Presymplectification Networks (PSNs)

• Presymplectification Networks (PSNs) are a machine learning framework designed to learn the dynamics of physical systems with holonomic constraints and dissipation.
• PSNs use Dirac structures to lift constrained phase space into a higher-dimensional manifold supporting a non-degenerate symplectic form, restoring geometric structure.
• Validated on complex robotics, PSNs enable accurate, drift-free dynamics prediction that preserves physical invariants from data, even in contact-rich environments.

Presymplectification Networks (PSNs) are a machine learning framework for learning the dynamics of physical systems with holonomic constraints and dissipation, where the canonical symplectic geometry of phase space is degenerate. PSNs introduce an end-to-end differentiable procedure, based on Dirac structures, to lift constrained phase space into a higher-dimensional manifold supporting a non-degenerate symplectic form. This allows structure-preserving learning of energy, momentum, and constraint-respecting trajectories even in contact-rich and dissipative mechanical environments.

1. Motivation and Conceptual Foundations

Physical systems such as legged robots, multibody mechanisms, and constrained dynamical systems exhibit complicated interactions between symmetries, conservation laws, and constraints. Traditional neural architectures for physics modeling, including Hamiltonian Neural Networks and Symplectic Neural Networks, rely fundamentally on the non-degeneracy of symplectic forms, which fail in systems with holonomic constraints or dissipation. In these systems, the symplectic two-form $\omega$ becomes degenerate, leading to the loss of invariants essential for long-term stability and accurate extrapolation.

PSNs address this by leveraging the geometric apparatus of Dirac structures to perform a symplectification, or presymplectification lift, embedding the degenerate phase space into an augmented, non-degenerate symplectic manifold. This process restores the mathematical structure required for structure-preserving dynamics learning.

2. Symplectification via Dirac Structures

A Dirac structure on a vector space $V$ is a maximal isotropic subspace $D \subset V \oplus V^*$ under the pairing

$$\langle (u, \alpha), (v, \beta) \rangle = \alpha(v) + \beta(u).$$

This construction generalizes both presymplectic and Poisson geometries, embodying both the motion constraints and the constraint forces.

The symplectification procedure involves:

1. Augmenting phase space: Introduce a clock variable $q^0 = t$ with conjugate momentum $p_0$, and Lagrange multipliers $\lambda_a$ for constraints with conjugate momenta $\pi_a$.
2. Defining the lifted bundle:

$$T^*\widetilde{Q} = T^*Q \times T^*(\mathbb{R} \times \mathbb{R}^m), \quad (\mathbf{Q},\mathbf{P}) = (q^0, q^i, \lambda_a; p_0, p_i, \pi_a).$$

1. Restoring the symplectic form: The canonical symplectic form becomes

$$\widetilde{\Omega} = dq^0 \wedge dp_0 + dq^i \wedge dp_i + d\lambda_a \wedge d\pi_a,$$

which is closed and non-degenerate.

1. Formulating the extended Hamiltonian:

$$\widetilde{H}(\mathbf{Q}, \mathbf{P}) = H(\mathbf{q}, \mathbf{p}) + p_0 + \lambda_a \phi^a(\mathbf{q}),$$

with gauge-fixing constraints $q^0 = t$, $p_0 + H + \lambda_a \phi^a = 0$, and $\pi_a = 0$.

The Dirac lift then specifies admissible motions and forces in this extended phase space, enabling the recovery of symplectic geometry for constrained, dissipative systems.

3. Core Components of the PSN Architecture

The PSN architecture integrates several key components for end-to-end learning of lifted dynamics:

• Recurrent encoder $\Psi_\theta$: A deep, 3-layer GRU processes the physical state $(\mathbf{q}, \mathbf{p})$, control input $\mathbf{u}$, and time $t$ to output lifted state variables $(\mathbf{Q}, \mathbf{P})$. The encoder predicts latent variables associated with constraints and dissipation (Lagrange multipliers, conjugate momenta).
• Implicit midpoint integration: The encoder predicts velocities $\mathbf{v}_t$, which are integrated with an implicit midpoint method:

$$\hat{\mathbf{z}}_{t+\Delta} = \hat{\mathbf{z}}_t + \mathbf{v}_{t+\Delta/2}\, \Delta.$$

• Flow-matching objective: Rather than requiring explicit knowledge of constraint forces, the architecture is trained to minimize a flow-matching loss

$$\mathcal{L}_{\mathrm{FM}} = \frac{1}{|\mathcal{D}|} \sum_t \left| \mathbf{v}^* - \Psi_\theta(\hat{\mathbf{z}}_t) \right|_2^2,$$

where $\mathbf{v}^*$ is the observed data velocity, and the neural network's velocity prediction is projected back onto observed space.

