Dynamics-Based Representations

• Dynamics-based representations are learning paradigms that encode a system's evolution in latent spaces to improve prediction and control.
• They integrate encoder networks with predictive modules, using techniques such as graph neural networks and physics-based priors for robust sequential modeling.
• These methods ensure identifiability and compositionality while enhancing sample efficiency and generalization across reinforcement learning, vision, and scientific applications.

Dynamics-based representations are a class of representation learning paradigms in which the structure or geometry of the learned latent space is explicitly shaped by the dynamics of the underlying system, often to improve prediction, policy learning, compositionality, or interpretability. Rather than learning representations solely through static similarity or task-specific objectives, dynamics-based representation learning enforces that the representations parameterize, predict, or otherwise reflect the temporal or stochastic evolution of the system—be that a physical environment, a neural population, a symbolic computation, or a graph evolution process. The resulting representations can encode not just instantaneous state or appearance, but also aspects of motion, controllability, uncertainty, and causal structure.

1. Foundations and Core Motivation

The foundational goal of dynamics-based representations is to encode in a latent space the key factors determining a system's future evolution—often through loss functions or inductive biases that reward faithful prediction of transitions, disentangle intrinsic state from noise, or regularize latent codes to respect system symmetries or physical laws.

In reinforcement learning (RL), for instance, models such as Unified Latent Dynamics (ULD) (Acharjee et al., 13 Feb 2026) and DR.Q (Lyu et al., 12 May 2026) learn state–action embeddings that must simultaneously explain value functions and enable accurate short-horizon transition prediction. In unsupervised or self-supervised settings (e.g., video representation, neural data analysis), dynamics-based losses such as predictive regularization, sliced Wasserstein matching to learnable latent dynamics priors, or constraints inspired by stochastic differential equations are used to force latent codes to reflect temporally coherent evolution (Wang et al., 2022, Hoang et al., 7 Oct 2025, Gosztolai et al., 2023). This guarantees that latent features are not only compact and predictive but also generalize under interventions or distributional shift.

Beyond the learning-theoretic perspective, dynamics-based representation has deep connections to mathematical systems theory, where invariant sets (e.g., attractors) and flows in continuous dynamical systems naturally serve as "representations" of environmental structure, category identity, or system memory (Hutson, 2021).

2. Methodological Architectures

Sequential and Predictive Models

A principal methodology is to couple encoder networks (e.g., convolutional or attention-based) with predictive modules—usually in the form of (a) direct next-step latent prediction, (b) multi-step rollout via a learned transition operator, or (c) structured propagators (e.g., Graph Neural Networks for entity-based representations). Classical examples include conditional variational autoencoders and transformer-based predictive architectures for visual dynamics (Luo et al., 2024, Hoang et al., 7 Oct 2025).

In object-centric or multi-entity regimes, SlotGNN (Rezazadeh et al., 2023) demonstrates an object-slot encoder followed by a GNN-based message-passing dynamics predictor, enforcing that slot latents both persistently identify entities and accurately evolve their states/actions according to physical interactions.

Dynamics Factorizations and Low-rank Structures

For efficiency and theoretical tractability, some approaches posit that the system transition kernel admits a low-rank factorization—embedding causes (state/action) and effects (next-state) in a shared low-dimensional space such that the transition probability equals a (regularized) bilinear form (Ma et al., 20 Aug 2025). In such cases, optimization is performed both over representation parameters and (in downstream tasks) over compact value-like dual functions.

Physics- and Geometry-Inspired Dynamics Priors

In physical and natural systems, imposing dynamics-priors—e.g., overdamped Langevin transitions with learnable force and diffusion fields (Wang et al., 2022), manifold-constrained vector-flow matching (Gosztolai et al., 2023), or energy-based attractor dynamics (Nam et al., 2023)—guarantees that representations satisfy domain-specific stochasticity, invariances, and eventually identifiability up to isometries. This contrasts with ad hoc priors like isotropic Gaussians.

