PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting

Shubham Bhardwaj; Chandrajit Bajaj

cs.LGFeb 20, 2026

PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting

Authors: Shubham Bhardwaj, Chandrajit Bajaj

Organizations: Department of Computer Science, Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, USA

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.

Figures & tables

Regime	Domain	System	Forecast $q$	$V(q)$ : landscape	$M(q)$ : geometry	$D(q)$ : dissipation	Ident.?
Known	Mechanics	Pendulum	Angle $\theta(t)$	$-g\cos\theta$	Scalar $m$	Air drag $d(\theta)$	Yes
	Mechanics	Spring–mass	Displacement $x(t)$	$\tfrac{1}{2}kx^{2}$	Scalar $m$	Viscous $b\dot{x}$	Yes
	Molecular	Lennard–Jones	Positions $r_{i}(t)$	$4\epsilon[(\sigma/r)^{12}{-}(\sigma/r)^{6}]$	Diagonal $mI$	Solvent friction $\gamma I$	Yes
	Electrical	RLC circuit	Charge $q_{C}(t)$	$q^{2}/2C$	Inductance $L$	Resistance $R$	Yes
	Thermal	Heat exchange	Temperature $T(t)$	$\tfrac{1}{2}c\,T^{2}$	Time constant $\tau$	Heat loss $\kappa$	Yes
Partial	Mechanics	Pendulum (unk. $m$ )	Angle $\theta(t)$	$-g\cos\theta$ (given)	$mI$ (learn $m$ )	Learned (bounded)	Yes

Table 1: Hamiltonian decomposition across knowledge regimes. Each system is described through the three components of the port-Hamiltonian template: the energy landscape

V(q)

(environmental forces and equilibria), the geometry

M(q)

(Riemannian metric governing the cost of motion), and the dissipation

D(q)

(energy loss structure). The regime determines which components are specified versus learned. Anchoring either

V(q)

M(q)

is sufficient to break gauge freedom and render the remaining parameters identifiable; leaving both unconstrained ( Unknown ) yields an underdetermined inverse problem even when forecasting is accurate. Our experiments cover all six domains; the port-Hamiltonian formulation applies to any system admitting a storage function and a conservative–dissipative split. † Identifiable up to an overall mass scale; fixing the gravitational constant

G

resolves the remaining one-parameter gauge.

Panel of Figure 1 — 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.

Panel of Figure 2 — 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 B describes the full suite of thirteen benchmarks, including conservative variants, the harmonic oscillator, and five non-mechanical systems (Tables 20 – 24 ).

Method	$R\neq 0$	Passivity	Spectral	Efficient
HNN	✗	✓	✗	✓
LNN	✗	✓	✗	✗
DHNN / Diss. SymODEN	✓	✗	✗	✓
SympNets / VPT	✗	✗	✗	✓
SSMs (S5, LinOSS, D-LinOSS)	—	✗	✗	✓
PHAST (ours)	✓	✓	✓	✓

Table 2: Comparison of dynamics learning approaches. PHAST combines dissipative modeling (

R\neq 0

), passivity-aware structure, spectral control, and efficient primitives for long-horizon rollouts. Passivity indicates a by-construction storage-function inequality.

Panel of Figure 3 — 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 $\tfrac{dH}{dt}\leq 0$ 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. ( 78 )). The $H_{k+1}\lesssim H_{k}$ annotations are schematic; discrete-time energy monotonicity depends on timestep and integrator (Appendix A.2 ).

Panel of Figure 4 — 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. 3 ); 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.

Model	Params	Cart-Pole (windy)
PHAST (ours)
PHAST (KNOWN)	3,589	$0.063\pm 0.019$
PHAST (PARTIAL)	14,283	$0.083\pm 0.022$
PHAST (UNKNOWN)	14,290	$0.109\pm 0.022$
S5 (best baseline)	17,218	$0.431\pm 0.077$

Table 3: Q-only open-loop forecasting (Cart-Pole). Mean

\pm

std of mixed-manifold rollout MSE at horizon

H{=}100

over 5 seeds. The mixed metric averages translation MSE and wrapped-angle MSE (Appendix F ). Lower is better.

Mechanical systems (rollout MSE at $H{=}100$ )
Benchmark	Best PHAST	Best baseline	Gain
Pendulum (cons)	$0.680\pm 0.043$ (PARTIAL)	$2.320\pm 0.224$ (Transformer)	$3.4\times$
Pendulum (damped)	$0.017\pm 0.005$ (KNOWN)	$0.450\pm 0.241$ (D-LinOSS)	$26.5\times$
Pendulum (windy)	$0.092\pm 0.014$ (PARTIAL)	$0.435\pm 0.239$ (D-LinOSS)	$4.7\times$
Cart-Pole (windy)	$0.063\pm 0.019$ (KNOWN)	$0.431\pm 0.077$ (S5)	$6.8\times$
Oscillator (cons)	$0.0010\pm 0.0002$ (KNOWN/PARTIAL)	$1.087\pm 0.299$ (Transformer)	$1.1\times 10^{3}$

Table 4: Suite summary across thirteen q-only benchmarks. For each benchmark, we report the best PHAST regime and the best baseline (mean

\pm

std over 5 seeds). The first eight rows use rollout MSE at horizon

H{=}100

; the last five report next-step MSE (Tables 20 – 24 ). Gains are per-row ratios and are not directly comparable across environments.

Model	$\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100)$ $\downarrow$	$R^{2}_{D}$ $\uparrow$	$\mathrm{MAE}_{D}$ $\downarrow$	$\mathrm{EbudRes}^{\mathrm{roll}}(H{=}100)$ $\downarrow$	Params
PHAST (KNOWN)	$0.106\pm 0.020$	$0.996\pm 0.004$	$0.007\pm 0.004$	$1.56\pm 0.02$	3,364
PHAST (PARTIAL)	$0.092\pm 0.014$	$0.654\pm 0.077$	$0.063\pm 0.011$	$1.50\pm 0.04$	13,736
PHAST (UNKNOWN)	$0.298\pm 0.048$	$-96.546\pm 15.897$	$1.343\pm 0.115$	$2.58\pm 0.19$	13,738
GRU	$1.796\pm 0.625$	—	—	$12.34\pm 6.58$	37,889
S5	$0.600\pm 0.047$	—	—	$11.71\pm 2.91$	17,089
LinOSS	$1.458\pm 0.324$	—	—	$56.14\pm 37.41$	17,089

Table 5: Windy Pendulum (q-only) identifiability and energy consistency. Mean

\pm

std over 5 seeds (same setup as Table 17 ). Baselines do not expose a damping field, so damping recovery metrics are not applicable.

