PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting

Abstract: Real physical systems are dissipative -- a pendulum slows, a circuit loses charge to heat -- and forecasting their dynamics from partial observations is a central challenge in scientific machine learning. We address the \emph{position-only} (q-only) problem: given only generalized positions~$q_t$ at discrete times (momenta~$p_t$ latent), learn a structured model that (a)~produces stable long-horizon forecasts and (b)~recovers physically meaningful parameters when sufficient structure is provided. The port-Hamiltonian framework makes the conservative-dissipative split explicit via $\dot{x}=(J-R)\nabla H(x)$, guaranteeing $dH/dt\le 0$ when $R\succeq 0$. We introduce \textbf{PHAST} (Port-Hamiltonian Architecture for Structured Temporal dynamics), which decomposes the Hamiltonian into potential~$V(q)$, mass~$M(q)$, and damping~$D(q)$ across three knowledge regimes (KNOWN, PARTIAL, UNKNOWN), uses efficient low-rank PSD/SPD parameterizations, and advances dynamics with Strang splitting. Across thirteen q-only benchmarks spanning mechanical, electrical, molecular, thermal, gravitational, and ecological systems, PHAST achieves the best long-horizon forecasting among competitive baselines and enables physically meaningful parameter recovery when the regime provides sufficient anchors. We show that identification is fundamentally ill-posed without such anchors (gauge freedom), motivating a two-axis evaluation that separates forecasting stability from identifiability.

• The paper demonstrates that PHAST effectively recovers physical parameters and ensures stable long-horizon rollouts across diverse dissipation benchmarks.
• It leverages a modular framework decomposing the system Hamiltonian into potential, mass, and damping components, enabling structured inference from position-only observations.
• Empirical evaluations confirm PHAST's superior forecasting accuracy and interpretability, even under high noise and partial observability scenarios.

Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting (PHAST)

Introduction and Problem Formulation

The paper "PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting" (2602.17998) presents a physically structured deep architecture for forecasting dissipative dynamical systems in partially observed (q-only) settings, where only positions are observed and momenta are latent. The key technical challenge addressed is learning stable, interpretable long-horizon rollouts and reconstructing physically meaningful parameters such as mass, damping, and potential even when parts of the governing laws are unknown or only partially anchored.

PHAST leverages the port-Hamiltonian (pH) framework, which expresses dissipative system dynamics as $\dot{x} = (J - R)\nabla H(x)$, providing explicit separation between conservative (Hamiltonian) and dissipative components. Critically, the framework ensures physical passivity ($dH/dt \le 0$) by construction if $R \succeq 0$, with $H$ the Hamiltonian, $J$ the canonical symplectic structure, and $R$ the momentum-directed dissipation operator. PHAST extends this methodologically to neural architectures, offering three structured knowledge regimes—KNOWN, PARTIAL, and UNKNOWN—distinguished by which physics components (potential, mass, damping) are anchored or learned.

Architecture and Methodology

PHAST's core is a modular architecture that decomposes the system Hamiltonian into three components:

• Potential $V(q)$: Encodes environmental forces and equilibria.
• Mass $M(q)$: Encodes configuration-dependent kinetic geometry.
• Damping $D(q)$: Encodes dissipation pathways.

The architecture employs a three-stage pipeline for q-only inference:

1. Observer: A finite difference augmented by a causal TCN corrector infers velocity from position history.
2. Canonicalizer: Maps $(q, \hat{\dot{q}})$ to an approximate $(q, \hat{p})$, enforcing mass-consistency when possible.
3. PHAST Core: Unrolls dynamics with a structure-preserving integrator (Strang splitting) that alternates between Hamiltonian (symplectic) and dissipative steps.

The architecture supports efficient low-rank parameterizations of both $M$ and $D$ (termed "Householder-style"), which maintain PSD/SPD constraints and enable efficient Woodbury-inverse computations for high-DOF systems. The framework allows physics-based caps on damping strength to block dissipation from absorbing all error, which would otherwise render system identification ill-posed.

