Learning coarse-step dynamics and internal mechanical response with graph networks

Vinay Sharma; Olga Fink

cs.LGSep 24, 2026

Learning coarse-step dynamics and internal mechanical response with graph networks

Authors: Vinay Sharma, Olga Fink

Organizations: Intelligent Maintenance and Operations Systems, EPFL, Lausanne, Switzerland

Abstract

Modern sensing records the motion of physical systems, but often leaves the forces and mechanical response governing that motion unobserved. Inferring these quantities from discretely sampled trajectories is especially difficult at coarse time scales, when mechanical response evolves between observations and interactions propagate across the system. Here we introduce Newmark-\b{eta}-DGN, a graph neural network-based framework that combines two structures inspired by computational mechanics. First, a semi-implicit update inspired by the Newmark-\b{eta} method uses learned momentum fluxes and matrix-valued response operators to advance the state over each observed interval. Second, an operator-weighted virtual hub provides system-wide coupling through a sparse set of connections. The learned quantities thus determine the predicted motion and remain accessible for mechanical analysis. Across a deformable beam, human motion and protein dynamics, Newmark-\b{eta}-DGN supports long-horizon prediction at time steps for which explicit learned simulators deteriorate. Without force, moment or constitutive relation supervision, forces inferred from walking kinematics track independently derived hip and knee joint moments, while response operators learned on the beam recover the relative spatial and directional structure of its finite-element stiffness tangent. Newmark-\b{eta}-DGN therefore links coarse-step prediction to the inference of mechanical quantities that were never observed during training.

Figures & tables

Figure 1: Observation-learned mechanics within a coarse graph update. a , Over a coarse observation interval, the loaded tip of a clamped rod must respond to the distant constraint within the same transition. b , Condensation of an implicit mechanical system produces dense coupling. Its rank-one approximation is represented by an operator-weighted virtual hub, giving a graph of diameter two with

\mathcal{O}(N)

edges. c , Each message-passing round is one semi-implicit Newmark substep: edge quantities are decoded and aggregated, the state is advanced and the resulting state enters the next round. d , Physical edges exchange non-central linear- and angular-momentum fluxes antisymmetrically, whereas virtual hub edges satisfy collective zero-sum balance. e , Positive-definite learned response operators enter the local translational and rotational Newmark systems. f , The quantities generating the transition remain accessible as internal-force, torque and response-operator readouts.

Figure 2: Newmark-

\beta

-DGN remains bounded over the clamped-beam rollout. Rollout position error against step, as a percentage of the beam length

L

, averaged over the 12 held-out test configurations. a , Tip-position error over the full rollout range (left) and zoomed to

\leq 10\%

(right). b , Whole-body error, the root-mean-square error over all body nodes, over the full range (left) and zoomed to

\leq 3\%

(right). Newmark-

\beta

-DGN (red) stays bounded and non-monotone, tracking the beam’s oscillation, whereas every baseline grows or diverges; a cross marks the first non-finite value. Line colour denotes architecture and line style the number of message-passing steps or the cluster count. Curves are means over the 12 test configurations at a single training seed (seed 42); no across-seed dispersion is shown.

Figure 3: Beam rollout under load-duration extrapolation. a , Reference and predicted configurations at four instants of free vibration for a held-out case whose load is removed at $4\,$ unit-time, against cut-off times of $2$ to $3\,$ units in training. b , Components of the tip displacement over the same trajectory: the transverse response follows the phase and decaying envelope of the reference while a small axial drift accumulates. Single seed ( $n=1$ , seed 42); no dispersion is shown.