Optimization-based Dynamics

In robotics and planning, systems where the discrete-time dynamics map is available only implicitly (e.g., as the solution to a physics- or contact-constrained optimization) can be embedded as representations using bi-level optimization and implicit differentiation. Here, the "representation" for each timestep is the optimal solution z∗(x,u), whose sensitivities drive trajectory optimization (Howell et al., 2021).

3. Key Losses, Regularization, and Identifiability

Predictive and Mutual Information Losses

Losses typically minimize the deviation between predicted and actual future state embeddings, with mutual information or InfoNCE terms to avoid representational collapse and to ensure informative, non-trivial alignment of features (Lyu et al., 12 May 2026). In the unsupervised context, distances (e.g., sliced Wasserstein (Wang et al., 2022), contrastive objectives (Gosztolai et al., 2023)) measure divergence between empirical and model-imposed trajectory distributions.

Multi-level and Hierarchical Structure

Hierarchical and multi-scale approaches (e.g., Midway Network (Hoang et al., 7 Oct 2025)) compute transition prediction and motion latents at multiple representational resolutions. This captures both coarse, global changes (e.g., camera motion) and fine, local object deformations, reflecting the multi-scale nature of natural scene dynamics.

Compositionality and Symbolic Structure

Advanced dynamics-based models treat symbolic formation itself as the convergence of stochastic flows to attractor basins; as in Nam et al. (Nam et al., 2023), compositional properties are enforced by energy-based objectives that couple continuous flow with discrete symbolic encoding, supporting systematic mapping of latent trajectories to symbolic codes.

Theoretical Identifiability

Where prior-based constraints are replaced by dynamics-matching (notably SDE-based priors), latent codes can be shown (under mild conditions) to be identifiable up to orthogonal transformations, eliminating many classical ambiguities of representation learning (Wang et al., 2022). In representation factorization (e.g., for offline IL), all dual functions in the corresponding saddle formulation lie in the span of the learned dynamics representations (Ma et al., 20 Aug 2025).

4. Empirical Results and Benchmarking

Dynamics-based representations consistently yield improved sample-efficiency, generalization, and stability across challenging benchmarks:

• In RL, DR.Q and ULD outperform strong model-free and model-based baselines across MuJoCo, DMC, HumanoidBench, and Atari, typically using a single hyperparameter set (Acharjee et al., 13 Feb 2026, Lyu et al., 12 May 2026).
• In offline imitation learning, pretraining low-rank dynamics embeddings enables near-expert-level policy transfer from as little as a single demonstration, and enables domain transfer from simulation to robot (Ma et al., 20 Aug 2025).
• In vision, jointly dynamics-constrained self-supervised models surpass classic SSL methods in segmentation (mIoU up to ≈55%) and optical flow (endpoint errors competitive with supervised) (Hoang et al., 7 Oct 2025, Luo et al., 2024).
• In neuroscience, dynamics-based manifold extraction (MARBLE) achieves state-of-the-art accuracy in decoding animal behavior and permits comparability across sessions and subjects (Gosztolai et al., 2023).
• For neural identity inference, time-invariant representations learned from population dynamics (NeuPRINT) are substantially more predictive of cell type than previous classifiers, especially in data-limited regimes (Mi et al., 2023).
• In meta-learning for dynamics, multiplicative hypernetwork-based conditioning enables real-time adaptation to environment variations, outperforming episodic meta-gradient baselines (Xian et al., 2021).

5. Domains of Application

Reinforcement and Imitation Learning

Dynamics-based representations are used as auxiliary tasks for encoder regularization, to improve policy learning from pixels (Luo et al., 2024, Hoang et al., 7 Oct 2025, Acharjee et al., 13 Feb 2026, Lyu et al., 12 May 2026), for imitation from varied or limited demonstrations (Ma et al., 20 Aug 2025), or for domain adaptation when action-annotated data is unavailable.

Video and Visual Understanding

Representations of visual dynamics (PVDR, Midway, SlotGNN) enable long-horizon video prediction, joint recognition and motion segmentation, and object-centric modeling with unsupervised slot discovery (Rezazadeh et al., 2023, Luo et al., 2024, Hoang et al., 7 Oct 2025).