Damping cap	WrapMSE θ (100) $\downarrow$	$R^{2}_{D}$ $\uparrow$	MAE D $\downarrow$	EbudRes(100) $\downarrow$
Unbounded	$0.059\pm 0.007$	$-90.15\pm 6.14$	$1.326\pm 0.044$	$1.50\pm 0.03$
$\sum_{i}\beta_{i}(q)\leq 0.5$	$0.071\pm 0.004$	$0.703\pm 0.072$	$0.061\pm 0.010$	$1.58\pm 0.04$

Table 6: PARTIAL regime: effect of bounding total damping strength (Windy Pendulum, q-only). Mean

\pm

std over 5 seeds on CPU (same data protocol as Sec. 4.1.1 ). Without the bound, damping becomes an “error sink” and is non-identifiable (

R^{2}_{D}\ll 0

) even when rollouts are accurate; bounding

\sum_{i=1}^{r}\beta_{i}(q)

recovers a physically meaningful damping field.

Figure 6: Damping identifiability and energy consistency across environments (q-only). Top row : learned damping field

d(\theta)

vs. ground truth (black). Pendulum : PHAST (KNOWN) recovers the sinusoidal profile

d(\theta){=}d_{0}{+}\Delta d\,|\!\sin\theta|

near-exactly (

R^{2}_{D}{\approx}1

); PARTIAL captures the shape but with a magnitude offset, illustrating the forecasting–identifiability trade-off. Double Pendulum : solid and dashed curves show per-joint damping (

d_{1}

d_{2}

). Despite chaotic dynamics and a coupled

M(q)

, KNOWN recovers both joints’ damping profiles to high accuracy. Cart-Pole : solid and dashed curves distinguish constant viscous cart friction (

d_{\mathrm{c}}

, flat) from angular wind damping on the pole (

d(\theta)

, sinusoidal) — two qualitatively different dissipation mechanisms that PHAST disentangles from q-only data. UNKNOWN shows poor identifiability across all systems (

R^{2}_{D}\ll 0

; Table 4 ). Baselines do not expose explicit damping fields (marked N/A). Bottom row : total energy

H(t)

during open-loop rollouts. In a dissipative system, energy must monotonically decrease; this is a necessary physical consistency check. PHAST (PARTIAL) most closely tracks the ground-truth energy decay across all three systems. Baselines either diverge (energy blowup) or collapse to zero, both indicating physically inconsistent dynamics. For PHAST, energy is computed using the observer’s learned velocity; for baselines, via finite differences.

Panel of Figure 7 — Figure 7: Energy–Casimir control as port-Hamiltonian interconnection. The plant (blue, left) and controller (green, right) are pH systems coupled through power ports: $u_{c}=y_{p}$ feeds plant velocity to the controller, $u_{p}=-y_{c}+v$ returns the shaped restoring force plus an auxiliary channel $v$ . When $v{=}0$ the cross-power cancels ( $y_{p}^{\top}u_{p}+y_{c}^{\top}u_{c}=0$ ), and the Casimir $\mathcal{C}=q-\xi$ is invariant. Damping injection $v=-d_{\mathrm{inj}}\hat{y}$ (red, dashed) ensures $\dot{H}_{\mathrm{cl}}\leq 0$ ; in q-only settings $\hat{y}\approx y_{p}$ is estimated, so estimation error can break exact power balance.

$\sigma$	Method	Velocity estimate	Success $\uparrow$	Final error $\downarrow$	Effort $\downarrow$	$\|\hat{\dot{q}}-\dot{q}\|$ $\downarrow$
0.00	Oracle	oracle	1.00	$1.31\times 10^{-4}$	262.9	0.000
0.00	Finite differences	FD	1.00	$1.21\times 10^{-4}$	268.6	0.007
0.00	MAP smoother	MAP smoother	1.00	$1.16\times 10^{-4}$	268.6	0.007
0.00	FD+TCN observer	FD+TCN	1.00	$1.03\times 10^{-3}$	270.7	0.008
0.00	PHAST-trained observer	PHAST observer	1.00	$5.71\times 10^{-3}$	272.4	0.019
0.01	Oracle	oracle	1.00	$5.49\times 10^{-3}$	262.9	0.000

Table 7: Energy–Casimir control on the true pendulum (q-only sensing). Mean over

100

trials (5 regimes

\times

20). All methods use the same predictor–corrector

\xi

update (

k_{\xi}=5

$\sigma$	Method	Velocity estimate	Success $\uparrow$	Final error $\downarrow$	Effort $\downarrow$	$\|\hat{\dot{q}}-\dot{q}\|$ $\downarrow$
0.00	Oracle	oracle	1.00	$1.31\times 10^{-4}$	262.9	0.000
0.00	PHAST model-based	$\partial\hat{H}/\partial p$	1.00	$5.46\times 10^{-5}$	245.0	0.011
0.01	Oracle	oracle	1.00	$5.49\times 10^{-3}$	262.9	0.000
0.01	PHAST model-based	$\partial\hat{H}/\partial p$	1.00	$5.32\times 10^{-3}$	245.1	0.011

Table 8: Energy–Casimir with PHAST-based velocity (q,p observed). Mean over

100

trials (5 regimes

\times

20). Both methods use the same predictor–corrector

\xi

update (

k_{\xi}=5

Method	Final error $\downarrow$	Effort $\downarrow$	$\|\hat{\dot{q}}-\dot{q}\|$ $\downarrow$
Oracle	0.0041	6.3	0.000
Finite differences	0.0170	80.1	1.117
MAP smoother	0.0133	10.3	0.095
FD+TCN observer	0.0137	8.6	0.130
PHAST-trained observer	0.0188	54.9	0.911

Table 9: Near-stable regime (

\sigma=0.01

). Learning a noise-aware q-only observer reduces control effort near the target by

\sim

\times

vs. finite differences.