Figure 4: Whole-body rollout error on the nine extrapolation cases. Each case extrapolates outside the training range in load, cut-off time, cross-section, length or mesh resolution. Bars show the whole-body error averaged over the 95-step rollout (

0

–

9.5\,\text{time-units}

), expressed as a percentage of the beam length

L

; the y-axis is logarithmic. Newmark-

\beta

-DGN uses four Newmark sub-steps, DGN and MGN twelve message-passing steps, and IGNS twenty-four internal steps, six times Newmark-

\beta

-DGN ’s budget. Single seed (

n=1

, seed 42);

Figure 5: Modal structure recovered from the rollout. a , Natural-frequency spectrum of the tip transverse displacement for the finite-element reference (black), Newmark-

\beta

-DGN (red) and DGN (blue dashed), with the fundamental frequencies annotated and the true fundamental marked. b – d , The first three proper-orthogonal mode shapes, finite element against Newmark-

\beta

-DGN , each labelled with its energy fraction and Modal Assurance Criterion. Newmark-

\beta

-DGN matches the finite-element fundamental and dominant mode; DGN peaks at a spurious lower frequency. No frequency or mode is supervised. Single seed (

n=1

, seed 42).

Figure 6: Motion-capture rollout accuracy and rotation robustness. a , Position rollout MSE on the unrotated test trajectories. Each group represents one model, and its five bars correspond to successive rollout steps of 30 frames each. DGN (global reference frame) uses Newmark-

\beta

-DGN ’s global reference frame in DGN ’s external-force channel; both are rotation-equivariant. b , Position rollout MSE after fixed rotations of the test trajectories about the vertical axis. The single Newmark-

\beta

-DGN group applies at every angle because its prediction is rotation-equivariant. Both panels are capped at

5

; triangles mark values beyond this limit, with the true value printed above. A step is marked as non-finite and omitted if any seed diverges. Bars show means over seeds

\{0,42,100\}

and error bars show one standard deviation (

n=3

Figure 7: Autoregressive rollout on the adenylate-kinase closed–open transition. Position mean squared error over five autoregressive steps of

15

frames each, for Newmark-

\beta

-DGN and the three baselines from their minimum-validation checkpoints. Newmark-

\beta

-DGN stays bounded across the horizon, while DGN diverges on the hub-free covalent graph and EGHN and EGNO operating on dense contact-distance based graphs diverge: the ordinate is capped at

1.2\,

Å 2 , triangles mark values above the cap with the true value printed, and crosses mark a non-finite (diverged) step. One seed (

n=1

, seed

42

). The full unclipped table and the mean-trajectory floor (the average-displacement baseline) are given in the Supplementary Information (Section 6.4).

Figure 8: Structural rollout of the closed–open transition. Newmark- $\beta$ -DGN ’s predicted backbone, coloured by domain (rigid core grey, NMP arm orange, LID blue), overlaid on the ground-truth conformation (light grey) at the closed start (frame $+0$ ), the transition midpoint ( $+45$ ) and the open end ( $+90$ ) of a held-out trajectory, superposed on the core. The two mobile domains swing away from the core through a collective motion of about $6\,$ Å, tracked with a root-mean-square C $\alpha$ position error of $1.07\,$ Å at the midpoint and $0.78\,$ Å at the open state.

Figure 9: Joint-moment demand inferred from marker kinematics alone. a , Representative autoregressive rollout of the marker graph for subject p2 for 8 steps, showing the recorded and predicted configurations. The joint-moment readout is evaluated over the five-step rollout. b , Normalised root-mean-square sagittal knee moment inferred from the decoded internal forces using Eq. ( 1 ), compared with the inverse-dynamics reference. Constant step length and constant step frequency are withheld gait strategies. Preferred walking at

2.0\,\mathrm{m\,s^{-1}}

is a withheld speed within a familiar strategy. The shaded step-frequency sweep at

1.25\,\mathrm{m\,s^{-1}}

contains training and validation conditions. Demand is normalised to the preferred

1.25\,\mathrm{m\,s^{-1}}

trial at unit subject mass and is therefore a ratio rather than a calibrated torque. Error bars show variation across deterministic evaluation windows for one trained model and one subject (

n=1

), not variation across training seeds. Neither joint moments nor ground-reaction forces enter the training objective.