Scientific Applications

Latent dynamics-based methods are employed for inferring the geometry and structure of neural trajectories, for molecular dynamics (e.g., recovering dihedral angles), and for decomposing complex wave phenomena—providing interpretable, low-dimensional embeddings respecting the system's stochastic or deterministic laws (Wang et al., 2022, Colas et al., 2019, Gosztolai et al., 2023).

Symbolic and Cognitive Modeling

Attractor dynamics frameworks bridge the gap between sub-symbolic continuous flows and emergent discrete codes, supporting compositional and interpretable symbolic computation within neural networks (Nam et al., 2023, Hutson, 2021).

6. Limitations, Challenges, and Extensions

A number of open questions remain. For factorizations, the required embedding rank can grow with system complexity, raising questions about scalability and automatic rank selection (Ma et al., 20 Aug 2025). In vision-oriented models, handling very long-horizon or high-frequency noise remains challenging; extending one-step prediction to full imagination remains an active research direction (Hoang et al., 7 Oct 2025, Luo et al., 2024).

Current frameworks typically assume Markovian or partially observed Markovian dynamics and may not seamlessly transfer to strongly nonstationary, adversarial, or multi-agent domains. Incorporation of additional priors—such as Hamiltonian or Schrödinger dynamics, non-Euclidean manifold constraints, or explicit object-compositionality—are promising directions (Wang et al., 2022, Gosztolai et al., 2023, Nam et al., 2023).

7. Summary Table: Representative Approaches and Domains

Approach/Paper	Domain	Key Mechanism/Objective
DR.Q (Lyu et al., 12 May 2026)	RL (continuous control)	Consistency + MI loss on φ(s,a), faded PER
ULD (Acharjee et al., 13 Feb 2026)	RL (multi-domain)	Linear value-aligned embedding + short-horizon loss
MARBLE (Gosztolai et al., 2023)	Neural population analysis	Manifold-based local flow matching
Midway Network (Hoang et al., 7 Oct 2025)	Vision (video)	Hierarchical latent dynamics for recognition & motion
PVDR (Luo et al., 2024)	Visual RL	Transformer CVAE + decoupled adaptation (video-based)
SlotGNN (Rezazadeh et al., 2023)	Visual dynamics	Unsupervised slot attention + relational GNN
Factorization IL (Ma et al., 20 Aug 2025)	Offline imitation learning	Low-rank bilinear transition factorization
Langevin-based DynAE (Wang et al., 2022)	Physics-inspired	SDE prior on latent transitions (SW distance)
Attractor/discrete symbolic (Nam et al., 2023)	Cognitive computation	Attractor dynamics with GFlowNet-EM discrimination
HyperDynamics (Xian et al., 2021)	Meta-dynamics/robotics	Hypernetwork meta-parameterization
DyRep (Trivedi et al., 2018)	Dynamic graphs	Multiscale point-process node embedding updates
Optimization-based (Howell et al., 2021)	Planning, control	KKT-constrained (implicit) dynamics representations

• DR.Q: (Lyu et al., 12 May 2026)
• ULD: (Acharjee et al., 13 Feb 2026)
• MARBLE: (Gosztolai et al., 2023)
• Midway Network: (Hoang et al., 7 Oct 2025)
• PVDR: (Luo et al., 2024)
• SlotGNN: (Rezazadeh et al., 2023)
• Dynamics factorization: (Ma et al., 20 Aug 2025)
• Langevin-based DynAE: (Wang et al., 2022)
• Attractor/symbolic: (Nam et al., 2023)
• HyperDynamics: (Xian et al., 2021)
• DyRep: (Trivedi et al., 2018)
• Optimization-based: (Howell et al., 2021)
• Cognitive dynamical systems: (Hutson, 2021)

Dynamics-based representations systematically encode temporal evolution and physical or statistical structure, achieving enhanced sample efficiency, generalization, interpretability, and cross-domain comparability—establishing such techniques as central to cutting-edge research in machine learning, reinforcement learning, scientific modeling, and cognitive computation.