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

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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~qtq_t at discrete times (momenta~ptp_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 x˙=(J−R)∇H(x)\dot{x}=(J-R)\nabla H(x), guaranteeing dH/dt≤0dH/dt\le 0 when R⪰0R\succeq 0. We introduce PHAST (Port-Hamiltonian Architecture for Structured Temporal dynamics), which decomposes the Hamiltonian into potentialV(q)V(q), massM(q)M(q), and damping~D(q)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

RegimeDomainSystemForecast qqV(q)V(q) : landscapeM(q)M(q) : geometryD(q)D(q) : dissipationIdent.?
KnownMechanicsPendulumAngle θ(t)\theta(t)−gcos⁡θ-g\cos\thetaScalar mmAir drag d(θ)d(\theta)Yes
MechanicsSpring–massDisplacement x(t)x(t)12kx2\tfrac{1}{2}kx^{2}Scalar mmViscous bx˙b\dot{x}Yes
MolecularLennard–JonesPositions ri(t)r_{i}(t)4ϵ[(σ/r)12−(σ/r)6]4\epsilon[(\sigma/r)^{12}{-}(\sigma/r)^{6}]Diagonal mImISolvent friction γI\gamma IYes
ElectricalRLC circuitCharge qC(t)q_{C}(t)q2/2Cq^{2}/2CInductance LLResistance RRYes
ThermalHeat exchangeTemperature T(t)T(t)12c T2\tfrac{1}{2}c\,T^{2}Time constant τ\tauHeat loss κ\kappaYes
PartialMechanicsPendulum (unk. mm )Angle θ(t)\theta(t)−gcos⁡θ-g\cos\theta (given)mImI (learn mm )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)V(q) (environmental forces and equilibria), the geometry M(q)M(q) (Riemannian metric governing the cost of motion), and the dissipation D(q)D(q) (energy loss structure). The regime determines which components are specified versus learned. Anchoring either V(q)V(q) or M(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 GG resolves the remaining one-parameter gauge.
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))(V,M(q)) ; dissipation D(q)D(q) is always learned. All regimes share the same continuous-time structure x˙=(J−R)∇H\dot{x}=(J{-}R)\nabla H and a structure-preserving discrete-time transition map ΦΔt\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)M(q) ), and Windy Cart-Pole ( R×S1\mathbb{R}\times\mathbb{S}^{1} , per-DOF damping, M(q)M(q) ). The Cart-Pole features two distinct damping mechanisms: constant viscous cart friction ( dcd_{\mathrm{c}} ) and position-dependent angular wind damping ( d(θ)d(\theta) ). (D) Wrapped-angle rollout MSE at H=100H{=}100 (lower == better) and damping identifiability RD2R^{2}_{D} (higher == better; bold == best per column). PHAST (KNOWN) achieves near-perfect identifiability ( RD2≈1R^{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 ).
MethodR≠0R\neq 0PassivitySpectralEfficient
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≠0R\neq 0 ), passivity-aware structure, spectral control, and efficient primitives for long-horizon rollouts. Passivity indicates a by-construction storage-function inequality.
Figure 3: PHAST core transition computation graph. Components VV , MM , and DD assemble into Hamiltonian HH and dissipation RR ; in the unforced case ( u=0u=0 ), the port-Hamiltonian form implies dHdt≤0\tfrac{dH}{dt}\leq 0 in continuous time. PHAST advances the phase state using a structure-preserving discrete-time map ΦΔt\Phi_{\Delta t} (Strang splitting) and optionally composes LL integration substeps per environment step (Eq. ( 78 )). The Hk+1≲HkH_{k+1}\lesssim H_{k} annotations are schematic; discrete-time energy monotonicity depends on timestep and integrator (Appendix A.2 ).
Figure 4: Two-axis evaluation (open-loop, conceptual). Forecasting accuracy (low rollout MSE) and physical identifiability (high damping R2R^{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=100H{=}100 steps. Column 1 (Windy Pendulum, θ\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, θ1\theta_{1} and θ2\theta_{2} ): a chaotic 2-DOF system with coupled configuration-dependent inertia M(q)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, θ\theta ): a mixed-manifold system ( R×S1\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 (θ,p)(\theta,p) with p=M(q)q˙p=M(q)\dot{q} during the open-loop segment. Momentum is latent in the q-only setting: for PHAST, q˙\dot{q} comes from the learned FD+TCN observer (Sec. 3 ); for baselines, q˙\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.
ModelParamsCart-Pole (windy)
PHAST (ours)
PHAST (KNOWN)3,5890.063±0.0190.063\pm 0.019
PHAST (PARTIAL)14,2830.083±0.0220.083\pm 0.022
PHAST (UNKNOWN)14,2900.109±0.0220.109\pm 0.022
S5 (best baseline)17,2180.431±0.0770.431\pm 0.077
Table 3: Q-only open-loop forecasting (Cart-Pole). Mean ±\pm std of mixed-manifold rollout MSE at horizon H=100H{=}100 over 5 seeds. The mixed metric averages translation MSE and wrapped-angle MSE (Appendix F ). Lower is better.
