ODE-Based Learning Dynamics

• ODE-based learning dynamics are methods that model continuous-time evolution using ODEs parameterized by machine learning models for system identification and forecasting.
• These techniques integrate physical priors and structured parameterizations, such as Hamiltonian and polynomial chaos approaches, to enhance interpretability and robustness.
• The frameworks enable end-to-end differentiable training via ODE solvers, supporting applications in control, multi-agent systems, and adaptive, real-time learning.

ODE-based learning dynamics constitute a broad and increasingly central methodology for modeling, system identification, control, forecasting, and representation learning with dynamic systems directly from data. At their core, these approaches parameterize the right-hand side of an ordinary differential equation (ODE) using machine learning models—typically neural networks—enabling the flexible, data-driven discovery or approximation of continuous-time dynamics, often with physics, structure, or task constraints built into the representation. This paradigm spans advances from black-box latent neural ODEs to highly structured, physically-informed or multi-agent ODE systems, and enables the seamless integration of learning, prediction, and optimal control in both physical and abstract dynamic environments.

1. Foundations: Parameterization and Structure of ODE-based Dynamics

ODE-based learning dynamics involve representing system evolution through an ODE of the form $\dot{x}(t) = f_\theta(x(t), t, u(t))$, where $x$ is the (possibly latent) state, $u$ is control or exogenous input, and $f_\theta$ is a neural or structured function parameterized by weights $\theta$—learned from trajectory data via end-to-end optimization through an ODE solver. This foundational approach is exemplified by the Neural ODE construct and its variants, which allow gradient-based training of arbitrary ODE RHS functions without requiring closed-form solutions or dense data (Hu et al., 2020).

Advanced ODE-based models introduce physically motivated inductive biases and structure. Hamiltonian Neural ODEs enforce energy conservation or symplecticity by parameterizing $f_\theta$ as the symplectic gradient of a scalar Hamiltonian $H_\theta(q, p)$, possibly with explicit mass, potential, and control terms: for instance, $\dot{q} = \partial H / \partial p$, $\dot{p} = -\partial H / \partial q + g(q)u$ as in Symplectic ODE-Net, which learns interpretable parameters such as the mass matrix $M^{-1}(q)$, potential $V(q)$, and input map $g(q)$ by embedding these as neural modules in the computation graph (Zhong et al., 2019). This design principle extends to multi-agent (Sanchez-Gonzalez et al., 2019), graph-based (Huang et al., 2023), and Lie-group-valued settings (Duong et al., 17 Jan 2024, Duong et al., 2021), where dynamics evolve on manifold-valued states with explicit geometry and symmetry constraints.

Polynomial and chaos-based parameterizations, such as the arbitrary Polynomial Chaos Expansion (aPCE) in ChaosODE, globally represent the vector field as an expansion $f(x) \approx \sum_i c_i \Psi_i(x)$ in orthonormal basis functions, emphasizing extrapolation and theoretical error control (Wildt et al., 19 Nov 2025). In contrast, ODENet leverages an explicit sparse polynomial ansatz $\theta\Lambda(x)$, enabling transparent dynamics discovery robust to noise and irregular sampling (Hu et al., 2020).

2. Learning, Optimization, and Algorithmic Pipelines

Modern ODE-based learning frameworks are characterized by end-to-end differentiable integration and optimization, leveraging both discrete and continuous backpropagation. Automatic differentiation through ODE solvers (e.g., Runge–Kutta, Dormand–Prince, implicit stiff integrators) enables joint training of the vector field parameters and, where present, auxiliary modules such as encoders, decoders, or controllers (Chi, 3 Jan 2024, Zhong et al., 2019, Yu et al., 6 Oct 2025).

Loss functions are typically based on multi-step prediction error (e.g., mean-squared error over rollout trajectories), KL or mutual information regularization for probabilistic or disentangled latent-variable models, and auxiliary penalties such as sparsity (L1) or kinetic energy controls. For hybrid or coupled architectures (e.g., Neural Control), dual loss structures are common: one aiming at dynamics identification and another at control or trajectory optimization, with alternating optimization schemes supporting simultaneous improvement of the model and the control policy (Chi, 3 Jan 2024).