Observer train noise	Final error $\downarrow$	Effort $\downarrow$	$\|\hat{\dot{q}}-\dot{q}\|$ $\downarrow$
$\sigma_{\text{train}}=0.00$ (mismatch)	0.0168	82.4	1.134
$\sigma_{\text{train}}=0.01$ (matched)	0.0120	8.5	0.132
$\sigma_{\text{train}}\sim\mathcal{U}[0,0.01]$ (anchored range)	0.0137	8.6	0.130

Table 10: Observer training noise vs. evaluation noise (

\sigma_{\text{eval}}=0.01

), near-stable regime. Q-only FD+TCN observer used inside the Energy–Casimir controller.

Method	Velocity estimate	Success $\uparrow$	Final error $\downarrow$	Effort $\downarrow$	$\|\hat{\dot{q}}-\dot{q}\|$ $\downarrow$
Oracle	oracle	1.00	$2.75\times 10^{-2}$	263.6	0.000
Finite differences	FD	0.00	$8.36\times 10^{-2}$	2109.0	5.582
FD+TCN observer	FD+TCN (noise)	0.26	$7.70\times 10^{-2}$	396.6	0.659

Table 11: High-noise stress test (

\sigma=0.05

). Mean over

100

trials. For the FD+TCN controller we use a noise-conditioned observer trained with

\sigma_{\text{train}}\sim\mathcal{U}[0,0.05]

Figure 8: Success rate breakdown at

\sigma=0.05

for q-only FD+TCN control. Each bar reports success over 20 trials for the given initial-condition regime, varying only the observer training noise distribution.

Appendix figures & tables39 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 9 — Figure 9: Single pendulum. A point mass $m$ at distance $\ell$ from a fixed pivot, with moment of inertia $I$ about the pivot. The angle $\theta$ is measured from the downward vertical. Position-dependent damping $d(\theta)$ acts at the pivot.

Panel of Figure 10 — Figure 10: Windy damping profile. The position-dependent damping coefficient $d(\theta)=d_{0}+\Delta d\,|\sin\theta|$ varies between $d_{0}=0.3$ (vertical) and $d_{0}+\Delta d=0.8$ (horizontal), modeling air resistance that depends on the pendulum’s cross-sectional area.

Panel of Figure 11 — Figure 11: Cart-pole system. A cart of mass $m_{c}$ moves on a horizontal rail while a pole of mass $m_{p}$ (concentrated at distance $\ell$ from the pivot) rotates about the cart. The angle $\theta$ is measured from the downward vertical (hanging configuration: $\theta=0$ ). Windy damping acts isotropically on $\dot{q}$ via a scalar coefficient $d(\theta)$ .

Panel of Figure 12 — Figure 12: Harmonic oscillator. Two independent harmonic oscillators with natural frequency $\omega$ and unit mass. Coordinates $q_{1},q_{2}$ measure displacements from equilibrium. Optional viscous damping $\gamma$ acts on each oscillator.

Panel of Figure 13 — Figure 13: Double pendulum (point-mass model). Two point masses $m_{1},m_{2}$ at the ends of massless rods of lengths $\ell_{1},\ell_{2}$ . The angles $\theta_{1}$ and $\theta_{2}$ are both measured from the downward vertical (absolute angles). Optional viscous damping $b$ acts at both joints.

Panel of Figure 14 — Figure 14: Spring–mass–damper. A mass $m$ connected to a wall by a spring (stiffness $k$ ) and dashpot (damping $b$ ) in parallel. Displacement $x$ is measured from equilibrium.

Panel of Figure 15 — Figure 15: Lennard–Jones cluster ( Jones, 1924 ) . $N{=}3$ particles interact pairwise via the LJ potential. Inset: the repulsive core ( $r^{-12}$ , red shading ) dominates at $r<\sigma$ ; the attractive well ( $-r^{-6}$ , blue shading ) has depth $\epsilon$ near the equilibrium distance $r_{\mathrm{eq}}=2^{1/6}\sigma$ . Dissipation is modeled by isotropic velocity-proportional damping ( $D=\gamma I$ ).

Panel of Figure 16 — Figure 16: Series RLC circuit ( Horowitz & Hill, 2015 ) . Charge $q_{C}$ on the capacitor is the generalized coordinate; flux linkage $p=Li$ is the conjugate momentum. Color coding shows the port-Hamiltonian mapping: capacitance $C$ $\to$ potential $V$ , inductance $L$ $\to$ mass $M$ , resistance $R$ $\to$ damping $D$ . The electrical energy $H=q_{C}^{2}/2C+Li^{2}/2$ is the storage function.

Panel of Figure 17 — Figure 17: Lumped heat exchange ( Incropera et al., 2007 ) . Two thermal masses with temperatures $T_{1}$ and $T_{2}$ are the generalized coordinates ( $n_{\mathrm{dof}}{=}2$ ). Stored thermal energy $V_{i}=\tfrac{1}{2}c_{i}T_{i}^{2}$ is the potential for each mass; the thermal time constants $\tau_{i}$ play the role of inertia; and the coupling coefficient $\kappa$ provides mutual heat exchange while individual losses $\kappa_{\mathrm{loss}}$ dissipate energy to the environment. This is a lumped-parameter model: temperature is assumed spatially uniform within each thermal mass (valid at low Biot number).

Panel of Figure 18 — Figure 18: Robot arm (2-DOF manipulator). Joint angles $\theta_{1},\theta_{2}$ are generalized coordinates. Link inertias and gravity torques are known from CAD; joint friction (red) is learned from trajectory data.