BenchmarkBest PHASTBest baselineGain
Mechanical systems (rollout MSE at H=100H{=}100 )
Pendulum (cons)0.680±0.0430.680\pm 0.043 (PARTIAL)2.320±0.2242.320\pm 0.224 (Transformer)3.4×3.4\times
Pendulum (damped)0.017±0.0050.017\pm 0.005 (KNOWN)0.450±0.2410.450\pm 0.241 (D-LinOSS)26.5×26.5\times
Pendulum (windy)0.092±0.0140.092\pm 0.014 (PARTIAL)0.435±0.2390.435\pm 0.239 (D-LinOSS)4.7×4.7\times
Cart-Pole (windy)0.063±0.0190.063\pm 0.019 (KNOWN)0.431±0.0770.431\pm 0.077 (S5)6.8×6.8\times
Oscillator (cons)0.0010±0.00020.0010\pm 0.0002 (KNOWN/PARTIAL)1.087±0.2991.087\pm 0.299 (Transformer)1.1×1031.1\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=100H{=}100 ; the last five report next-step MSE (Tables 20 – 24 ). Gains are per-row ratios and are not directly comparable across environments.
ModelWrapMSEθroll(H=100)\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100) ↓\downarrowRD2R^{2}_{D} ↑\uparrowMAED\mathrm{MAE}_{D} ↓\downarrowEbudResroll(H=100)\mathrm{EbudRes}^{\mathrm{roll}}(H{=}100) ↓\downarrowParams
PHAST (KNOWN)0.106±0.0200.106\pm 0.0200.996±0.0040.996\pm 0.0040.007±0.0040.007\pm 0.0041.56±0.021.56\pm 0.023,364
PHAST (PARTIAL)0.092±0.0140.092\pm 0.0140.654±0.0770.654\pm 0.0770.063±0.0110.063\pm 0.0111.50±0.041.50\pm 0.0413,736
PHAST (UNKNOWN)0.298±0.0480.298\pm 0.048−96.546±15.897-96.546\pm 15.8971.343±0.1151.343\pm 0.1152.58±0.192.58\pm 0.1913,738
GRU1.796±0.6251.796\pm 0.625——12.34±6.5812.34\pm 6.5837,889
S50.600±0.0470.600\pm 0.047——11.71±2.9111.71\pm 2.9117,089
LinOSS1.458±0.3241.458\pm 0.324——56.14±37.4156.14\pm 37.4117,089
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 capWrapMSE θ (100) ↓\downarrowRD2R^{2}_{D} ↑\uparrowMAE D ↓\downarrowEbudRes(100) ↓\downarrow
Unbounded0.059±0.0070.059\pm 0.007−90.15±6.14-90.15\pm 6.141.326±0.0441.326\pm 0.0441.50±0.031.50\pm 0.03
∑iβi(q)≤0.5\sum_{i}\beta_{i}(q)\leq 0.50.071±0.0040.071\pm 0.0040.703±0.0720.703\pm 0.0720.061±0.0100.061\pm 0.0101.58±0.041.58\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 ( RD2≪0R^{2}_{D}\ll 0 ) even when rollouts are accurate; bounding ∑i=1rβi(q)\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(θ)d(\theta) vs. ground truth (black). Pendulum : PHAST (KNOWN) recovers the sinusoidal profile d(θ)=d0+Δd ∣ ⁣sin⁡θ∣d(\theta){=}d_{0}{+}\Delta d\,|\!\sin\theta| near-exactly ( RD2≈1R^{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 ( d1d_{1} , d2d_{2} ). Despite chaotic dynamics and a coupled M(q)M(q) , KNOWN recovers both joints’ damping profiles to high accuracy. Cart-Pole : solid and dashed curves distinguish constant viscous cart friction ( dcd_{\mathrm{c}} , flat) from angular wind damping on the pole ( d(θ)d(\theta) , sinusoidal) — two qualitatively different dissipation mechanisms that PHAST disentangles from q-only data. UNKNOWN shows poor identifiability across all systems ( RD2≪0R^{2}_{D}\ll 0 ; Table 4 ). Baselines do not expose explicit damping fields (marked N/A). Bottom row : total energy H(t)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.
Figure 7: Energy–Casimir control as port-Hamiltonian interconnection. The plant (blue, left) and controller (green, right) are pH systems coupled through power ports: uc=ypu_{c}=y_{p} feeds plant velocity to the controller, up=−yc+vu_{p}=-y_{c}+v returns the shaped restoring force plus an auxiliary channel vv . When v=0v{=}0 the cross-power cancels ( yp⊤up+yc⊤uc=0y_{p}^{\top}u_{p}+y_{c}^{\top}u_{c}=0 ), and the Casimir C=q−ξ\mathcal{C}=q-\xi is invariant. Damping injection v=−dinjy^v=-d_{\mathrm{inj}}\hat{y} (red, dashed) ensures H˙cl≤0\dot{H}_{\mathrm{cl}}\leq 0 ; in q-only settings y^≈yp\hat{y}\approx y_{p} is estimated, so estimation error can break exact power balance.