Multiple-shooting and global optimization routines—such as particle swarm and CMA-ES—are often invoked in global polynomial or kernel-based models to address the non-convexity and ensure stability under sparse or noisy data (Wildt et al., 19 Nov 2025). In buffer-free, streaming, or continual-learning scenarios, ODE-based models such as ODEStream eschew data storage and update parameters in an online fashion via per-sample gradients, achieving robust tracking and adaptation to concept drift, irregular sampling, and evolving data distributions (Abushaqra et al., 11 Nov 2024, Zhang et al., 30 Jun 2024).

3. Structured and Physics-Informed Models

A defining development in ODE-based learning dynamics is the incorporation of physical and structural priors into the parameterization, yielding models that are interpretable, energy-preserving, and sample-efficient. Symplectic ODE networks explicitly encode Hamiltonian or port-Hamiltonian structure, learning energy functions whose symplectic gradients guarantee conservation laws and support model-based control via energy shaping and damping injection (Zhong et al., 2019, Duong et al., 17 Jan 2024, Duong et al., 2021).

Graph-based ODEs, including Hamiltonian Graph Networks and their generalizations (e.g., GG-ODE, MS-GODE), capture both the continuous-time evolution and the interaction graph structure, enabling modeling of multi-agent or multi-particle systems. These architectures support environment- and mode-dependent adaptation through exogenous latent variables, mask-based sub-networks, and regularizers for disentanglement and continual learning (Huang et al., 2023, Zhang et al., 30 Jun 2024). Structured latent ODEs and single-cell ODE models similarly use explicit parameterizations (e.g., local linear operators, input-disentangled latent codes) to guarantee biological interpretability and scenario-based counterfactual generation (Bassewitz et al., 3 Oct 2025, Chapfuwa et al., 2022).

In stiff parametric regimes—such as chemical kinetics—physics-informed ODE learning integrates stiff-aware solvers, multi-stage training (ODE + CRNN), and normalization techniques, ensuring robust estimation of interpretable coefficients under extreme stiffness ratios (Peng et al., 8 May 2025). This encapsulates a general trend: first learning a smooth, possibly latent, ODE trajectory, then extracting or fine-tuning interpretable parameters in a structured or physics-informed model.

4. Applications: Control, Forecasting, and Sequence Modeling

ODE-based learning dynamics have been successfully applied across a spectrum of domains:

• Control and system identification: Neural Control (NC) and port-Hamiltonian architectures jointly learn both the dynamics and an optimal feedback policy in continuous time for tasks such as linear-quadratic regulation and CartPole stabilization. Model-based control is enabled via direct synthesis from the learned energy or Hamiltonian, using energy shaping and damping injection (Chi, 3 Jan 2024, Duong et al., 17 Jan 2024, Duong et al., 2021, Yu et al., 6 Oct 2025).
• Reinforcement learning for POMDPs: Continuous-time latent ODEs (e.g., GRU-ODE) robustly capture dynamics under partial observability and irregular sampling, providing smooth context encodings for policy and value networks, and improving sample efficiency and final returns over discrete RNN/RL baselines (Zhao et al., 2023).
• Time series and streaming prediction: ODE-based forecasts excel in handling irregularly spaced time series, concept drift, and nonstationarity, thanks to the continuous-time formulation and online update mechanisms (Abushaqra et al., 11 Nov 2024).
• Biological and physical reasoning: From learning explicit, interpretable gene-regulatory dynamics in single-cell differentiation (Bassewitz et al., 3 Oct 2025) to zero-shot, scenario-driven generation of biological trajectories under novel interventions (Chapfuwa et al., 2022), ODE learning circumnavigates the limitations of black-box models and costly optimal-transport-based preprocessing.
• Video modeling and generative modeling: Modeling motion in continuous time enables arbitrary frame-rate video generation with higher FID quality, dynamic interpolation, and cross-domain motion transfer (Kim et al., 2021).

