PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting

TL;DR

Across thirteen q-only benchmarks spanning mechanical, electrical, molecular, thermal, gravitational, 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.

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.

Figures (23)

• Figure 1: PHAST unifies three knowledge regimes under one port-Hamiltonian template. Regimes differ only in what is given vs. learned for $(V,M(q))$; dissipation $D(q)$ is always learned. All regimes share the same continuous-time structure $\dot x=(J{-}R)\nabla H$ and a structure-preserving discrete-time transition map $\Phi_{\Delta t}$ (Strang splitting with a symplectic core), ensuring definiteness constraints by construction.
• Figure 2: Benchmark suite overview. (A--C) Schematics of the three representative dissipative systems with position-dependent damping: Windy Pendulum, Windy Double Pendulum (configuration-dependent $M(q)$), and Windy Cart-Pole ($\mathbb{R}\times\mathbb{S}^1$, per-DOF damping, $M(q)$). The Cart-Pole features two distinct damping mechanisms: constant viscous cart friction ($d_{\mathrm{c}}$) and position-dependent angular wind damping ($d(\theta)$). (D) Wrapped-angle rollout MSE at $H{=}100$ (lower$=$ better) and damping identifiability $R^2_D$ (higher$=$ better; bold$=$ best per column). PHAST (KNOWN) achieves near-perfect identifiability ($R^2_D \approx 1$) on Pendulum and Double Pendulum; PHAST (PARTIAL) achieves the best forecasting across all three systems. All models use 3k--9k parameters (exact counts vary by state dimension; see Appendix). Baselines do not expose explicit damping fields (---). Appendix \ref{['app:environments']} describes the full suite of thirteen benchmarks, including conservative variants, the harmonic oscillator, and five non-mechanical systems (Tables \ref{['tab:qonly_rollout_rlc']}--\ref{['tab:qonly_rollout_predprey']}).
• Figure 3: PHAST core transition computation graph. Components $V$, $M$, and $D$ assemble into Hamiltonian $H$ and dissipation $R$; in the unforced case ($u=0$), the port-Hamiltonian form implies $\tfrac{dH}{dt}\le 0$ in continuous time. PHAST advances the phase state using a structure-preserving discrete-time map $\Phi_{\Delta t}$ (Strang splitting) and optionally composes $L$ integration substeps per environment step (Eq. \ref{['eq:phast_substeps']}). The $H_{k+1}\lesssim H_k$ annotations are schematic; discrete-time energy monotonicity depends on timestep and integrator (Appendix \ref{['app:math:strang']}).
• Figure 4: Two-axis evaluation (open-loop, conceptual). Forecasting accuracy (low rollout MSE) and physical identifiability (high damping $R^2$) are distinct objectives; model/regularizer choices induce trade-offs.
• Figure 5: Open-loop rollouts and phase-space portraits across three dissipative benchmarks (q-only).Top row: a single test trajectory is teacher-forced through a short burn-in window (grey region, vertical dashed line), then predicted open-loop for $H{=}100$ steps. Column 1 (Windy Pendulum, $\theta$): the simplest system --- a single angle with position-dependent damping. PHAST (KNOWN/PARTIAL) tracks the decaying oscillation almost exactly; all three baselines diverge within a few periods. Columns 2--3 (Windy Double Pendulum, $\theta_1$ and $\theta_2$): a chaotic 2-DOF system with coupled configuration-dependent inertia $M(q)$. This is the hardest benchmark: small errors grow exponentially, yet PHAST maintains trajectory coherence on both joints far longer than baselines. Showing both angles reveals that the model captures inter-joint coupling, not just marginal statistics. Column 4 (Windy Cart-Pole, $\theta$): a mixed-manifold system ($\mathbb{R}{\times}\mathbb{S}^1$) with two qualitatively different damping mechanisms (constant cart friction $+$ angular wind damping). PHAST (PARTIAL) achieves the tightest tracking here. Bottom row: canonical phase-space portraits $(\theta, p)$ with $p = M(q)\dot{q}$ during the open-loop segment. Momentum is latent in the q-only setting: for PHAST, $\dot{q}$ comes from the learned FD+TCN observer (Sec. \ref{['sec:methods']}); for baselines, $\dot{q}$ is approximated by finite differences. The closed orbits visible for PHAST confirm that the model has learned physically consistent Hamiltonian structure, whereas baseline phase portraits collapse or spiral outward.

Theorems & Definitions (4)

• Theorem 1.1: Energy Balance and Passivity
• proof
• Proposition 1.2
• proof