σ\sigmaMethodVelocity estimateSuccess ↑\uparrowFinal error ↓\downarrowEffort ↓\downarrow∣q˙^−q˙∣|\hat{\dot{q}}-\dot{q}| ↓\downarrow
0.00Oracleoracle1.001.31×10−41.31\times 10^{-4}262.90.000
0.00Finite differencesFD1.001.21×10−41.21\times 10^{-4}268.60.007
0.00MAP smootherMAP smoother1.001.16×10−41.16\times 10^{-4}268.60.007
0.00FD+TCN observerFD+TCN1.001.03×10−31.03\times 10^{-3}270.70.008
0.00PHAST-trained observerPHAST observer1.005.71×10−35.71\times 10^{-3}272.40.019
0.01Oracleoracle1.005.49×10−35.49\times 10^{-3}262.90.000
Table 7: Energy–Casimir control on the true pendulum (q-only sensing). Mean over 100100 trials (5 regimes ×\times 20). All methods use the same predictor–corrector ξ\xi update ( kξ=5k_{\xi}=5 ).
σ\sigmaMethodVelocity estimateSuccess ↑\uparrowFinal error ↓\downarrowEffort ↓\downarrow∣q˙^−q˙∣|\hat{\dot{q}}-\dot{q}| ↓\downarrow
0.00Oracleoracle1.001.31×10−41.31\times 10^{-4}262.90.000
0.00PHAST model-based∂H^/∂p\partial\hat{H}/\partial p1.005.46×10−55.46\times 10^{-5}245.00.011
0.01Oracleoracle1.005.49×10−35.49\times 10^{-3}262.90.000
0.01PHAST model-based∂H^/∂p\partial\hat{H}/\partial p1.005.32×10−35.32\times 10^{-3}245.10.011
Table 8: Energy–Casimir with PHAST-based velocity (q,p observed). Mean over 100100 trials (5 regimes ×\times 20). Both methods use the same predictor–corrector ξ\xi update ( kξ=5k_{\xi}=5 ).
MethodFinal error ↓\downarrowEffort ↓\downarrow∣q˙^−q˙∣|\hat{\dot{q}}-\dot{q}| ↓\downarrow
Oracle0.00416.30.000
Finite differences0.017080.11.117
MAP smoother0.013310.30.095
FD+TCN observer0.01378.60.130
PHAST-trained observer0.018854.90.911
Table 9: Near-stable regime ( σ=0.01\sigma=0.01 ). Learning a noise-aware q-only observer reduces control effort near the target by ∼\sim 9 ×\times vs. finite differences.
Observer train noiseFinal error ↓\downarrowEffort ↓\downarrow∣q˙^−q˙∣|\hat{\dot{q}}-\dot{q}| ↓\downarrow
σtrain=0.00\sigma_{\text{train}}=0.00 (mismatch)0.016882.41.134
σtrain=0.01\sigma_{\text{train}}=0.01 (matched)0.01208.50.132
σtrain∼U[0,0.01]\sigma_{\text{train}}\sim\mathcal{U}[0,0.01] (anchored range)0.01378.60.130
Table 10: Observer training noise vs. evaluation noise ( σeval=0.01\sigma_{\text{eval}}=0.01 ), near-stable regime. Q-only FD+TCN observer used inside the Energy–Casimir controller.
MethodVelocity estimateSuccess ↑\uparrowFinal error ↓\downarrowEffort ↓\downarrow∣q˙^−q˙∣|\hat{\dot{q}}-\dot{q}| ↓\downarrow
Oracleoracle1.002.75×10−22.75\times 10^{-2}263.60.000
Finite differencesFD0.008.36×10−28.36\times 10^{-2}2109.05.582
FD+TCN observerFD+TCN (noise)0.267.70×10−27.70\times 10^{-2}396.60.659
Table 11: High-noise stress test ( σ=0.05\sigma=0.05 ). Mean over 100100 trials. For the FD+TCN controller we use a noise-conditioned observer trained with σtrain∼U[0,0.05]\sigma_{\text{train}}\sim\mathcal{U}[0,0.05] .
Figure 8: Success rate breakdown at σ=0.05\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

Figure 9: Single pendulum. A point mass mm at distance ℓ\ell from a fixed pivot, with moment of inertia II about the pivot. The angle θ\theta is measured from the downward vertical. Position-dependent damping d(θ)d(\theta) acts at the pivot.
Figure 10: Windy damping profile. The position-dependent damping coefficient d(θ)=d0+Δd ∣sin⁡θ∣d(\theta)=d_{0}+\Delta d\,|\sin\theta| varies between d0=0.3d_{0}=0.3 (vertical) and d0+Δd=0.8d_{0}+\Delta d=0.8 (horizontal), modeling air resistance that depends on the pendulum’s cross-sectional area.
Figure 11: Cart-pole system. A cart of mass mcm_{c} moves on a horizontal rail while a pole of mass mpm_{p} (concentrated at distance ℓ\ell from the pivot) rotates about the cart. The angle θ\theta is measured from the downward vertical (hanging configuration: θ=0\theta=0 ). Windy damping acts isotropically on q˙\dot{q} via a scalar coefficient d(θ)d(\theta) .
Figure 12: Harmonic oscillator. Two independent harmonic oscillators with natural frequency ω\omega and unit mass. Coordinates q1,q2q_{1},q_{2} measure displacements from equilibrium. Optional viscous damping γ\gamma acts on each oscillator.
Figure 13: Double pendulum (point-mass model). Two point masses m1,m2m_{1},m_{2} at the ends of massless rods of lengths ℓ1,ℓ2\ell_{1},\ell_{2} . The angles θ1\theta_{1} and θ2\theta_{2} are both measured from the downward vertical (absolute angles). Optional viscous damping bb acts at both joints.