Panel of Figure 19 — Figure 19: N-body gravitational system ( Binney & Tremaine, 2008 ) . $N{=}3$ point masses interact pairwise via Newtonian gravity. The gravitational potential $V$ is known in functional form but individual masses $m_{i}$ are learned (PARTIAL regime). A softened potential $r_{ij}^{2}+\epsilon_{\mathrm{soft}}^{2}$ replaces $r_{ij}^{2}$ to avoid the singularity at close approach. Drag $D=\gamma I$ models residual dissipation (e.g., tidal friction or numerical damping); it is learned with tight bounds since point-mass gravitational systems are nearly conservative.

Panel of Figure 20 — Figure 20: Predator–prey (Lotka–Volterra) ( Murray, 2002 ; Strogatz, 2018 ) . Prey density $n_{1}$ is the generalized coordinate $q$ ; predator density $n_{2}$ serves as the conjugate momentum $p$ ( $n_{\mathrm{dof}}{=}1$ ). All port-Hamiltonian components are learned (UNKNOWN regime): $V(q)$ is a neural potential generalizing the classical Lotka–Volterra conserved quantity, $M$ is a learned SPD “ecological inertia,” and $D$ is a learned PSD dissipation encoding mortality, disease, and resource depletion. The port-Hamiltonian structure guarantees bounded dynamics even without parameter recovery.

Component	Parameters	Shared across $t$ ?	Shared across substeps $L$ ?
Observer $o_{\phi}$ (FD+TCN)	$\phi$	✓	—
Canonicalizer $C_{\psi}$	$\psi$	✓	—
Physics $(V,M,D)$	$\theta$	✓	✓
Substep timesteps $\{\delta t_{s}\}_{s=1}^{L}$	$\tau$	✓	per-substep

Appendix

Table 12: What is learned vs. shared in the q-only PHAST pipeline. PHAST core transition substeps

L

reuse the same physics components

(V,M,D)

and differ only through intermediate states (and, if enabled, per-substep timesteps). Regime choices determine which physics components are learned (Fig. 1 ).

Hyperparameter	KNOWN	PARTIAL	UNKNOWN
Damping terms $r$	2	2	2
Damping bound $\bar{\beta}$ (Windy)	unbounded	$\Delta d\,(=0.5)$	unbounded
$d_{0}$ policy (Windy)	learned	fixed	learned
Potential hidden dim	—	64	64
Potential layers	—	3	3
Mass terms (if learned)	—	—	$\min(4,n)$

Appendix

Table 13: Hyperparameters for the q-only experiments (Sec. 4 ).

Baseline	Key settings
GRU	dropout=0.0
S5	$d_{\text{state}}=64$ , internal $\Delta t_{\text{ssm}}=0.01$
LinOSS	internal $\Delta t_{\text{linoss}}=1.0$ (absorbed into learned eigenvalues)
D-LinOSS	internal state dim $=\lfloor d/2\rfloor$ , learnable per-oscillator timestep
Transformer	$n_{\text{heads}}=4$ , FF dim $=4d$ , dropout=0.1
VPT	$n_{\text{ff\_sublayers}}=2$ , skew-symmetric attention

Appendix

Table 14: Baseline architecture hyperparameters (q-only experiments). All baselines are trained as causal seq2seq predictors mapping

q_{0:t}\mapsto\hat{q}_{t+1}

with shared global settings hidden dimension

d=64

and depth

2

(unless otherwise noted).

Benchmark	$\Delta t$	Initial conditions	Damping
Pendulum (cons)	0.05	$\theta_{0}\sim\mathrm{Unif}[-\pi,\pi]$ , $p_{0}\sim\mathcal{N}(0,3^{2})$	none
Pendulum (damped)	0.05	$\theta_{0}\sim\mathrm{Unif}[-\pi,\pi]$ , $p_{0}\sim\mathcal{N}(0,3^{2})$	constant ( $\gamma{=}0.5$ )
Pendulum (windy)	0.05	$\theta_{0}\sim\mathrm{Unif}[-\pi,\pi]$ , $p_{0}\sim\mathcal{N}(0,4^{2})$	$d(\theta)=d_{0}+\Delta d\|\sin\theta\|$ ( $d_{0}{=}0.3$ , $\Delta d{=}0.5$ )
Cart-Pole (windy)	0.02	$(x_{0},\theta_{0})\sim\mathrm{Unif}([-1,1]\times[-\pi,\pi])$ , $p_{0}\sim\mathrm{Unif}[-2,2]^{2}$	windy on $\theta$ ( $d_{0}{=}0.3$ , $\Delta d{=}0.5$ )
Oscillator (cons)	0.02	$q_{0}\sim\mathcal{N}(0,I)$ , $p_{0}\sim\mathcal{N}(0,I)$	none
Oscillator (damped)	0.02	$q_{0}\sim\mathcal{N}(0,I)$ , $p_{0}\sim\mathcal{N}(0,I)$	constant ( $\gamma{=}0.1$ )

Appendix

Table 15: Simulator initial conditions for the thirteen q-only benchmarks.

Paper metric	Implementation key(s)
One-step wrapped-angle MSE ( $\mathrm{WrapMSE}$ )	theta_wrap_mse
One-step Euclidean MSE ( $\mathrm{MSE}$ )	mse
Rollout wrapped-angle MSE at horizon $H$ ( $\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H)$ )	rollout_theta_wrap_mse_h{H}
Rollout Euclidean MSE at horizon $H$ ( $\mathrm{MSE}^{\mathrm{roll}}(H)$ )	rollout_mse_h{H}
Rollout mixed-manifold MSE at horizon $H$ ( $\mathrm{MixedMSE}^{\mathrm{roll}}(H)$ )	rollout_mixed_mse_h{H}
Takeover rollouts (PHAST only; Eq. ( 8 ))	rollout_takeover_*

Appendix

Table 16: Paper metric names and their corresponding metric identifiers used in our implementation.

Panel of Figure 21 — Figure 21: Q-only computation graph and diagnostic probes. Training is teacher-forced (ground-truth $q$ history), while evaluation is open-loop and therefore sensitive to compounding error. Autoregressive rollouts keep the observer in the loop; takeover rollouts isolate the PHAST core transition given a single inferred boundary state.

