ordinary differential equations

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 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.

This study addresses initial-value problems for systems of first-order ordinary differential equations (ODEs), aiming to automatically discover implicit algebraic constraints among numerical solution components. We propose a data-driven method based on sparse identification: a candidate function library is constructed, and L₁-regularized sparse regression is applied to high-accuracy numerical solutions to directly learn concise, interpretable implicit relations—without requiring prior knowledge of the governing equations or explicit symbolic solving. Unlike conventional system identification approaches, our method eliminates reliance on structural assumptions by embedding sparsity priors directly into solution-space analysis. The approach is validated on canonical dynamical systems—including the Lorenz, Van der Pol, and chemical reaction models—demonstrating robustness and effectiveness in recovering physically meaningful conservation laws or dimensional-reduction relationships. This work establishes a new paradigm for structural analysis and reduced-order modeling of ODE systems through purely data-informed constraint 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.

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 lack of theoretical understanding regarding the online learning dynamics and generalization mechanisms of high-dimensional controlled nonlinear dynamical systems, such as neural ordinary differential equations (neural ODEs). For the first time, it systematically applies dynamical mean-field theory to analyze online stochastic gradient descent training of neural ODEs. In the high-dimensional limit, the framework rigorously solves the coupled dynamics of training and inference and analytically derives the associated learning curves. This study establishes the first tractable theoretical framework for understanding both the training dynamics and generalization capabilities of deep continuous models, revealing the precise evolution laws governing high-dimensional neural ODEs under online learning.

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.

This study addresses whether the failure of ordinary differential equation (ODE) solvers stems from implementation limitations or intrinsic computational barriers. By employing the framework of reverse mathematics, it establishes—for the first time in a representation-independent manner—a precise correspondence between the solvability of initial value problems for ODEs and the regularity of the right-hand side function \( f \), aligning this relationship with the hierarchy of the "Big Five" subsystems of reverse mathematics. Integrating computable analysis, complexity theory, and formal ODE theory, the work reveals regularity as an inherent algorithmic invariant and quantifies how existence, radius of convergence, and modulus of continuity govern computational feasibility. This yields a complete stratification ranging from polynomial-time computability to transfinite computation, identifies key parameters for restoring feasibility, and provides a unified theoretical foundation for diagnosing and designing ODE solvers.

Existing formal verification methods for neural ordinary differential equations (neural ODEs) suffer from limited accuracy and scalability, supporting only single-shot reachability analysis. This work proposes the first end-to-end verification framework for neural ODEs, integrating continuous-time mixed-monotonicity theory, interval-based reachability analysis, a counterexample-guided iterative refinement loop for input sets, and a parallel scheduling mechanism. The framework further introduces three heuristic strategies for input set partitioning. It supports diverse neural ODE architectures and safety specifications, demonstrating significant improvements over state-of-the-art tools NNV 2.0 and CORA on benchmark problems. Experimental results show that the approach substantially enhances both the precision and efficiency of verifying safety set inclusion and classification robustness for neural ODEs.

This work addresses the problem of automatically inferring symbolic ordinary differential equations (ODEs) from multiple observed trajectories to enhance generalization in modeling dynamical systems. We propose MIO, a novel framework that integrates multi-instance learning with Transformer-based symbolic regression, where features from multiple trajectory instances are aggregated to strengthen symbolic expression learning. MIO robustly discovers closed-form ODEs across varying noise levels and for systems of dimension one to four. Experiments demonstrate that MIO significantly outperforms existing symbolic regression baselines on multiple benchmark dynamical systems; even with the simplest mean aggregation strategy, performance gains are statistically significant. The core contribution is the first systematic incorporation of the multi-instance learning paradigm into symbolic dynamical modeling—leveraging redundant observations to improve both generalization and robustness.