Figure 14: Spring–mass–damper. A mass mm connected to a wall by a spring (stiffness kk ) and dashpot (damping bb ) in parallel. Displacement xx is measured from equilibrium.
Figure 15: Lennard–Jones cluster ( Jones, 1924 ) . N=3N{=}3 particles interact pairwise via the LJ potential. Inset: the repulsive core ( r−12r^{-12} , red shading ) dominates at r<σr<\sigma ; the attractive well ( −r−6-r^{-6} , blue shading ) has depth ϵ\epsilon near the equilibrium distance req=21/6σr_{\mathrm{eq}}=2^{1/6}\sigma . Dissipation is modeled by isotropic velocity-proportional damping ( D=γID=\gamma I ).
Figure 16: Series RLC circuit ( Horowitz & Hill, 2015 ) . Charge qCq_{C} on the capacitor is the generalized coordinate; flux linkage p=Lip=Li is the conjugate momentum. Color coding shows the port-Hamiltonian mapping: capacitance CC →\to potential VV , inductance LL →\to mass MM , resistance RR →\to damping DD . The electrical energy H=qC2/2C+Li2/2H=q_{C}^{2}/2C+Li^{2}/2 is the storage function.
Figure 17: Lumped heat exchange ( Incropera et al., 2007 ) . Two thermal masses with temperatures T1T_{1} and T2T_{2} are the generalized coordinates ( ndof=2n_{\mathrm{dof}}{=}2 ). Stored thermal energy Vi=12ciTi2V_{i}=\tfrac{1}{2}c_{i}T_{i}^{2} is the potential for each mass; the thermal time constants τi\tau_{i} play the role of inertia; and the coupling coefficient κ\kappa provides mutual heat exchange while individual losses κloss\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).
Figure 18: Robot arm (2-DOF manipulator). Joint angles θ1,θ2\theta_{1},\theta_{2} are generalized coordinates. Link inertias and gravity torques are known from CAD; joint friction (red) is learned from trajectory data.
Figure 19: N-body gravitational system ( Binney & Tremaine, 2008 ) . N=3N{=}3 point masses interact pairwise via Newtonian gravity. The gravitational potential VV is known in functional form but individual masses mim_{i} are learned (PARTIAL regime). A softened potential rij2+ϵsoft2r_{ij}^{2}+\epsilon_{\mathrm{soft}}^{2} replaces rij2r_{ij}^{2} to avoid the singularity at close approach. Drag D=γID=\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.
Figure 20: Predator–prey (Lotka–Volterra) ( Murray, 2002 ; Strogatz, 2018 ) . Prey density n1n_{1} is the generalized coordinate qq ; predator density n2n_{2} serves as the conjugate momentum pp ( ndof=1n_{\mathrm{dof}}{=}1 ). All port-Hamiltonian components are learned (UNKNOWN regime): V(q)V(q) is a neural potential generalizing the classical Lotka–Volterra conserved quantity, MM is a learned SPD “ecological inertia,” and DD is a learned PSD dissipation encoding mortality, disease, and resource depletion. The port-Hamiltonian structure guarantees bounded dynamics even without parameter recovery.
ComponentParametersShared across tt ?Shared across substeps LL ?
Observer oϕo_{\phi} (FD+TCN)ϕ\phi✓—
Canonicalizer CψC_{\psi}ψ\psi✓—
Physics (V,M,D)(V,M,D)θ\theta✓✓
Substep timesteps {δts}s=1L\{\delta t_{s}\}_{s=1}^{L}τ\tau✓per-substep
Appendix
Table 12: What is learned vs. shared in the q-only PHAST pipeline. PHAST core transition substeps LL reuse the same physics components (V,M,D)(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 ).
HyperparameterKNOWNPARTIALUNKNOWN
Damping terms rr222
Damping bound βˉ\bar{\beta} (Windy)unboundedΔd (=0.5)\Delta d\,(=0.5)unbounded
d0d_{0} policy (Windy)learnedfixedlearned
Potential hidden dim—6464
Potential layers—33
Mass terms (if learned)——min⁡(4,n)\min(4,n)
Appendix
Table 13: Hyperparameters for the q-only experiments (Sec. 4 ).
BaselineKey settings
GRUdropout=0.0
S5dstate=64d_{\text{state}}=64 , internal Δtssm=0.01\Delta t_{\text{ssm}}=0.01
LinOSSinternal Δtlinoss=1.0\Delta t_{\text{linoss}}=1.0 (absorbed into learned eigenvalues)
D-LinOSSinternal state dim =⌊d/2⌋=\lfloor d/2\rfloor , learnable per-oscillator timestep
Transformernheads=4n_{\text{heads}}=4 , FF dim =4d=4d , dropout=0.1
VPTnff_sublayers=2n_{\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 q0:t↦q^t+1q_{0:t}\mapsto\hat{q}_{t+1} with shared global settings hidden dimension d=64d=64 and depth 22 (unless otherwise noted).