PHAST (ours)
Model	Params	Conservative	Damped	Windy
PHAST (KNOWN)	3,364	$0.738\pm 0.143$	$0.017\pm 0.005$	$0.106\pm 0.020$
PHAST (PARTIAL)	13,736	$0.680\pm 0.043$	$0.018\pm 0.001$	$0.092\pm 0.014$
PHAST (UNKNOWN)	13,738	$0.993\pm 0.128$	$0.275\pm 0.083$	$0.298\pm 0.048$
Baselines
GRU	37,889	$2.939\pm 0.302$	$1.310\pm 0.743$	$1.796\pm 0.625$

Appendix

Table 17: Q-only open-loop forecasting (1-DOF pendulum environments). Mean

\pm

std of wrapped-angle rollout MSE at horizon

H{=}100

over 5 seeds. Parameter counts correspond to the Windy setting (largest; position-dependent damping introduces additional parameters). Lower is better.

PHAST (ours)
Model	Params	Double (cons)	Double (damped)
PHAST (KNOWN)	3,588	$0.421\pm 0.061$	$0.332\pm 0.030$
PHAST (PARTIAL)	12,166	$0.402\pm 0.047$	$0.320\pm 0.032$
PHAST (UNKNOWN)	13,067	$0.629\pm 0.060$	$0.529\pm 0.047$
Baselines
GRU	38,146	$1.300\pm 0.143$	$1.346\pm 0.127$

Appendix

Table 18: Q-only open-loop forecasting (2-DOF double pendulum environments). Mean

\pm

std of wrapped-angle rollout MSE at horizon

H{=}100

over 5 seeds.

PHAST (ours)
Model	Params	Oscillator (cons)	Oscillator (damped)
PHAST (KNOWN)	3,587	$0.0010\pm 0.0003$	$0.0012\pm 0.0004$
PHAST (PARTIAL)	12,165	$0.0010\pm 0.0002$	$0.0011\pm 0.0003$
PHAST (UNKNOWN)	12,171	$0.0072\pm 0.0031$	$0.0101\pm 0.0046$
Baselines
GRU	38,146	$1.8470\pm 0.0754$	$1.4565\pm 0.0811$

Appendix

Table 19: Q-only open-loop forecasting (Oscillator environments). Mean

\pm

std of rollout MSE at horizon

H{=}100

over 5 seeds.

Model	Params	RLC (damped)
PHAST (ours)
PHAST (KNOWN)	3,364	$2.64\times 10^{-5}\pm 1.5\times 10^{-8}$
PHAST (PARTIAL)	11,878	$2.64\times 10^{-5}\pm 1.1\times 10^{-8}$
PHAST (UNKNOWN)	11,879	$2.63\times 10^{-5}\pm 3.1\times 10^{-8}$
Baselines
S5	17,089	$1.84\times 10^{-3}\pm 6.69\times 10^{-4}$

Appendix

Table 20: Q-only open-loop forecasting (RLC circuit, KNOWN regime). Mean

\pm

std of next-step MSE over 5 seeds. The damped series RLC circuit (

L{=}1

C{=}1

R{=}0.5

) is the electrical analog of a damped spring–mass system. Lower is better.

Model	Params	LJ-3 (damped)
PHAST (ours)
PHAST (KNOWN)	4,490	$4.61\times 10^{-10}\pm 4.20\times 10^{-12}$
PHAST (PARTIAL)	13,324	$4.59\times 10^{-10}\pm 2.04\times 10^{-12}$
PHAST (UNKNOWN)	13,355	$6.93\times 10^{-10}\pm 1.38\times 10^{-11}$
Baselines
S5	17,734	$2.05\times 10^{-4}\pm 4.33\times 10^{-5}$

Appendix

Table 21: Q-only open-loop forecasting (Lennard–Jones 3-particle cluster, KNOWN regime). Mean

\pm

std of next-step MSE over 5 seeds. Three particles in 2D with pairwise LJ potential (

\epsilon{=}1

\sigma{=}1

) and Langevin friction (

\gamma{=}0.1

). Lower is better.

Model	Params	Heat Exch. (damped)
PHAST (ours)
PHAST (KNOWN)	3,592	$2.42\times 10^{-6}\pm 9.74\times 10^{-10}$
PHAST (PARTIAL)	12,170	$2.42\times 10^{-6}\pm 2.63\times 10^{-9}$
PHAST (UNKNOWN)	12,173	$2.48\times 10^{-6}\pm 1.85\times 10^{-8}$
Baselines
S5	17,218	$1.00\times 10^{-3}\pm 1.62\times 10^{-4}$

Appendix

Table 22: Q-only open-loop forecasting (Heat exchange, KNOWN regime). Mean

\pm

std of next-step MSE over 5 seeds. Two coupled thermal masses (

c_{1}{=}c_{2}{=}1

, coupling

\kappa{=}0.5

, loss

\kappa_{\mathrm{loss}}{=}0.1

). Lower is better.

Model	Params	N-body 3 (damped)
PHAST (ours)
PHAST (KNOWN)	4,489	$4.28\times 10^{-8}\pm 1.28\times 10^{-11}$
PHAST (PARTIAL)	13,323	$4.27\times 10^{-8}\pm 2.25\times 10^{-11}$
PHAST (UNKNOWN)	13,355	$4.28\times 10^{-8}\pm 6.17\times 10^{-11}$
Baselines
S5	17,734	$3.03\times 10^{-3}\pm 5.98\times 10^{-4}$

Appendix

Table 23: Q-only open-loop forecasting (N-body gravity, PARTIAL regime). Mean

\pm

std of next-step MSE over 5 seeds. Three bodies in 2D with softened gravity (

G{=}1

\epsilon_{\mathrm{soft}}{=}0.1

) and drag (

\gamma{=}0.05

). Gravitational potential template is given; masses and damping are learned. Lower is better.

