability: artifact rejection), the Ergodic Domain S (asymptotic statistical invariants; native capability: stability estimation), and the Variational Domain V (finite-time Lyapunov exponent field; native capability: hemodynamic inference). We prove a Domain Sufficiency Theorem (the Parseval analog for attractor information) and establish that three domains are necessary and sufficient. Geometry Domain validation via the SCSI framework across 176,742 PPG segments from four datasets yields AUC = 0.757 [0.686-0.828] and NPV = 0.966 after correcting three systematic evaluation artifacts (+0.179 net inflation). Ablation confirms C_NL as the dominant Geometry Domain component (Delta AUC = -0.413) and intra-domain redundancy across five components.
Nikhil Saran, Sushant Pokhriyal, Stefan Klus +2math.DS cs.LG
This manuscript proposes an integral formulation of the newly defined quadratic embedding method for identifying nonlinear systems (QENDy). In the original algorithm, trajectory data points along with their time derivatives are used. Methods for calculating time derivatives make the algorithm sensitive to noise. Our integral formulation does not use the time derivatives. This results in a more robust method to learn the dynamics.
S. V. Manivelan, Andrei Velichko, I. Manimehannlin.CD cs.LG physics.data-an
Detecting transient chaos from scalar observations without governing equations represents a fundamental challenge in nonlinear dynamics. We propose a geometry-guided machine learning framework that unifies predictive trajectory divergence with macroscopic attractor morphology to track abrupt regime shifts. The methodology extracts a local instability scale via out-of-sample k-nearest neighbor forecast errors to establish the ML-FTLE estimator, subsequently mapping this temporal divergence onto a structural closeness matrix derived from a minimal dictionary of Poincare occupancy grids. By employing partial least squares regression, we extract a latent geometric component calibrated directly to the empirical finite-time Lyapunov spectrum, yielding the Poincare-based geometric-guided FTLE. Validation against analytical QR-FTLE baselines confirms that fusing topological state spaces with predictive divergence systematically improves continuous transition tracking. The Structural Similarity Index optimally resolves gradual damping, while Hausdorff Distance exhibits extreme resilience during abrupt phase-space collapses. Furthermore, macroscopic spatial discretization acts as a robust topological regularizer against additive Gaussian noise, preserving deterministic signatures even at moderate signal thresholds. This equation-free framework provides a highly accurate, noise-resilient diagnostic for monitoring structural transitions in complex non-stationary systems.
Real-time data analysis requires the ability to accurately and adaptively address nonlinear dynamics in a nonstationary data stream while preserving computational efficiency. However, nonlinear dynamics are so complex that capturing dynamically changing nonlinear patterns and utilizing them for downstream tasks under strict time constraints is nontrivial. To bridge the gap between nonlinear complexity and computational tractability, this study applies Koopman operator theory, which states that nonlinear dynamics can be represented as linear transitions in an infinite-dimensional space. Building upon finite-dimensional approximations of this operator, we present AdaKoop, an efficient streaming algorithm for modeling nonlinear dynamics over nonstationary data streams. Our approach utilizes a probabilistic framework grounded in Koopman operator theory, treating both raw observations and reproducing kernel Hilbert space (RKHS) features as emissions from latent vectors. This dual-view formulation allows nonlinear dynamics to be expressed as a tractable linear system. Therefore, AdaKoop enables the efficient and stable modeling of nonlinear dynamics in a streaming fashion, avoiding the prohibitive computational costs of iterative nonlinear optimization. Furthermore, to address nonstationarity in data streams, AdaKoop adaptively detects the switching of patterns via statistical hypothesis testing for abrupt pattern shifts and incrementally updates model parameters to handle continuous changes. Extensive experiments on a total of 71 practical benchmark datasets across various domains demonstrate that AdaKoop outperforms state-of-the-art methods in terms of real-time forecasting accuracy and computational efficiency.
Data assimilation (DA) integrates observational information with model predictions to improve state estimation in complex systems. While filtering provides the basis for online forecasts by using only past and present observations, it can exhibit delays and biases when the underlying dynamics evolve rapidly or undergo regime transitions. Smoothing, which additionally incorporates future observations, provides a natural pipeline for hindcasting and reanalysis that yields an uncertainty reduction beyond the filter. This paper introduces an ensemble Kalman-Bucy smoother (EnKBS) for continuous-time DA of nonlinear dynamical systems, where the smoother's conditional distributions are reconstructed using ensemble moments. The result is a derivative-free framework that does not require explicit computation of tangent-linear or adjoint models, which converges to the exact smoother solution at the infinite-ensemble limit for a wide class of complex systems. Incorporating standard regularization techniques for high-dimensional systems, such as covariance localization and inflation, the skill of the EnKBS is demonstrated in various important scientific problems. By integrating future observations, which reveal the underlying causal mechanisms for retrospective state updates, the EnKBS is used for Bayesian-based inference of causal relationships and their temporal influence range in a dyadic trigger-feedback model and the development of a causality-driven iterative learning algorithm that identifies the structure and recovers the hidden parameters of a nonlinear reduced-order model mimicking midlatitude atmospheric circulation. Notably, both tasks remain effective with an ensemble size of $O(10)$ under partial observations, suggesting that EnKBS can support the instantaneous discovery of high-dimensional complex systems over time.