BenchmarkΔt\Delta tInitial conditionsDamping
Pendulum (cons)0.05θ0∼Unif[−π,π]\theta_{0}\sim\mathrm{Unif}[-\pi,\pi] , p0∼N(0,32)p_{0}\sim\mathcal{N}(0,3^{2})none
Pendulum (damped)0.05θ0∼Unif[−π,π]\theta_{0}\sim\mathrm{Unif}[-\pi,\pi] , p0∼N(0,32)p_{0}\sim\mathcal{N}(0,3^{2})constant ( γ=0.5\gamma{=}0.5 )
Pendulum (windy)0.05θ0∼Unif[−π,π]\theta_{0}\sim\mathrm{Unif}[-\pi,\pi] , p0∼N(0,42)p_{0}\sim\mathcal{N}(0,4^{2})d(θ)=d0+Δd∣sin⁡θ∣d(\theta)=d_{0}+\Delta d|\sin\theta| ( d0=0.3d_{0}{=}0.3 , Δd=0.5\Delta d{=}0.5 )
Cart-Pole (windy)0.02(x0,θ0)∼Unif([−1,1]×[−π,π])(x_{0},\theta_{0})\sim\mathrm{Unif}([-1,1]\times[-\pi,\pi]) , p0∼Unif[−2,2]2p_{0}\sim\mathrm{Unif}[-2,2]^{2}windy on θ\theta ( d0=0.3d_{0}{=}0.3 , Δd=0.5\Delta d{=}0.5 )
Oscillator (cons)0.02q0∼N(0,I)q_{0}\sim\mathcal{N}(0,I) , p0∼N(0,I)p_{0}\sim\mathcal{N}(0,I)none
Oscillator (damped)0.02q0∼N(0,I)q_{0}\sim\mathcal{N}(0,I) , p0∼N(0,I)p_{0}\sim\mathcal{N}(0,I)constant ( γ=0.1\gamma{=}0.1 )
Appendix
Table 15: Simulator initial conditions for the thirteen q-only benchmarks.
Paper metricImplementation key(s)
One-step wrapped-angle MSE ( WrapMSE\mathrm{WrapMSE} )theta_wrap_mse
One-step Euclidean MSE ( MSE\mathrm{MSE} )mse
Rollout wrapped-angle MSE at horizon HH ( WrapMSEθroll(H)\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H) )rollout_theta_wrap_mse_h{H}
Rollout Euclidean MSE at horizon HH ( MSEroll(H)\mathrm{MSE}^{\mathrm{roll}}(H) )rollout_mse_h{H}
Rollout mixed-manifold MSE at horizon HH ( MixedMSEroll(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.
Figure 21: Q-only computation graph and diagnostic probes. Training is teacher-forced (ground-truth qq 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.
ModelParamsConservativeDampedWindy
PHAST (ours)
PHAST (KNOWN)3,3640.738±0.1430.738\pm 0.1430.017±0.0050.017\pm 0.0050.106±0.0200.106\pm 0.020
PHAST (PARTIAL)13,7360.680±0.0430.680\pm 0.0430.018±0.0010.018\pm 0.0010.092±0.0140.092\pm 0.014
PHAST (UNKNOWN)13,7380.993±0.1280.993\pm 0.1280.275±0.0830.275\pm 0.0830.298±0.0480.298\pm 0.048
Baselines
GRU37,8892.939±0.3022.939\pm 0.3021.310±0.7431.310\pm 0.7431.796±0.6251.796\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=100H{=}100 over 5 seeds. Parameter counts correspond to the Windy setting (largest; position-dependent damping introduces additional parameters). Lower is better.
ModelParamsDouble (cons)Double (damped)
PHAST (ours)
PHAST (KNOWN)3,5880.421±0.0610.421\pm 0.0610.332±0.0300.332\pm 0.030
PHAST (PARTIAL)12,1660.402±0.0470.402\pm 0.0470.320±0.0320.320\pm 0.032
PHAST (UNKNOWN)13,0670.629±0.0600.629\pm 0.0600.529±0.0470.529\pm 0.047
Baselines
GRU38,1461.300±0.1431.300\pm 0.1431.346±0.1271.346\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=100H{=}100 over 5 seeds.
ModelParamsOscillator (cons)Oscillator (damped)
PHAST (ours)
PHAST (KNOWN)3,5870.0010±0.00030.0010\pm 0.00030.0012±0.00040.0012\pm 0.0004
PHAST (PARTIAL)12,1650.0010±0.00020.0010\pm 0.00020.0011±0.00030.0011\pm 0.0003
PHAST (UNKNOWN)12,1710.0072±0.00310.0072\pm 0.00310.0101±0.00460.0101\pm 0.0046
Baselines
GRU38,1461.8470±0.07541.8470\pm 0.07541.4565±0.08111.4565\pm 0.0811
Appendix
Table 19: Q-only open-loop forecasting (Oscillator environments). Mean ±\pm std of rollout MSE at horizon H=100H{=}100 over 5 seeds.
ModelParamsRLC (damped)
PHAST (ours)
PHAST (KNOWN)3,3642.64×10−5±1.5×10−82.64\times 10^{-5}\pm 1.5\times 10^{-8}
PHAST (PARTIAL)11,8782.64×10−5±1.1×10−82.64\times 10^{-5}\pm 1.1\times 10^{-8}
PHAST (UNKNOWN)11,8792.63×10−5±3.1×10−82.63\times 10^{-5}\pm 3.1\times 10^{-8}
Baselines
S517,0891.84×10−3±6.69×10−41.84\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=1L{=}1 , C=1C{=}1 , R=0.5R{=}0.5 ) is the electrical analog of a damped spring–mass system. Lower is better.