Model	Params	Predator–Prey (damped)
PHAST (ours)
PHAST (KNOWN)	3,364	$0.0203\pm 0.0002$
PHAST (PARTIAL)	11,878	$0.0203\pm 0.0002$
PHAST (UNKNOWN)	11,879	$0.0199\pm 0.0004$
Baselines
S5	17,089	$6.12\pm 0.58$

Appendix

Table 24: Q-only open-loop forecasting (Predator–prey, UNKNOWN regime). Mean

\pm

std of next-step MSE over 5 seeds. Dissipative Lotka–Volterra (

\alpha{=}1

\beta{=}0.1

\gamma{=}0.4

\delta{=}0.1

K{=}100

\mu{=}0.01

). All PHAST components are neural (non-canonical Hamiltonian structure). Lower is better.

Model	Params	$\mathrm{WrapMSE}_{\theta}$ $\downarrow$	$\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(100)$ $\downarrow$	$R^{2}_{D}$ $\uparrow$	$\mathrm{MAE}_{D}$ $\downarrow$
PHAST (PARTIAL)	10,376	$1.49{\times}10^{-5}\pm 2.86{\times}10^{-6}$	$0.089\pm 0.005$	$0.798\pm 0.034$	$0.045\pm 0.005$
PHAST (UNKNOWN)	10,378	$6.35{\times}10^{-5}\pm 1.47{\times}10^{-5}$	$0.128\pm 0.003$	$0.583\pm 0.082$	$0.081\pm 0.010$
DHNN	107,124	$0.186\pm 0.061$	$2.264\pm 0.160$	—	—

Appendix

Table 25: Full-state Windy Pendulum (q,p observed): PHAST vs DHNN. Mean

\pm

std over 5 seeds (dataset seed fixed to 42). Lower is better. DHNN does not expose an explicit damping field

D(q)

, so

R^{2}_{D}

\mathrm{MAE}_{D}

are not reported.

Model	Params	$\mathrm{MixedMSE}^{\mathrm{roll}}(H{=}100)$ $\downarrow$	$\mathrm{EbudRes}^{\mathrm{roll}}(H{=}100)$ $\downarrow$	$\mathrm{PassViol}^{\mathrm{roll}}(H{=}100)$ $\downarrow$	$R^{2}_{D}$ $\uparrow$	$\mathrm{MAE}_{D}$ $\downarrow$
PHAST (KNOWN, constant $M$ )	3,589	$0.096\pm 0.016$	$3.31\pm 0.39$	$0.124\pm 0.007$	$0.956\pm 0.004$	$0.028\pm 0.001$
PHAST (KNOWN, $M(q)$ )	3,589	$0.031\pm 0.006$	$1.30\pm 0.06$	$0.016\pm 0.007$	$0.986\pm 0.002$	$0.013\pm 0.001$

Appendix

Table 26: Nonseparable mass ablation (Windy Cart-Pole, q-only). Mean

\pm

std over 3 seeds (dataset seed fixed to 42). Both rows use the same implicit-midpoint conservative core; only the mass model differs. Lower is better.

Model	Params	$\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100)$ $\downarrow$	$\mathrm{EbudRes}^{\mathrm{roll}}(H{=}100)$ $\downarrow$	$\mathrm{PassViol}^{\mathrm{roll}}(H{=}100)$ $\downarrow$
PHAST (KNOWN, constant $M$ )	3,589	$0.367\pm 0.083$	$32.71\pm 4.40$	$0.408\pm 0.030$
PHAST (KNOWN, $M(q)$ )	3,589	$0.284\pm 0.029$	$17.16\pm 1.00$	$0.343\pm 0.026$

Appendix

Table 27: Nonseparable mass ablation (Damped Double Pendulum, q-only). Mean

\pm

std over 3 seeds (dataset seed fixed to 42). Both rows use the same implicit-midpoint conservative core; only the mass model differs. Lower is better.

Environment	Metric	$\Delta t$	$L{=}1$	$L{=}4$
Windy Cart-Pole	$\mathrm{MixedMSE}^{\mathrm{roll}}(H{=}100)$	0.05	$0.134\pm 0.020$	$0.126\pm 0.016$
Double Pendulum	$\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100)$	0.04	$2.412\pm 0.071$	$2.366\pm 0.088$

Appendix

Table 28: Effect of PHAST core transition substeps at coarse sampling. Mean

\pm

std over 5 seeds for PHAST (PARTIAL, q-only) with fixed timesteps

\delta t_{s}=\Delta t/L

. Metrics are rollout error at horizon

H{=}100

Figure 22: CPU microbenchmark of PHAST structured primitives. Average wall-clock time per call for Householder damping application and Woodbury-based mass solve, sweeping degrees of freedom

n

with fixed rank

r{=}2

Setting	$\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100)$ $\downarrow$	$R^{2}_{D}$ $\uparrow$	$\mathrm{MAE}_{D}$ $\downarrow$	$\mathrm{EbudRes}(H{=}100)$ $\downarrow$	$\mathrm{PassViol}^{\mathrm{roll}}(H{=}100)$ $\downarrow$
fixed $\Delta t_{\mathrm{model}}$ , fixed $d_{0}$	$0.175\pm 0.009$	$0.465\pm 0.038$	$0.084\pm 0.003$	$1.34\pm 0.04$	$0.020\pm 0.004$
learnable $\Delta t_{\mathrm{model}}$ , fixed $d_{0}$	$0.153\pm 0.013$	$0.472\pm 0.040$	$0.085\pm 0.003$	$1.28\pm 0.04$	$0.015\pm 0.004$
fixed $\Delta t_{\mathrm{model}}$ , learned $d_{0}$	$0.175\pm 0.009$	$0.476\pm 0.047$	$0.082\pm 0.004$	$1.34\pm 0.04$	$0.020\pm 0.004$
learnable $\Delta t_{\mathrm{model}}$ , learned $d_{0}$	$0.153\pm 0.013$	$0.478\pm 0.047$	$0.084\pm 0.004$	$1.28\pm 0.04$	$0.015\pm 0.004$

Appendix

Table 29: Learnable timestep and base damping policy (Windy Pendulum, q-only; PARTIAL). Mean

\pm

std over 3 seeds on CPU (20 epochs; dataset seed fixed to 42). All runs use Householder damping with a fixed cap

\sum_{i}\beta_{i}(q)\leq\bar{\beta}

(

\bar{\beta}=0.5

) and report horizon-

H{=}100

diagnostics under the standard q-only rollout protocol (Sec. 4.1.1 ).

