phase diagram mapping

Standard supervised learning introduces arbitrary and discontinuous implicit selectors in problems with multiple solutions by enforcing a single label assignment. This work proposes an equilibrium dynamics framework grounded in weight-sharing dynamical systems, where distinct initial states converge to different stable attractors, collectively representing the solution set through the attractor landscape. The framework employs generalized set-valued mappings to model locally Lipschitz branches and proves that the induced selector is almost everywhere regular—outperforming handcrafted alternatives. Coupled with a diversity control mechanism, the method successfully discovers multiple valid solutions in frustrated Ising models without requiring branch-specific labels, surpassing single-branch supervised approaches. Further experiments on the Allen–Cahn equation elucidate the inherent trade-off between solution accuracy and diversity.

High-dimensional parameter optimization in nonlinear dynamical systems is hindered by the non-differentiability of topological responses (e.g., attractors, periodic orbits) and the lack of gradient information for structural features. Method: This paper introduces a topology-guided gradient descent method grounded in the differentiability of persistent homology. It integrates differentiable persistence diagrams into a parametric optimization framework, constructing a differentiable topological loss function. The approach unifies persistent homology, differentiable topological data analysis, and numerical dynamical simulation (e.g., Runge–Kutta), enabling end-to-end, gradient-driven parameter path planning. Contribution/Results: Evaluated on canonical chaotic and bifurcating systems—including Lorenz, Rössler, and Hindmarsh–Rose—the method achieves precise control over target topological invariants (e.g., number of loops, connected components) and yields optimization trajectories with explicit topological interpretability.

This work addresses two key challenges in multiscale dynamical systems: the difficulty of initializing slow manifolds and the high computational cost of computing steady-state solutions for bifurcation diagrams. We propose a geometry-driven inverse modeling framework based on conditional score-based generative models (cSGMs). For the first time, conditional generative modeling is applied to slow manifold sampling and bifurcation diagram interpolation—enabling high-fidelity steady-state initial conditions to be generated directly from prescribed slow-variable values or new parameter configurations, without explicitly solving differential equations. The method integrates manifold learning, dynamical system dimensionality reduction, and generative inverse modeling to achieve label-controllable, mesh-free, data-driven slow manifold initialization and bifurcation diagram extrapolation and completion. Extensive validation across multiple ODE and PDE systems demonstrates substantial acceleration in steady-state acquisition while preserving accuracy, generalizability, and computational efficiency.

Existing mathematical modeling lacks a rigorous, unambiguous ontological foundation, hindering a unified characterization of the mapping between models and real-world phenomena. This paper introduces, for the first time, an axiomatic definition of mathematical models grounded in Hilbert-space operator theory: a model is formalized as a computable operator acting on random variables, systematically unifying theoretical derivation, experimental implementation, and statistical identification. We further establish a geometric correspondence between the model manifold and the prediction surface, exposing intrinsic structural properties and the fundamental nature of model computability. This framework fills a critical gap in the formal ontology of modeling, providing a unified mathematical foundation for interdisciplinary model construction. It significantly enhances the logical rigor of theoretical inference and the reliability of empirical validation.

This work addresses the problem of unsupervised learning of low-dimensional, manipulable dynamical system representations—namely, compact and smooth state variables coupled with differentiable vector fields—directly from raw video, without prior physical knowledge or domain-specific assumptions. We propose the first end-to-end, video-driven framework for manipulable dynamics discovery, integrating neural implicit state modeling, contrastive spatiotemporal regularization, and differential-geometric constraints to jointly ensure state interpretability, dynamical differentiability, and behavioral analyzability. Evaluated across diverse dynamical systems—including chaotic, limit-cycle, stable fixed-point, and natural oscillatory regimes—the method accurately recovers essential dynamical features (e.g., attractors, bifurcations, conserved quantities) and achieves significantly higher long-horizon prediction accuracy than existing baselines.

This study investigates atypical bifurcation diagram phenomena arising in a class of one-dimensional discrete maps. Distinct from conventional bifurcation structures, these patterns exhibit complex dynamical behaviors rarely documented or entirely absent in standard bifurcation theory. By integrating dynamical systems theory, numerical simulations, and advanced visualization techniques, the work systematically uncovers and classifies a range of singular dynamical regimes within this map family. The findings not only reveal a rich repertoire of novel bifurcation scenarios but also deepen the understanding of complexity-generating mechanisms in nonlinear systems, offering fresh theoretical insights and empirical instances of nonstandard bifurcation phenomena.

This work addresses the limitations of traditional Markov switching models, which rely on fixed parametric transition functions and thus struggle to capture nonlinearities and context dependence in latent state dynamics. The authors propose a semiparametric state-space model in which state transition probabilities are defined via sigmoid-transformed nonlinear functions learned in either a reproducing kernel Hilbert space or a spline space. A generalized expectation-maximization algorithm is developed to jointly estimate the transition function and observation parameters. By relaxing rigid parametric assumptions, the approach substantially enhances the ability to model complex temporal latent-state dynamics. Theoretical guarantees—including identifiability and consistency of the estimators—are established. Empirical results demonstrate that the model more accurately recovers nonlinear transition mechanisms on synthetic data and achieves superior state classification and earlier regime detection on financial time series.

This study addresses the challenge of practical identifiability in parameter estimation for ordinary differential equation models of dynamical systems, which often arises due to limited, noisy, and partially observable data. To this end, the authors propose a Practical Identifiability Index (PII) that quantifies marginal parameter uncertainty via the logarithmic span of positive-parameter confidence intervals, offering a concise measure of how strongly observational data constrain each parameter. The PII enables consistent, order-of-magnitude comparisons across parameters, models, and experimental designs and complements existing approaches such as coverage probability and profile likelihood. Validated through parametric bootstrapping, sensitivity analysis, and structural identifiability theory on growth and compartmental epidemic models, the framework demonstrates that parameter uncertainty diminishes with more informative calibration windows, lower noise levels, and weaker parameter coupling; early-dynamics observability accelerates convergence, and additional measurements substantially enhance identifiability of latent-variable-associated parameters.

This work addresses the fundamental limits of symbolic discovery of dynamical system governing equations, demonstrating that these limits are governed by the geometric structure of attractors rather than solely by algorithmic choices or data volume. The study proposes the smallest eigenvalue, λ_min(M), of the moment matrix of the invariant measure as a universal identifiability bound, revealing for the first time that attractor geometry fundamentally constrains equation discovery through this quantity. Leveraging the Birkhoff ergodic theorem to compute λ_min(M), the authors validate its algorithm-agnostic nature using SINDy and PySR on Lorenz-84 and Lorenz-96 systems, and introduce a Soft F1-weighted structural scoring metric to discern performance differences invisible to conventional metrics. Results show that while chaos enhances λ_min(M), noise sensitivity varies across algorithms, and the proposed framework enables cross-system transferability without retraining.

This work addresses the challenge of modeling dynamical systems subject to state-dependent, non-i.i.d., and non-Gaussian noise by proposing a general identification framework. By integrating dynamical system embedding theory with random feature mappings, the method extends classical noise-free system identification approaches to complex stochastic environments. It establishes that only \(2p+1\) random features are sufficient to uniquely identify continuous or discrete-time dynamical models containing \(p\) parameters. Theoretical analysis provides identifiability guarantees for a broad class of stochastic dynamical systems, while numerical experiments on the Lorenz-63 system and Hénon map demonstrate the method’s efficacy in accurately recovering underlying system structures from observations corrupted by strongly correlated, non-Gaussian noise.