Figure 10: Edge scaling and computational cost. a , Directed edges per graph for the beam mesh (left) and the protein backbone (right). Newmark-

\beta

-DGN replaces a distance-based contact graph with

2N

hub edges, which reduces the protein edge count by a factor of

16.3

. On the fixed beam mesh, no model uses a distance cutoff and no corresponding reduction arises. b , Peak memory against wall-clock time per training iteration, using a batch size of four and latent widths of

128

for the protein and

64

for the beam. Two protein configurations are exceptions: EGNO and DGN with global edges require

9622\,\mathrm{MB}

and

10\,048\,\mathrm{MB}

, respectively, at width

128

and therefore do not fit on the

11\,\mathrm{GB}

device; their plotted points are measured at width

64

. Five forward-and-backward iterations are timed after one discarded warm-up iteration, with each model profiled in a separate process pinned to a single device and retaining its published edge topology. Single measurement per configuration.

Setting	Value
Optimiser	Adam
MLP layers per block	2
Newmark parameters $\gamma,\beta$	$\tfrac{1}{2},\tfrac{1}{4}$ (fixed)
Newmark substeps $S$	4
Message-passing rounds per prediction span	4 (one per substep)
Response operators $\mathbf{K}_{ij},\mathbf{D}_{ij},\mathbf{K}^{\mathrm{rot}}_{ij},\mathbf{D}^{\mathrm{rot}}_{ij}$	per-edge SPD matrices, Cholesky parameterisation

Supplementary Table 1: Settings shared by Newmark-

\beta

-DGN across the four benchmarks.

Model	Head	Equivariant vector inputs	Invariant scalar inputs
EGHN	EquivariantScalarNet or EGMN	$[\mathbf{l}_{nf},\mathbf{x}-\mathbf{l}_{X},\mathbf{v}-\mathbf{l}_{V},\mathbf{nf}]$	$\mathrm{cat}(h,\mathbf{l}_{H})$
EGHNO	EquivariantScalarNet or EGMN	$[\mathbf{l}_{nf},\mathbf{x}_{0r}-\mathbf{l}_{X},\mathbf{v}_{0r}-\mathbf{l}_{V},\mathbf{nf}]$	$\mathrm{cat}(h_{\mathrm{low}},\mathbf{l}_{H})$
EGNO	EquivariantScalarNet	$[\mathbf{v}_{\mathrm{egnn}},\mathbf{x}_{\mathrm{out}}-\mathbf{x}_{\mathrm{init}}]$	$h_{\mathrm{out}}$

Supplementary Table 2: Velocity heads added to the position-only baselines. Each head uses the same network class, hidden width and activation as the corresponding position decoder.

\mathbf{l}_{nf}

and

\mathbf{nf}

are the pooled and unpooled equivariant vectors of the hierarchical block;

\mathbf{l}_{X}

and

\mathbf{l}_{V}

are the pooled coordinate and velocity;

h

and

\mathbf{l}_{H}

are the low- and high-level invariant features; and

h_{\mathrm{out}}

is the hidden state after temporal convolution.

Property	Value
Nodes	mesh vertices
Physical edges	mesh connectivity
Hub edges	$2N$ virtual edges to one hub
Node features	position, velocity, free/clamped type, applied load
Edge features	scalar attribute ( $-1$ on hub edges)
Edge-local frame	yes ( Dynami-CAL GraphNet )

Supplementary Table 3: Beam representation supplied to Newmark-

\beta

-DGN .

N

is the number of mesh vertices and depends on the configuration. Baselines receive the same observed physical state on the hub-free graph.

Model	latent $nf$	capacity parameter	batch	learning rate	weight decay
Newmark- $\beta$ -DGN	64	$S=4$ sub-steps	32	$5\times 10^{-4}$	$10^{-10}$
Dynami-CAL GraphNet	64	$8$ and $12$ messages	32	$5\times 10^{-4}$	$10^{-10}$
MGN	64	$4$ , $8$ , $12$ messages	64	$5\times 10^{-4}$	$10^{-10}$
EGHN	64	$15$ clusters	32	$5\times 10^{-4}$	$10^{-10}$
EGHNO	64	$5$ clusters	32	$5\times 10^{-4}$	$10^{-10}$
EGNO	64	$12$ layers	32	$5\times 10^{-4}$	$10^{-10}$

Supplementary Table 4: Beam elastodynamics: hyperparameters. Capacity parameters have architecture-specific meanings, as explained in Section 2.2 .