Model (scaled)	Params	$\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100)$ $\downarrow$
GRU ( $d{=}128$ )	149,505	$2.130\pm 0.566$
S5 ( $d{=}128$ )	34,049	$0.654\pm 0.015$
Transformer ( $d{=}128$ )	396,929	$0.868\pm 0.237$

Appendix

Table 30: Baseline capacity scaling (Windy Pendulum, q-only). Mean

\pm

std of wrapped-angle rollout MSE at horizon

H{=}100

over 3 seeds (dataset seed fixed to 42) on CPU. Increasing baseline hidden dimension does not close the gap to PHAST (Table 17 ).

Panel of Figure 23 — Figure 23: Low-rank PSD damping with spectral bound. Top: $D(q)=d_{0}I+\sum_{i=1}^{r}\beta_{i}(q)\,k_{i}(q)k_{i}(q)^{\top}$ is PSD by construction. Bottom: Bounding $\sum_{i=1}^{r}\beta_{i}(q)\leq\bar{\beta}$ certifies $\lambda_{\max}(D(q))\leq d_{0}+\bar{\beta}$ (Eq. ( 11 )).

Cap on $\sum_{i=1}^{r}\beta_{i}(q)$	$\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100)$ $\downarrow$	$R^{2}_{D}$ $\uparrow$	$\mathrm{MAE}_{D}$ $\downarrow$	$\mathrm{EbudRes}(H{=}100)$ $\downarrow$	$\mathrm{PassViol}^{\mathrm{roll}}(H{=}100)$ $\downarrow$
Unbounded	$0.162\pm 0.015$	$-40.940\pm 8.630$	$0.897\pm 0.089$	$1.43\pm 0.04$	$0.022\pm 0.006$
$\sum_{i=1}^{r}\beta_{i}(q)\leq 0.25$	$0.180\pm 0.012$	$-0.080\pm 0.081$	$0.105\pm 0.004$	$1.32\pm 0.03$	$0.019\pm 0.004$
$\sum_{i=1}^{r}\beta_{i}(q)\leq 0.5$	$0.175\pm 0.009$	$0.465\pm 0.038$	$0.084\pm 0.003$	$1.34\pm 0.04$	$0.020\pm 0.004$
$\sum_{i=1}^{r}\beta_{i}(q)\leq 1.0$	$0.175\pm 0.014$	$-1.506\pm 0.582$	$0.211\pm 0.026$	$1.38\pm 0.05$	$0.022\pm 0.004$

Appendix

Table 31: Cap sensitivity sweep (Windy Pendulum, q-only; PARTIAL). Mean

\pm

std over 3 seeds on CPU (20 epochs; dataset seed fixed to 42). All runs use Householder damping with fixed

\Delta t=0.05

, fixed

\Delta t_{\mathrm{model}}=\Delta t

, fixed

d_{0}

, and vary only the total-strength cap

\sum_{i}\beta_{i}(q)\leq\bar{\beta}

. We report horizon-

H{=}100

diagnostics under the standard q-only rollout protocol (Sec. 4.1.1 ).

Cap on $\sum_{i=1}^{r}\beta_{i}(q)$	$\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100)$ $\downarrow$	$R^{2}_{D}$ $\uparrow$	$\mathrm{MAE}_{D}$ $\downarrow$	$\mathrm{EbudRes}(H{=}100)$ $\downarrow$
Unbounded	$0.298\pm 0.054$	$-96.546\pm 17.774$	$1.343\pm 0.129$	$2.581\pm 0.214$
$\sum_{i=1}^{r}\beta_{i}(q)\leq 0.5$	$0.416\pm 0.071$	$-0.234\pm 0.268$	$0.134\pm 0.019$	$2.713\pm 0.207$
$\sum_{i=1}^{r}\beta_{i}(q)\leq 1.0$	$0.396\pm 0.076$	$-6.457\pm 1.376$	$0.363\pm 0.034$	$2.666\pm 0.214$

Appendix

Table 32: UNKNOWN regime: effect of bounding total damping strength (Windy Pendulum, q-only). Mean

\pm

std over 5 seeds on CPU (same data protocol as Sec. 4.1.1 ). Bounding

\sum_{i=1}^{r}\beta_{i}(q)

substantially improves damping recovery (identifiability) but can degrade long-horizon open-loop forecasting, illustrating a forecasting–identifiability trade-off in the UNKNOWN regime.

Setup	Mass	$\mathrm{FrobErr}_{M}$ $\downarrow$	$\mathrm{MAE}_{\nabla V}$ $\downarrow$	Conclusion
Learn $M$ (known $V$ ) + $\mathcal{L}_{\mathrm{res}}$	SPD	$0.011$	—	Identifiable
Learn $M$ (known $V$ ) + $\mathcal{L}_{\mathrm{res}}$	NeuralMass	$0.306$	—	Identifiable (high var.)
Learn $V$ (known $M$ ) + $\mathcal{L}_{\mathrm{res}}$	Known $M$	—	$0.094$	Identifiable
Learn both ( $M$ + neural $V$ ) + $\mathcal{L}_{\mathrm{res}}$	SPD	$2.37$	$6.93$	Non-identifiable
Learn both ( $M$ + neural $V$ ) + $\mathcal{L}_{\mathrm{res}}$	NeuralMass	$2.23$	$6.73$	Non-identifiable
Learn $M$ + structured $V$ + $\mathcal{L}_{\mathrm{res}}$	SPD	$0.035$	$0.082$	Identifiable

Appendix

Table 33: Gauge-freedom ablation (conservative double pendulum). Mean over 2 runs (50 epochs). Lower is better. “Structured

V

” means the functional form is fixed but coefficients are learned (analogous to PHAST PARTIAL). Errors are averaged over held-out configurations:

\mathrm{FrobErr}_{M}:=\mathbb{E}_{q}\|M(q)-\hat{M}(q)\|_{F}

and

\mathrm{MAE}_{\nabla V}:=\frac{1}{n}\mathbb{E}_{q}\|\nabla V(q)-\nabla\hat{V}(q)\|_{1}

Index	Range	Meaning
$b$	$1,\ldots,B$	Trajectory index (batching)
$t$	$0,\ldots,T-1$	Time index (transitions use $t{=}0,\ldots,T{-}2$ )
$h$	$0,\ldots,H-1$	Rollout step index (horizon $H$ )
$s$	$1,\ldots,L$	PHAST substep index
$i$	$1,\ldots,r$	Rank-1 term index in low-rank updates
$j$	$1,\ldots,n$	Coordinate index

Appendix

Table 34: Index ranges.