ModelParamsLJ-3 (damped)
PHAST (ours)
PHAST (KNOWN)4,4904.61×10−10±4.20×10−124.61\times 10^{-10}\pm 4.20\times 10^{-12}
PHAST (PARTIAL)13,3244.59×10−10±2.04×10−124.59\times 10^{-10}\pm 2.04\times 10^{-12}
PHAST (UNKNOWN)13,3556.93×10−10±1.38×10−116.93\times 10^{-10}\pm 1.38\times 10^{-11}
Baselines
S517,7342.05×10−4±4.33×10−52.05\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 ( ϵ=1\epsilon{=}1 , σ=1\sigma{=}1 ) and Langevin friction ( γ=0.1\gamma{=}0.1 ). Lower is better.
ModelParamsHeat Exch. (damped)
PHAST (ours)
PHAST (KNOWN)3,5922.42×10−6±9.74×10−102.42\times 10^{-6}\pm 9.74\times 10^{-10}
PHAST (PARTIAL)12,1702.42×10−6±2.63×10−92.42\times 10^{-6}\pm 2.63\times 10^{-9}
PHAST (UNKNOWN)12,1732.48×10−6±1.85×10−82.48\times 10^{-6}\pm 1.85\times 10^{-8}
Baselines
S517,2181.00×10−3±1.62×10−41.00\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 ( c1=c2=1c_{1}{=}c_{2}{=}1 , coupling κ=0.5\kappa{=}0.5 , loss κloss=0.1\kappa_{\mathrm{loss}}{=}0.1 ). Lower is better.
ModelParamsN-body 3 (damped)
PHAST (ours)
PHAST (KNOWN)4,4894.28×10−8±1.28×10−114.28\times 10^{-8}\pm 1.28\times 10^{-11}
PHAST (PARTIAL)13,3234.27×10−8±2.25×10−114.27\times 10^{-8}\pm 2.25\times 10^{-11}
PHAST (UNKNOWN)13,3554.28×10−8±6.17×10−114.28\times 10^{-8}\pm 6.17\times 10^{-11}
Baselines
S517,7343.03×10−3±5.98×10−43.03\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=1G{=}1 , ϵsoft=0.1\epsilon_{\mathrm{soft}}{=}0.1 ) and drag ( γ=0.05\gamma{=}0.05 ). Gravitational potential template is given; masses and damping are learned. Lower is better.
ModelParamsPredator–Prey (damped)
PHAST (ours)
PHAST (KNOWN)3,3640.0203±0.00020.0203\pm 0.0002
PHAST (PARTIAL)11,8780.0203±0.00020.0203\pm 0.0002
PHAST (UNKNOWN)11,8790.0199±0.00040.0199\pm 0.0004
Baselines
S517,0896.12±0.586.12\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 ( α=1\alpha{=}1 , β=0.1\beta{=}0.1 , γ=0.4\gamma{=}0.4 , δ=0.1\delta{=}0.1 , K=100K{=}100 , μ=0.01\mu{=}0.01 ). All PHAST components are neural (non-canonical Hamiltonian structure). Lower is better.
ModelParamsWrapMSEθ\mathrm{WrapMSE}_{\theta} ↓\downarrowWrapMSEθroll(100)\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(100) ↓\downarrowRD2R^{2}_{D} ↑\uparrowMAED\mathrm{MAE}_{D} ↓\downarrow
PHAST (PARTIAL)10,3761.49×10−5±2.86×10−61.49{\times}10^{-5}\pm 2.86{\times}10^{-6}0.089±0.0050.089\pm 0.0050.798±0.0340.798\pm 0.0340.045±0.0050.045\pm 0.005
PHAST (UNKNOWN)10,3786.35×10−5±1.47×10−56.35{\times}10^{-5}\pm 1.47{\times}10^{-5}0.128±0.0030.128\pm 0.0030.583±0.0820.583\pm 0.0820.081±0.0100.081\pm 0.010
DHNN107,1240.186±0.0610.186\pm 0.0612.264±0.1602.264\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)D(q) , so RD2R^{2}_{D} / MAED\mathrm{MAE}_{D} are not reported.
ModelParamsMixedMSEroll(H=100)\mathrm{MixedMSE}^{\mathrm{roll}}(H{=}100) ↓\downarrowEbudResroll(H=100)\mathrm{EbudRes}^{\mathrm{roll}}(H{=}100) ↓\downarrowPassViolroll(H=100)\mathrm{PassViol}^{\mathrm{roll}}(H{=}100) ↓\downarrowRD2R^{2}_{D} ↑\uparrowMAED\mathrm{MAE}_{D} ↓\downarrow
PHAST (KNOWN, constant MM )3,5890.096±0.0160.096\pm 0.0163.31±0.393.31\pm 0.390.124±0.0070.124\pm 0.0070.956±0.0040.956\pm 0.0040.028±0.0010.028\pm 0.001
PHAST (KNOWN, M(q)M(q) )3,5890.031±0.0060.031\pm 0.0061.30±0.061.30\pm 0.060.016±0.0070.016\pm 0.0070.986±0.0020.986\pm 0.0020.013±0.0010.013\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.