• Context window: A temporal window of length 10 is employed for temporal inpainting, supporting robust velocity estimation.

This combination enables the network to "inpaint" missing constraint information and internal forces from observed phase-space sequences.

4. Symplectic Network (SympNet) Integration

After encoding the lifted state, a Symplectic Network (SympNet) is used as a step predictor in the lifted, non-degenerate phase space. The SympNet advances the system in time as follows: $$\mathbf{z}_{t+\Delta} = \mathbf{z}_t \oplus \Delta S_\phi(\mathbf{z}_t), \qquad S_\phi^*\widetilde{\Omega} = \widetilde{\Omega},$$ where $S_\phi$ is the symplectic flow and $\oplus$ denotes integration on a Lie group.

Key features include:

• Symplecticity by design: The SympNet implements exact Lie-Trotter splitting through its internal $S$-block structure.
• Parameter freezing: During SympNet training, the PSN encoder parameters are frozen.
• One-step prediction objective: The loss is

$$\mathcal{L}_{\mathrm{pred}} = \frac{1}{|\mathcal{D}|} \sum_t \left| \mathbf{z}_{t+\Delta} \ominus \mathbf{x}_{t+\Delta} \right|_2^2.$$

This procedure ensures that the advanced trajectories preserve energy, momentum, and remain on the constraint manifold.

5. Application to High-Dimensional Constrained Robotics

PSNs were validated on the ANYmal quadruped, a complex, torque-controlled robot characterized by high-dimensional hybrid phase spaces and multiple holonomic constraints (e.g., foot-ground contact). The pipeline proceeds as follows:

• Dirac-lifted data: Raw state sequences of the robot are augmented by the PSN encoder, embedding them into the lifted phase space with latent variables for control, dissipation, and constraints.
• Training: No ground-truth constraint forces are required; the system is trained purely on state and control data.
• Interpretability: The latent coordinates after the Dirac lift correspond to physically interpretable quantities: $p_0$ (nonconservative energy/control), $\pi$ (constraint actions), $\lambda$ (contact multipliers), and $\mathbf{p}$ (generalized momenta).
• Empirical results: The PSN accurately predicts conjugate momenta with low error and enables integration by the SympNet without drift or blow-up, preserving geometric invariants even over long rollouts.

This approach allows contact-rich, dissipative, and highly constrained robots to be modeled from data while maintaining physically consistent predictions.

6. Research Impact and Future Methodological Directions

PSNs constitute the first reported machine learning framework to restore non-degenerate symplectic geometry for constrained, dissipative mechanical systems, bridging a significant gap in structure-preserving dynamics learning. The framework allows for the inclusion of first-principles geometric constraints without requiring explicit analytical force computation.

Planned extensions and open research avenues include:

1. Symplectic flow matching: Extending the flow-matching loss to lifted symplectic space, potentially obviating the need for explicit integration layers.
2. Multi-step prediction: Recurrent or transformer-based symplectic networks could provide stable long-term rollout in conservative systems.
3. Modularity and scalability: Applying the framework to articulated collectives and modular robots, and analyzing memory requirements as constraint topology evolves.
4. Autonomous constraint discovery: Automating the inference of evolving contact structures to handle real-world manipulation and locomotion tasks more generally.

A plausible implication is that PSNs will facilitate structure-preserving learning in many previously inaccessible domains, including large-scale robotics, multibody simulation, and complex biological systems.

7. Mathematical Summary Table

Component	Purpose
Dirac Lift / Symplectification	Embed degenerate phase space into higher-dimensional symplectic
PSN / GRU Encoder	Learn and inpaint mapping to lifted, constraint-encoded space
Flow-Matching Objective	Match observed and predicted velocity fields
SympNet Predictor	Advance dynamics while preserving symplectic structure
Application (ANYmal quadruped)	Demonstrate PSN viability in high-DOF, contact-driven settings

Key mathematical objects include the canonical symplectic form $\omega = dq^i \wedge dp_i$, the lifted symplectic form $\widetilde{\Omega}$, Dirac structure $D(\mathbf{q})$, and the related loss functions for flow matching and symplectic integration.

Presymplectification Networks provide a unification of deep learning and geometric mechanics for constrained systems, forming a principled and extensible foundation for structure-preserving modeling of complex physical systems.