ode analysis

This study addresses the limited model interpretability and physical consistency in ordinary differential equation (ODE) discovery from noisy data. We propose a fully data-driven method that requires neither predefined basis functions nor explicit regularization. Our key innovation is the first incorporation of the analytical general solution form of linear, constant-coefficient ODEs into the modeling framework, coupled with spline-based transformation for smooth derivative estimation and unsupervised gradient matrix construction to directly solve for ODE coefficients via linear systems. By embedding the structural prior of linear ODEs, the method inherently ensures physical interpretability and structural fidelity, eliminating reliance on hand-crafted bases common in sparse regression approaches. Experiments demonstrate high-accuracy recovery of ground-truth dynamics even under severe noise, significantly improving robustness, generalization, and model fidelity compared to state-of-the-art methods.

Real-analytic differential-algebraic dynamic logic (DA-DL) lacks a sound proof calculus, hindering formal verification of high-index differential-algebraic equations (DAEs). Method: (1) Leveraging real-analytic function theory and index reduction, we establish a logically sound transformation of DAEs into equivalent ordinary differential equations (ODEs); (2) We introduce “ghost switching”, the first mechanism enabling exact decomposition of multimodal DAE systems into hybrid systems under precise conditions; (3) We design a DA-DL proof calculus compatible with the axiomatic foundation of differential dynamic logic (dL). Contribution/Results: Our framework guarantees logical equivalence throughout DAE transformations. We formally verify the Euclidean pendulum model, proving semantic equivalence between its original DAE formulation and the reduced ODE system. This work establishes a novel paradigm for modeling, reasoning about, and verifying high-index DAE systems—bridging a critical gap between symbolic DAE reduction and formal verification.

This work addresses the problem of jointly identifying dynamical laws and causal structure from observed time-series data generated by ordinary differential equation (ODE)-driven systems, while enabling counterfactual prediction under interventions. We propose a novel neural ODE framework that—uniquely—integrates lightweight sparsity and symmetry regularization to achieve robust dynamics modeling and causal graph learning even under non-identifiable conditions. The method unifies the representation of inter-variable dynamics and causal dependencies, supporting explicit intervention inference on both variables and system parameters. Evaluated across diverse synthetic benchmarks—including linear and nonlinear first- and second-order ODE systems—as well as real-world datasets, our approach significantly improves dynamical reconstruction accuracy and counterfactual prediction reliability, while enhancing causal interpretability through structured, sparse, and symmetric Jacobian estimation.

This work addresses the structural identifiability of parameters in stochastic differential equation (SDE) models under multiple interventions—i.e., whether SDE parameters can be uniquely recovered from samples of post-intervention stationary distributions. Theoretically, we establish the first uniqueness guarantee for SDE parameter recovery under multi-intervention settings; for linear SDEs, we derive a tight lower bound on the minimum number of required interventions; for weak-noise nonlinear SDEs, we obtain an upper bound on identifiability. Methodologically, we propose a parametric framework featuring learnable activation functions, integrating intervention modeling, stationary distribution analysis, and weak-noise asymptotic theory. Experiments on synthetic data demonstrate that our approach accurately recovers ground-truth parameters, and the theory-guided learnable architecture significantly improves both estimation accuracy and robustness.

Modeling individualized local dynamics in high-noise, multivariate longitudinal clinical data using ordinary differential equations (ODEs) remains challenging due to poor robustness and identifiability. Method: We propose a local ODE modeling paradigm that initializes ODE solutions at each observed time point, generating and aggregating multiple local ODE trajectories per observation. The framework jointly learns a low-dimensional dynamic latent space and patient-specific ODE parameters via a variational autoencoder architecture, differentiable programming, and baseline-feature-driven ODE parameterization—enabling end-to-end interpretable modeling. Contribution/Results: Evaluated on real-world spinal muscular atrophy cohort data and synthetic experiments, our method significantly improves short-term local prediction accuracy of health-state changes over global regression models (p < 0.01) and enhances dynamic interpretability. It establishes a novel paradigm for personalized disease progression modeling grounded in mechanistic, observation-driven ODE learning.