ModelParamsWrapMSEθroll(H=100)\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100) ↓\downarrowEbudResroll(H=100)\mathrm{EbudRes}^{\mathrm{roll}}(H{=}100) ↓\downarrowPassViolroll(H=100)\mathrm{PassViol}^{\mathrm{roll}}(H{=}100) ↓\downarrow
PHAST (KNOWN, constant MM )3,5890.367±0.0830.367\pm 0.08332.71±4.4032.71\pm 4.400.408±0.0300.408\pm 0.030
PHAST (KNOWN, M(q)M(q) )3,5890.284±0.0290.284\pm 0.02917.16±1.0017.16\pm 1.000.343±0.0260.343\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.
EnvironmentMetricΔt\Delta tL=1L{=}1L=4L{=}4
Windy Cart-PoleMixedMSEroll(H=100)\mathrm{MixedMSE}^{\mathrm{roll}}(H{=}100)0.050.134±0.0200.134\pm 0.0200.126±0.0160.126\pm 0.016
Double PendulumWrapMSEθroll(H=100)\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100)0.042.412±0.0712.412\pm 0.0712.366±0.0882.366\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 δts=Δt/L\delta t_{s}=\Delta t/L . Metrics are rollout error at horizon H=100H{=}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 nn with fixed rank r=2r{=}2 .
SettingWrapMSEθroll(H=100)\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100) ↓\downarrowRD2R^{2}_{D} ↑\uparrowMAED\mathrm{MAE}_{D} ↓\downarrowEbudRes(H=100)\mathrm{EbudRes}(H{=}100) ↓\downarrowPassViolroll(H=100)\mathrm{PassViol}^{\mathrm{roll}}(H{=}100) ↓\downarrow
fixed Δtmodel\Delta t_{\mathrm{model}} , fixed d0d_{0}0.175±0.0090.175\pm 0.0090.465±0.0380.465\pm 0.0380.084±0.0030.084\pm 0.0031.34±0.041.34\pm 0.040.020±0.0040.020\pm 0.004
learnable Δtmodel\Delta t_{\mathrm{model}} , fixed d0d_{0}0.153±0.0130.153\pm 0.0130.472±0.0400.472\pm 0.0400.085±0.0030.085\pm 0.0031.28±0.041.28\pm 0.040.015±0.0040.015\pm 0.004
fixed Δtmodel\Delta t_{\mathrm{model}} , learned d0d_{0}0.175±0.0090.175\pm 0.0090.476±0.0470.476\pm 0.0470.082±0.0040.082\pm 0.0041.34±0.041.34\pm 0.040.020±0.0040.020\pm 0.004
learnable Δtmodel\Delta t_{\mathrm{model}} , learned d0d_{0}0.153±0.0130.153\pm 0.0130.478±0.0470.478\pm 0.0470.084±0.0040.084\pm 0.0041.28±0.041.28\pm 0.040.015±0.0040.015\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 ∑iβi(q)≤βˉ\sum_{i}\beta_{i}(q)\leq\bar{\beta} ( βˉ=0.5\bar{\beta}=0.5 ) and report horizon- H=100H{=}100 diagnostics under the standard q-only rollout protocol (Sec. 4.1.1 ).
Model (scaled)ParamsWrapMSEθroll(H=100)\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100) ↓\downarrow
GRU ( d=128d{=}128 )149,5052.130±0.5662.130\pm 0.566
S5 ( d=128d{=}128 )34,0490.654±0.0150.654\pm 0.015
Transformer ( d=128d{=}128 )396,9290.868±0.2370.868\pm 0.237
Appendix
Table 30: Baseline capacity scaling (Windy Pendulum, q-only). Mean ±\pm std of wrapped-angle rollout MSE at horizon H=100H{=}100 over 3 seeds (dataset seed fixed to 42) on CPU. Increasing baseline hidden dimension does not close the gap to PHAST (Table 17 ).
Figure 23: Low-rank PSD damping with spectral bound. Top: D(q)=d0I+∑i=1rβi(q) ki(q)ki(q)⊤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 ∑i=1rβi(q)≤βˉ\sum_{i=1}^{r}\beta_{i}(q)\leq\bar{\beta} certifies λmax⁡(D(q))≤d0+βˉ\lambda_{\max}(D(q))\leq d_{0}+\bar{\beta} (Eq. ( 11 )).