(Figure 1)

Figure 1: Open-loop rollouts and phase-space portraits across three dissipative benchmarks (q-only), showing the stability of PHAST's long-horizon predictions under various knowledge regimes.

PHAST's optimizer supports end-to-end objectives that blend data likelihood, passivity regularization, energy accounting, and rollout consistency. This leads naturally to a two-axis evaluation protocol separating long-horizon forecasting error from physical parameter identifiability.

Empirical Results

Forecasting Accuracy:

PHAST is evaluated on thirteen benchmarks—including mechanical, electrical, molecular, thermal, gravitational, and ecological systems—representing a spectrum from fully specified (KNOWN) physics to fully black-box (UNKNOWN). In all cases, only $q$ is observed; baselines include geometric, dissipative, and sequence models without explicit port-Hamiltonian structure.

Stability and Parameter Recovery:

In all environments, PHAST demonstrates superior long-horizon forecasting stability. On the Windy Pendulum, for instance, PHAST with full physics anchors (KNOWN) achieves a 100-step rollout MSE of 0.106 (baseline best: 0.435) and recovers the true position-dependent damping with $R^2 = 0.996$. When damping strength is not physically bounded, PHAST can maintain accurate rollouts while fitting non-physical damping fields—a critical point argued in the paper and validated empirically.

(Figure 2)

Figure 2: Damping identifiability and energy budget consistency for several representative systems. PHAST (KNOWN) recovers the true damping field nearly exactly; physics-informed bounds in PARTIAL mitigate error-sink issues.

The architecture scales efficiently with DOF due to its low-rank structure primitives; CPU timings confirm near-linear scaling for Householder damping application and Woodbury mass inversion.

(Figure 3)

Figure 3: CPU microbenchmark of PHAST structured primitives, demonstrating that wall-clock per-step cost remains computationally tractable for increasing system dimension.

Success Under Noise and Partial Observability:

Closed-loop stabilization studies using Energy–Casimir control show that PHAST maintains performance in feedback settings. Velocity estimation remains the practical bottleneck under high observation noise; noise-matched observers lead to a tenfold reduction in unnecessary control effort relative to unfiltered finite differences.

(Figure 4)

Figure 4: Success rate breakdown at high observer noise ($\sigma=0.05$) for q-only FD+TCN control, confirming that proper observer training is necessary for robust, stable feedback control.

Theoretical and Practical Significance

PHAST formalizes and empirically validates the importance of explicit physical structure—more so than mere expressive capacity—for stable, interpretable forecasting in dissipative systems. Theoretical implications include:

• Demonstrating regime-anchored parameter identifiability, and formalizing the ill-posedness of general inverse problems in absence of physical anchors (gauge freedom).
• Establishing the broad reach of port-Hamiltonian learning outside classical mechanics, including multi-domain applications such as circuit, ecological, and molecular systems, provided a storage function and dissipative splits exist.
• Presenting a two-axis evaluation methodology reflecting the decoupling of rollout accuracy and physical interpretability.

Practically, PHAST's structured neural framework sets a foundation for robust, interpretable, and generalizable forecasting and control in scientific domains where partial prior knowledge is widespread (robotics, molecular simulation, control of engineered systems, etc.).

Future Directions

Open avenues include scaling PHAST to high-DOF and continuum-level dynamical models, data-driven inference of optimal parameter bounds, integration into model-predictive control and online adaptive planning, extension to more complex forced settings (closed-loop stabilization at non-natural equilibria), and meta-learning for regime-switching or system identification during deployment.

Conclusion

PHAST provides a unified, structure-preserving neural architecture for learning dissipative dynamics from partial observations in the port-Hamiltonian formalism. Its explicit regime-aware decomposition enables state-of-the-art long-horizon stability and interpretable identification of physical parameters, subject to well-posedness conditions provided by prior knowledge anchoring. The work evidences the necessity of structure, not mere capacity, for robust generalization in physics-informed sequence modeling and control.