Symbol	Space / shape	Meaning
Problem setting & knowledge regimes Sec. 1
$n$	$\mathbb{N}$	Degrees of freedom
$\mathcal{Q}$	manifold	Configuration space (e.g., $\mathbb{R}^{n_{e}}\times(\mathbb{S}^{1})^{n_{a}}$ )
$q$	$\mathcal{Q}$	Generalized configuration (forecast target)
$p$	$\mathbb{R}^{n}$	Conjugate momentum (latent in q-only setting)
$x=(q,p)$	$\mathcal{Q}\times\mathbb{R}^{n}$	Phase-space state

Appendix

Table 35: Symbol reference. Symbols are grouped by the section in which they are introduced. Gray rows mark group headers indicating the originating section.

Time Series Forecasting

Time series forecasting predicts future values from ordered observations using statistical, neural, and probabilistic models to support decisions under uncertainty.

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May 15, 2026

Identify Then Project: Contrastive Learning of Latent Dynamics from Partial Observations with Port-Hamiltonian Structure

Identifying latent state representations and dynamics is essential when direct modeling in observation space is infeasible, particularly under partial and high-dimensional observations. In such settings, representation learning and physics-aware modeling are inherently coupled. We study this problem for latent port-Hamiltonian systems, a structured class encompassing both conservative and dissipative dynamics. We propose a two-stage identify-then-project framework. First, a contrastive teacher learns continuous-time latent dynamics from partial observations. Then, a student projects the identified teacher representation and dynamics onto a port-Hamiltonian submanifold via a learned affine chart, yielding a physically consistent realization. As a conceptual counterfactual, we also consider a single-stage variant that jointly learns latent identification and port-Hamiltonian structure, but find it to be less reliable, motivating the proposed two-stage teacher-student framework. We show theoretically that affine projection is the natural bridge between the affine gauge of contrastive latent identification and the port-Hamiltonian systems. Empirically, we demonstrate that the proposed two-stage approach preserves the teacher's dynamics while enforcing physical structure, and performs more reliably than the single-stage alternative, particularly in dissipative regimes and high-dimensional visual settings.

Peilun Li, Kaiyuan Tan, Daniel Moyer +1

Department of Computer Science Vanderbilt University Nashville, TN 37235

Contrastive Learning Latent Dynamics Modeling System Identification Physics-Informed ML State Representation Learning Port-Hamiltonian Systems+1

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May 22, 2026

Learning partially observed systems with neural Hamiltonian ordinary differential equations

When learning dynamical systems from data, embedding physical structure can constrain the solution space and improve generalization, but many physics-informed models assume access to the full system state. This limits their use in partially observed settings, where some state variables are completely unobserved and must be inferred without direct supervision. Here, we present neural Hamiltonian ordinary differential equations (NHODE), a framework that combines Hamiltonian neural networks (HNNs) with neural ordinary differential equations (neural ODEs) to learn partially observed dynamical systems from data. The Hamiltonian structure enforces energy conservation by construction, while the neural ODE framework enables a flexible training procedure that allows the loss to be defined only on observed variables. We also incorporate additional physical constraints through symmetry-aware coordinate transformations and separable energy formulations. The framework is evaluated on systems of increasing complexity, from linear and nonlinear mass-spring systems to the chaotic three-body problem. Across all examples, increasing the amount of embedded physical structure improves the accuracy and long-horizon stability of the predictions. Even in the most challenging regimes, the NHODE framework captures both observed and latent dynamics, whereas purely data-driven baselines become unstable.

Sunniva Meltzer, Sølve Eidnes, Alexander Johannes Stasik

Department of Mathematics and Cybernetics, SINTEF Digital, Norway · Department of Physics, University of Oslo, Norway · Department of Data Science, Norwegian University of Life Science, Norway

Dynamical Systems Latent Dynamics Modeling Nonlinear System Identification Hamiltonian Neural Networks Physics-Informed ML Neural ODEs+1

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Aug 13, 2026

History-informed Lagrangian Neural Networks

Forecasting the long-horizon evolution of mechanical systems from position-only observations is a pivotal yet difficult task, as hidden velocities and trajectory-specific physical properties must be inferred simultaneously. Although physics-guided neural networks like Lagrangian Neural Networks (LNNs) guarantee physical plausibility, they generally require complete state inputs and lack adaptability to changing system parameters. To break these limitations, we introduce History-informed Lagrangian Neural Networks (HiLNN). Grounded in the insight that temporal position sequences implicitly encode underlying dynamics, HiLNN employs a recurrent encoder to extract a latent context from history. This context not only reconstructs the unobserved initial velocity but also adaptively modulates the mass matrix, potential energy, and damping coefficients of a structured Lagrangian system. By leveraging a differentiable RK4 rollout scheme, the entire pipeline is optimized end-to-end under multi-step trajectory supervision and energy-consistency regularization. Empirical evaluations across conservative, dissipative, and heterogeneous variable-parameter systems show that HiLNN delivers superior long-term prediction accuracy and maintains precise energy profiles compared to state-of-the-art baselines. The source code is publicly available at https://github.com/yingtian22/History-informed-LNN.

Tianshuo Zhang, Xianglei Xing, Wenzhe Zhai +2

College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China

Latent Dynamics Modeling Physics-Informed ML Lagrangian Neural Networks

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Abstract

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