Cap on ∑i=1rβi(q)\sum_{i=1}^{r}\beta_{i}(q)WrapMSEθroll(H=100)\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100) ↓\downarrowRD2R^{2}_{D} ↑\uparrowMAED\mathrm{MAE}_{D} ↓\downarrowEbudRes(H=100)\mathrm{EbudRes}(H{=}100) ↓\downarrowPassViolroll(H=100)\mathrm{PassViol}^{\mathrm{roll}}(H{=}100) ↓\downarrow
Unbounded0.162±0.0150.162\pm 0.015−40.940±8.630-40.940\pm 8.6300.897±0.0890.897\pm 0.0891.43±0.041.43\pm 0.040.022±0.0060.022\pm 0.006
∑i=1rβi(q)≤0.25\sum_{i=1}^{r}\beta_{i}(q)\leq 0.250.180±0.0120.180\pm 0.012−0.080±0.081-0.080\pm 0.0810.105±0.0040.105\pm 0.0041.32±0.031.32\pm 0.030.019±0.0040.019\pm 0.004
∑i=1rβi(q)≤0.5\sum_{i=1}^{r}\beta_{i}(q)\leq 0.50.175±0.0090.175\pm 0.0090.465±0.0380.465\pm 0.0380.084±0.0030.084\pm 0.0031.34±0.041.34\pm 0.040.020±0.0040.020\pm 0.004
∑i=1rβi(q)≤1.0\sum_{i=1}^{r}\beta_{i}(q)\leq 1.00.175±0.0140.175\pm 0.014−1.506±0.582-1.506\pm 0.5820.211±0.0260.211\pm 0.0261.38±0.051.38\pm 0.050.022±0.0040.022\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 Δt=0.05\Delta t=0.05 , fixed Δtmodel=Δt\Delta t_{\mathrm{model}}=\Delta t , fixed d0d_{0} , and vary only the total-strength cap ∑iβi(q)≤βˉ\sum_{i}\beta_{i}(q)\leq\bar{\beta} . We report horizon- H=100H{=}100 diagnostics under the standard q-only rollout protocol (Sec. 4.1.1 ).
Cap on ∑i=1rβi(q)\sum_{i=1}^{r}\beta_{i}(q)WrapMSEθroll(H=100)\mathrm{WrapMSE}_{\theta}^{\mathrm{roll}}(H{=}100) ↓\downarrowRD2R^{2}_{D} ↑\uparrowMAED\mathrm{MAE}_{D} ↓\downarrowEbudRes(H=100)\mathrm{EbudRes}(H{=}100) ↓\downarrow
Unbounded0.298±0.0540.298\pm 0.054−96.546±17.774-96.546\pm 17.7741.343±0.1291.343\pm 0.1292.581±0.2142.581\pm 0.214
∑i=1rβi(q)≤0.5\sum_{i=1}^{r}\beta_{i}(q)\leq 0.50.416±0.0710.416\pm 0.071−0.234±0.268-0.234\pm 0.2680.134±0.0190.134\pm 0.0192.713±0.2072.713\pm 0.207
∑i=1rβi(q)≤1.0\sum_{i=1}^{r}\beta_{i}(q)\leq 1.00.396±0.0760.396\pm 0.076−6.457±1.376-6.457\pm 1.3760.363±0.0340.363\pm 0.0342.666±0.2142.666\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 ∑i=1rβi(q)\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.
SetupMassFrobErrM\mathrm{FrobErr}_{M} ↓\downarrowMAE∇V\mathrm{MAE}_{\nabla V} ↓\downarrowConclusion
Learn MM (known VV ) + Lres\mathcal{L}_{\mathrm{res}}SPD0.0110.011—Identifiable
Learn MM (known VV ) + Lres\mathcal{L}_{\mathrm{res}}NeuralMass0.3060.306—Identifiable (high var.)
Learn VV (known MM ) + Lres\mathcal{L}_{\mathrm{res}}Known MM—0.0940.094Identifiable
Learn both ( MM + neural VV ) + Lres\mathcal{L}_{\mathrm{res}}SPD2.372.376.936.93Non-identifiable
Learn both ( MM + neural VV ) + Lres\mathcal{L}_{\mathrm{res}}NeuralMass2.232.236.736.73Non-identifiable
Learn MM + structured VV + Lres\mathcal{L}_{\mathrm{res}}SPD0.0350.0350.0820.082Identifiable
Appendix
Table 33: Gauge-freedom ablation (conservative double pendulum). Mean over 2 runs (50 epochs). Lower is better. “Structured VV ” means the functional form is fixed but coefficients are learned (analogous to PHAST PARTIAL). Errors are averaged over held-out configurations: FrobErrM:=Eq∥M(q)−M^(q)∥F\mathrm{FrobErr}_{M}:=\mathbb{E}_{q}\|M(q)-\hat{M}(q)\|_{F} and MAE∇V:=1nEq∥∇V(q)−∇V^(q)∥1\mathrm{MAE}_{\nabla V}:=\frac{1}{n}\mathbb{E}_{q}\|\nabla V(q)-\nabla\hat{V}(q)\|_{1} .
IndexRangeMeaning
bb1,…,B1,\ldots,BTrajectory index (batching)
tt0,…,T−10,\ldots,T-1Time index (transitions use t=0,…,T−2t{=}0,\ldots,T{-}2 )
hh0,…,H−10,\ldots,H-1Rollout step index (horizon HH )
ss1,…,L1,\ldots,LPHAST substep index
ii1,…,r1,\ldots,rRank-1 term index in low-rank updates
jj1,…,n1,\ldots,nCoordinate index
Appendix
Table 34: Index ranges.
SymbolSpace / shapeMeaning
Problem setting & knowledge regimes Sec. 1
nnN\mathbb{N}Degrees of freedom
Q\mathcal{Q}manifoldConfiguration space (e.g., Rne×(S1)na\mathbb{R}^{n_{e}}\times(\mathbb{S}^{1})^{n_{a}} )
qqQ\mathcal{Q}Generalized configuration (forecast target)
ppRn\mathbb{R}^{n}Conjugate momentum (latent in q-only setting)
x=(q,p)x=(q,p)Q×Rn\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.
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