PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting

Shubham Bhardwaj^1,2, Chandrajit Bajaj¹

¹Department of Computer Science & Oden Institute, The University of Texas at Austin ²Mihawk.ai

Paper   |   arXiv   |   Project Page

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100-step open-loop rollouts across 8 physical systems. Blue = PHAST (Known), Green = PHAST (Partial), Red = GRU, Gold = S5, Purple = LinOSS, White = ground truth. PHAST tracks ground truth where baselines diverge.

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TL;DR

We embed the port-Hamiltonian energy structure $\dot{x}=(J{-}R)\nabla H(x)$ into a neural architecture that decomposes dynamics into potential $V(q)$, mass $M(q)$, and damping $D(q)$. Across 13 benchmarks spanning 6 physical domains, this structure alone yields the best long-horizon stability — outperforming S5, GRU, Transformers, and other baselines by up to $10^5\times$.

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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 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 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.

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Three Knowledge Regimes, One Template

PHAST unifies three levels of prior knowledge under a single port-Hamiltonian template. The architecture is identical across regimes — only the component instantiations change.

KNOWN — $V(q)$: given from physics | $M(q)$: given from physics | $D(q)$: learned (PSD) | Identifiable E.g., pendulum, RLC circuit, Lennard-Jones, heat exchange

PARTIAL — $V(q)$: template + learned $\tilde{V}$ | $M(q)$: given | $D(q)$: learned (bounded) | Partially identifiable E.g., double pendulum, N-body gravity, cart-pole

UNKNOWN — $V(q)$: neural network | $M(q)$: neural SPD net | $D(q)$: neural PSD net | Not identifiable E.g., predator-prey, black-box dynamics

Shared across all regimes: port-Hamiltonian state $x=(q,p)$ with conservative-dissipative split | passivity guarantee $dH/dt \le 0$ by construction | Strang-split symplectic integration | causal velocity observer for $q$-only input

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Key Insight: Forecasting ≠ Identifiability

These three systems form a progression of structural complexity: Windy Pendulum (1-DOF, scalar damping $d(\theta)$), Double Pendulum (2-DOF chain with configuration-dependent mass $M(q)$), and Cart-Pole (mixed topology $\mathbb{R} \times S^1$, per-DOF damping with two distinct mechanisms: constant cart friction $d_c$ and position-dependent angular wind damping $d(\theta)$).

A model that forecasts well need not recover the true physics — damping $D(q)$ can act as a "stabilizer" absorbing errors in $V$ or $M$, making the inverse problem non-identifiable without sufficient structure. PHAST exposes this via two evaluation axes:

• Forecasting — rollout MSE at $H{=}100$: PHAST (PARTIAL) achieves the best forecasting across all three systems
• Identifiability — damping $R^2_D$ vs. ground truth: PHAST (KNOWN) achieves near-perfect recovery ($R^2_D \approx 1$) on Pendulum and Double Pendulum

Baselines (S5, LinOSS, VPT) do not expose explicit damping fields — they can forecast, but cannot tell you why the system dissipates.

Rollouts & phase portraits: PHAST tracks ground truth in state and phase space.

Damping recovery & energy passivity: KNOWN regime recovers true $D(q)$.

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Benchmark Systems

Evaluated across 13 benchmarks spanning 6 physical domains. The 8 systems below appear in the animated comparison above.

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Results

Suite summary across thirteen $q$-only benchmarks. Best PHAST regime vs. best baseline (mean $\pm$ std over 5 seeds).

Mechanical systems (rollout MSE at $H=100$)

Non-mechanical systems (next-step MSE)

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BibTeX

@article{bhardwaj2026phast,
  title   = {PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting},
  author  = {Bhardwaj, Shubham and Bajaj, Chandrajit},
  journal = {arXiv preprint arXiv:2602.17998},
  year    = {2026},
  url     = {https://arxiv.org/abs/2602.17998}
}

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People

• Shubham Bhardwaj
• Chandrajit Bajaj