5. Theoretical Properties, Extrapolation, and Robustness

ODE-based learning dynamics offer distinct theoretical and empirical properties:

• Generalization and extrapolation: Global polynomial chaos parameterizations (e.g., CODE (Wildt et al., 19 Nov 2025)) demonstrate superior interpolation and extrapolation performance, especially in low-data or high-noise regimes, when the true dynamics are amenable to polynomial expansion. ChaosODE achieves near machine-precision on known polynomial systems and robust extrapolation to unseen initial conditions.
• Gradient stability: Nested ODE-to-ODE constructions (ODEtoODE (Choromanski et al., 2020)) constrain parameter rhythms (e.g., on $O(d)$) to eliminate vanishing/exploding gradients, yielding depth-independent convergence bounds and supporting deep ODE-based architectures.
• Interpretability and identifiability: Structured models, especially those with explicit parameterization of physical quantities, afford direct interpretability and access to bio-physical models (e.g., gene interaction matrices, ODE Jacobians (Bassewitz et al., 3 Oct 2025)), scenario control (structured latent ODEs (Chapfuwa et al., 2022)), and energy-based controllers (SymODEN, Port-Hamiltonian nets (Zhong et al., 2019, Duong et al., 17 Jan 2024)).
• Robustness to noise and sparsity: ODE learning with integral- or integral-matching loss (ODENet), and the use of buffer-free state updates (ODEStream), provide resilience to measurement noise, irregular sampling intervals, and nonstationary sequence characteristics (Hu et al., 2020, Abushaqra et al., 11 Nov 2024).

6. Advanced Directions: Continual, Adaptive, and Hierarchical ODE Learning

Emerging research extends classic ODE learning to adaptive, hierarchical, and continual learning contexts:

• Continual dynamics learning: Mode-switching Graph ODEs (MS-GODE) incorporate mask-based sub-network selection and storage, yielding new trajectories via a quick mask-matching step and achieving near-zero catastrophic forgetting across shifting environments (Zhang et al., 30 Jun 2024).
• Adaptive dynamics and environment inference: AD-NODE enables mobile robots to infer latent environmental factors directly from state/action history, supporting real-time model predictive control in the presence of unknown, time-varying operational conditions. A two-phase training procedure first fits a privileged model with known environment variables, then trains an adaptive module for online inference (Yu et al., 6 Oct 2025).
• Cross-environment generalization: GG-ODE generalizes dynamics across multi-agent systems and environments by factorizing common physics laws via a shared GNN-ODE and learning per-environment exogenous factors, enforced by mutual information minimization and contrastive regularization (Huang et al., 2023).

7. Practical Considerations and Limitations

Key guidelines and caveats for ODE-based learning dynamics include:

• Choice of parameterization: Structured or global polynomial models favor extrapolation and interpretability but may face limitations in high dimensions; neural or kernel approaches offer flexibility but risk overfitting and poor out-of-distribution performance (Wildt et al., 19 Nov 2025).
• Solver selection: Stiff regimes (e.g., chemical kinetics) necessitate implicit or stiff-aware solvers and specialized loss terms; explicit methods suffice for benign cases but may fail under disparate timescales (Peng et al., 8 May 2025).
• Optimization pipeline: Multiple-shooting schemes, careful initialization (e.g., from derivative surrogates), and hybrid global-local optimization are critical for stable convergence, especially under sparse or noisy measurement (Wildt et al., 19 Nov 2025).
• Scalability: Polynomial and locally linear ODEs scale cubically with latent dimension, which may necessitate sparse priors or dimension reduction (Bassewitz et al., 3 Oct 2025).
• Limitations: Global polynomial models can be ill-conditioned outside the data domain; energy-based and physics-informed models require careful parameterization to avoid pathological behavior (e.g., non-positive definite mass matrices); adjoint-based backpropagation incurs memory overhead for long sequence or deep ODE networks (Zhong et al., 2019, Wildt et al., 19 Nov 2025).

ODE-based learning dynamics constitute a foundational, unifying paradigm for modern continuous-time modeling, control, and adaptive learning. Their blend of flexibility, structure, and theoretical rigor enables both state-of-the-art performance and principled interpretability across increasingly complex and heterogeneous dynamic systems.