ode-based modeling

This work addresses the core challenge of interpretable and efficient discovery of ordinary differential equations (ODEs) from observational data to model complex physical system dynamics. We propose an end-to-end automated discovery framework integrating formal grammars, PCA-based dimensionality reduction, and stochastic search. Formal grammars constrain the symbolic expression space and embed domain knowledge, while dimensionality reduction enhances search efficiency; together, they enable structured, interpretable dynamical law identification. Compared to state-of-the-art approaches—including Transformer-based and genetic programming methods—our framework achieves significantly improved sample and parameter efficiency across multiple benchmark systems, notably structural dynamics. The resulting ODE models exhibit both higher predictive accuracy and greater structural simplicity. By unifying symbolic regression with physics-informed constraints and scalable optimization, our approach establishes a novel paradigm for physics-guided scientific discovery.

To address the limited generalization and long-term prediction capability of ordinary differential equation (ODE) dynamics modeling under sparse and noisy time-series data, this paper proposes CODE: a method that employs arbitrary polynomial chaos expansion (aPCE) to globally model the ODE right-hand side via orthogonal polynomials—replacing unstructured fitting with neural networks or kernel functions. This design endows the model with strong regularization and physical consistency, significantly enhancing extrapolation to unseen initial conditions and robustness in long-horizon forecasting under data scarcity and noise corruption. CODE integrates aPCE representation, ODE-constrained learning, and optimization strategies tailored for sparse time-series observations, and provides a reproducible, robust training protocol. Experiments on the Lotka–Volterra system demonstrate that CODE consistently outperforms state-of-the-art methods—including NeuralODE and KernelODE—across diverse noise levels, broad initial-condition domains, and extended prediction horizons.

Conventional dynamical system modeling relies on complex closed-form ordinary differential equations (ODEs), suffering from poor interpretability, limited editability, and difficulty in enforcing prior behavioral constraints (e.g., non-negativity, asymptotic decay). Method: We propose “direct semantic modeling”—a novel paradigm that bypasses symbolic equation derivation and instead learns semantic representations of system behavior directly from data. Behavioral specifications (e.g., physical or biological plausibility) are encoded as differentiable optimization objectives within a neural ODE framework, supported by behavior-feature parameterization, physics-informed end-to-end training, and a differentiable behavioral verification module. Contribution/Results: Evaluated on low-dimensional systems such as pharmacokinetics, our approach achieves 98.7% behavioral compliance, improves debugging efficiency by 5×, and substantially reduces dependence on mathematical domain experts. It is the first method enabling behavior-level editability, formal verifiability, and high-fidelity dynamic modeling.

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 optimal control problem for high-dimensional, nonlinear, and analytically intractable dynamical systems—including discrete/continuous-time and deterministic/stochastic settings. We propose an end-to-end learning framework based on neural ordinary differential equations (Neural ODEs) and differentiable parameterization. The method jointly models system dynamics via Neural ODEs, represents control policies using deep neural networks, and leverages automatic differentiation and gradient-based optimization to enable implicit, differentiable parameterization of control inputs and efficient backpropagation through time. Compared to conventional numerical or analytical approaches, our framework significantly reduces computational overhead, avoids biases introduced by model simplification, and supports data-driven control under black-box dynamics. We validate its high accuracy, strong generalization, and cross-domain applicability across diverse real-world applications—including biological regulation, engineering systems, physical simulation, and medical intervention—establishing a scalable deep learning paradigm for computationally intensive dynamic system control.

Nonstationarity in the solutions of parametric stiff ordinary differential equations (ODEs) degrades Gaussian process (GP) modeling accuracy. To address this, we propose a data-driven solution-path reparameterization method: leveraging gradient information from numerical ODE solvers, we apply a preprocessing transformation to the original solution trajectories—without altering the GP structure—yielding approximately stationary representations in the reparameterized space. This strategy incurs negligible computational overhead while substantially enhancing the GP’s capacity to capture multiscale and stiff-slow dynamics, improving both fidelity and generalization. Experiments across multiple canonical stiff ODE benchmarks demonstrate that our approach significantly outperforms standard GPs and existing nonstationary GP methods. Crucially, it achieves high-accuracy parametric response prediction while preserving the computational efficiency inherent to surrogate modeling.

This work addresses the challenge of modeling complex dynamical systems and discovering their governing equations from noisy, sparse observations. To this end, it proposes a novel framework that integrates Neural Ordinary Differential Equations (Neural ODEs) with Symbolic Regression. By leveraging the strong extrapolation capabilities of Neural ODEs under dynamically similar conditions, the method generates high-fidelity trajectory data, which in turn enriches the input for Symbolic Regression and enables efficient recovery of interpretable analytical governing equations. Experimental results demonstrate that, using only 10% of the original data, the approach accurately reconstructs the complete governing equations for two out of three real-world systems and yields a highly accurate approximation for the third, substantially enhancing the ability to uncover physical laws from limited and noisy measurements.

Conventional ODE numerical methods often suffer from poor convergence in complex scenarios involving high stiffness, discontinuous boundaries, or singular perturbations. To address this, this paper proposes an enhanced Physics-Informed Neural Network (PINN) framework. Methodologically, it integrates structural priors with hard physical constraints, formulates a weighted composite loss comprising data fidelity, initial-condition satisfaction, and PDE residual minimization, and incorporates adaptive spatiotemporal sampling, multi-activation-function co-optimization, and systematic hyperparameter tuning. Key innovations include a novel loss-balancing mechanism, improved embedding of physical constraints via hard enforcement, and enhanced training stability. Extensive experiments on diverse classical ODE benchmarks demonstrate substantial improvements in solution accuracy and convergence robustness—particularly for strongly nonlinear and irregular dynamical systems—while exhibiting superior generalization capability.