MetaSym: A Symplectic Meta-learning Framework for Physical Intelligence

TL;DR

MetaSym addresses the challenge of scalable physics-aware dynamics modeling by embedding a symplectic prior into a neural encoder and coupling it with a meta-learned autoregressive decoder. The SymplecticEncoder preserves the symplectic form $d\text{Φ}_{\theta}(\mathbf{x})^{\top} J\, d\text{Φ}_{\theta}(\mathbf{x}) = J$ and, with time-reversal training, mitigates energy drift, while the ActiveDecoder uses meta-attention to adapt to nonconservative forces and system variations. The framework is validated on three domains—high-dimensional spring meshes, open quantum systems, and quadrotor dynamics—showing superior few-shot adaptation and long-horizon accuracy using a model smaller than baselines. These results suggest that combining structure-preserving priors with targeted meta-learning yields robust, data-efficient physical predictions suitable for real-time decision-making and control.

Abstract

Scalable and generalizable physics-aware deep learning has long been considered a significant challenge with various applications across diverse domains ranging from robotics to molecular dynamics. Central to almost all physical systems are symplectic forms, the geometric backbone that underpins fundamental invariants like energy and momentum. In this work, we introduce a novel deep learning framework, MetaSym. In particular, MetaSym combines a strong symplectic inductive bias obtained from a symplectic encoder, and an autoregressive decoder with meta-attention. This principled design ensures that core physical invariants remain intact, while allowing flexible, data-efficient adaptation to system heterogeneities. We benchmark MetaSym with highly varied and realistic datasets, such as a high-dimensional spring-mesh system (Otness et al., 2021), an open quantum system with dissipation and measurement backaction, and robotics-inspired quadrotor dynamics. Our results demonstrate superior performance in modeling dynamics under few-shot adaptation, outperforming state-of-the-art baselines that use larger models.

Figures (5)

• Figure 1: MetaSym integrates a SymplecticEncoder (light-green), ActiveDecoder (light-purple), and ControlNet (pink). The SymplecticEncoder is pre-trained in isolation on conservative state-space data using shared forward/inverse networks receiving $(\mathbf{q}^{(i)}_t, \mathbf{p}^{(i)}_t, \Delta t)$ and $(\mathbf{q}^{(i)}_{t+1}, \mathbf{p}^{(i)}_{t+1}, \Delta t)$ respectively, and thus explicitly enforcing time-reversibility. Subsequently, with the SymplecticEncoder frozen, the ActiveDecoder and ControlNet are then trained autoregressively via teacher forcing, where system-specific adaptation is achieved by fine-tuning the cross-attention’s query/value parameters (purple dots in cross attention) with few-shot gradient steps. During inference, the ControlNet processes a sequence of non-conservative coordinates and control signals $\{\mathbf{\tilde{q}}^{(i)}_{t:T}, \mathbf{\tilde{p}}^{(i)}_{t:T}, \Delta t, \mathbf{\tilde{u}}^{(i)}_{t:T}\}$, while the SymplecticEncoder projects them onto the conservative manifold and integrates them on it, producing $\mathbf z_c$. The ActiveDecoder using its cross-attention, perturbs $\mathbf z_c$ to predict the dynamics of the system for future timesteps, autoregressively. We indicate the next-step predictions of our network with $\left(\mathbf{\hat{q}}^{(i)}_{t+1}, \mathbf{\hat{p}}^{(i)}_{t+1}\right)$.
• Figure 2: (Left) Time evolution of the system's position and momentum variables for a representative set of masses in the spring mesh. The orange curves represent the ground-truth trajectories for each phase-space coordinate type (i.e., $q_x, q_y, p_x$, and $p_y$), while the blue curves depict the model’s predictions. The close alignment between these trajectories underscores the model’s capacity to accurately capture the underlying long-term dynamics (600 timesteps) of the coupled spring system using a context window of 30 timesteps. (Right) Plots illustrating the mean squared error (MSE) of the time evolution of each phase-space coordinate type (dots) for five (A-E) spring-mesh systems in the test set. Each column encapsulates the spread of errors observed across all masses in the spring-mesh for a given phase-space coordinate across multiple timesteps, with the boxes marking their median values. The uniformly low median errors across all components demonstrate that the model generalizes effectively to different spring-mesh systems for all phase-space coordinates.
• Figure 3: (Left) Time evolution of the two quadratures measured via heterodyne detection for a representative quantum system in the test set, characterized by a measurement efficiency $\eta = 0.86$ and measurement strength $\kappa$. The orange lines indicate the true measurement trajectories, while the blue lines display the model’s predictions (MetaSym). The close overlap between these trajectories highlights the model’s effectiveness in capturing the underlying quantum dynamics from heterodyne signals. The context window of the model is 30 timesteps. (Right) Plots showing the mean squared error (MSE) of the predicted quadratures (dots) for five randomly selected test systems. The consistently low median errors (boxes) across all tested systems underscore the robustness and generalization capabilities of the model.
• Figure 4: (Left) Represents the translational and angular phase-space evolution of the quadrotor, with training and test trajectories generated using the Crocoddyl trajectory optimization package mastalli20crocoddyl. Each task is initialized with randomized initial conditions and a randomized terminal position over a 1.5s horizon (150 timesteps) with a context window of 30 timesteps. The ground-truth test trajectory (orange line) overlaps with MetaSym's predictions (blue line) indicating the excellent predictive capabilites of our model. (Right) Plots summarizing the mean squared error (MSE) of the phase-space coordinates evolution for five randomly generated test systems (dots). The consistently low median errors (boxes) across all components underscore the robustness and generalization capabilities of the model.
• Figure 5: Inner Adaptation convergence for early training stage (left) and close-to-convergence training stage (right) for a quadrotor.

Theorems & Definitions (3)

• Definition C.1: Symplectic Property
• Theorem C.2: $O(\rho)$ Near-Symplectic Composition
• proof : Proof