Existing machine learning approaches for dynamical systems are limited by discrete-time modeling and local analysis, failing to capture coexisting local and global bifurcations. To address this, we propose a continuous-time modeling paradigm based on Neural Ordinary Differential Equations (Neural ODEs). Our method directly learns parameter-dependent vector fields from noisy, sparse time-series data, enabling differentiable modeling and extrapolation across bifurcations. This work is the first to apply Neural ODEs to bifurcation structure prediction—overcoming constraints of the training parameter domain—and accurately reconstructs complex bifurcation diagrams in predator–prey systems. It demonstrates robust generalization under data scarcity and noise corruption. The core contribution is the establishment of the first data-driven, continuous-time framework for cross-bifurcation dynamics, uniquely integrating physical interpretability with global predictive capability.

To address the challenge of jointly achieving high expressivity, numerical stability, and computational efficiency in time-varying weight modeling for continuous-time deep learning (e.g., neural ODEs), this paper proposes a low-dimensional parameterization method based on Legendre orthogonal polynomials. Specifically, time-varying weights are constrained to a subspace spanned by a finite number of Legendre basis functions. This formulation preserves strong representational capacity while significantly improving training stability and reducing memory and computational overhead. Integrated with the ODE-ResNet architecture, the approach supports both discrete-then-optimize and optimize-then-discrete training paradigms. Empirical evaluation on three high-dimensional physics benchmark tasks demonstrates that, compared to unconstrained models and monomial-basis parameterizations, our method achieves comparable or superior accuracy while accelerating training convergence and reducing average computational cost by 30–50%.

This study systematically compares MATLAB, Mathematica, and Maple in solving ordinary differential equations (ODEs), partial differential equations (PDEs), and systems of differential equations. A unified benchmark suite—grounded in analytically tractable reference solutions—is employed to empirically evaluate the tools across five dimensions: syntactic usability, numerical accuracy, computational efficiency, visualization capability, and specialized solver functionality. Crucially, the work introduces a novel, problem-driven software selection framework that classifies tasks by intrinsic characteristics—including equation type, stiffness, and boundary condition complexity. Results indicate that Mathematica excels in symbolic solution derivation and medium-scale ODE accuracy; MATLAB demonstrates superior performance in large-scale numerical simulation and engineering-oriented PDE applications; and Maple offers distinctive advantages in special-function handling and analytic derivation. This is the first systematic, multidimensional comparative study of these major mathematical software platforms, thereby bridging a critical gap in computational tool evaluation and providing actionable, evidence-based guidance for scientific and engineering practice.

This work addresses the challenge of constructing globally consistent Koopman eigenfunction representations for continuous-time dynamical systems exhibiting singularities—such as multistability, limit cycles, or separatrices—when only sparse, local observations are available. Conventional approaches struggle to achieve this efficiently. Leveraging the algebraic structure that non-zero Koopman eigenfunctions form a multiplicative group, the authors propose generating an expanded feature space via polynomial combinations of a small set of principal eigenfunctions. They further introduce a cross-singularity matching and continuation strategy that substantially enriches the repertoire of usable eigenfunctions. This framework enables high-fidelity, globally coherent modeling of dynamics from sparse data and significantly enhances the representation of key observables in complex systems.

This work addresses the challenge of modeling high-dimensional nonlinear physical systems in the absence of explicit governing equations by proposing a novel approach that integrates neural implicit fields with spectral decomposition of the Koopman operator. By factorizing and decoupling spatial modes from temporal dynamics, the method constructs a generalizable, parameterized flow operator capable of explicitly learning the system’s spectral structure without requiring prior knowledge of the underlying dynamics. It represents the first integration of neural implicit representations with dynamic mode decomposition, enabling stable long-term prediction, interpolation across parameters, and accurate identification of eigenmodes, eigenvalues, and stability characteristics. The framework demonstrates high accuracy and strong generalization across diverse spatiotemporal dynamical systems.