Supplementary Figure 1: Training and validation loss for Newmark- $\beta$ -DGN on the beam. Both terms of Equation ( 1 ) are shown. Early stopping selects the checkpoint at the minimum of the validation curve. Seed 42.

Panel of Figure 16 — Supplementary Figure 2: Rollout of Newmark- $\beta$ -DGN on four held-out configurations. Top left, the refined mesh ( $\texttt{res}=8$ ), the only extrapolation that leaves the training envelope of the stiffness spectral radius. Top right, the beam extrapolated to three times the training length, where the rank-one hub begins to be an inadequate surrogate for the long-range coupling. Bottom, the two cross-sections outside the training product, $W=1.5$ , $D=0.4$ on the left and $W=0.4$ , $D=1.5$ on the right, which exchange the bending stiffness of the two transverse axes. The remaining five configurations behave comparably and are not reproduced here.

Configuration	Sub-steps	Whole-body error (%)	factor
Dynami-CAL GraphNet (hub off, explicit)	$12$	$3.00$	—
$+$ operator-weighted hub (explicit, $\mathbf{A}_{i}=\mathbf{I}$ )	$4$	$1.26$	$2.4\times$
$+$ semi-implicit solve ( $\beta=0$ )	$4$	$0.70$	$1.8\times$
$+$ stiffness term $\beta$ ( Newmark- $\beta$ -DGN )	$4$	$0.54$	$1.3\times$

Supplementary Table 5: Whole-body error decomposition (mean over

21

configurations, % of

L

). The hub is the dominant ingredient; the semi-implicit solve and the stiffness term

\beta

are refinements. Cells are cumulative: each row adds one ingredient to the row above. The first hub row also reduces the sub-step count from

12

4

, so its factor bundles the two changes. Seed 42.

Configuration	Model error coarse reference	Model error converged reference	Reference discretisation error
$L1.0\ W0.5\ D1.0$ res4 (in-distribution)	$0.14$	$0.19$	$0.09$
$L1.5\ W1.0\ D1.0$ res4 (in-distribution)	$0.08$	$0.24$	$0.19$
$L2.0\ W0.5\ D0.5$ res4 (in-distribution)	$0.70$	$0.70$	$0.49$
$L1.75\ T_{c}4$ res4 (extrapolation)	$0.38$	$0.51$	$0.38$
$L1.75\ W1.5\ D0.4$ res4 (cross-section)	$0.21$	$0.26$	$0.14$
$L1.5\ W0.75\ D0.75$ res8	$0.52$	$0.52$	$0.35$

Supplementary Table 6: Whole-body error against a temporally converged reference (

\delta t=10^{-3}\,

, mean over 95 steps, % of

L

). The first two error columns compare Newmark-

\beta

-DGN with the indicated reference; the final column reports the coarse-reference discretisation error. Seed 42.

Quantity	training mesh (res4)	refined mesh (res8)
$\omega_{\max}$	$442$	$876$
$\Delta t\,\omega_{\max}$ (outer step)	$44$	$88$
multiple of explicit limit	$22\times$	$44\times$
symplectic-Euler amplification / sub-step	$109\times$	$455\times$
Newmark amplification / step	$1.0$	$1.0$

Supplementary Table 7: Spectral stability of the beam operator.

\omega_{\max}

is the largest natural frequency of the linearised system; the explicit limit is

\Delta t\,\omega\leq 2

. The Newmark amplification is unity at every step size; the symplectic-Euler amplification is per sub-step (

S=4

Internal steps	compliant res4	long beams	refined res8
$1$ (native)	$10^{6}$ – $10^{7}$	$10^{6}$	$10^{7}$
$4$ ( $=$ Newmark- $\beta$ -DGN )	$11$ – $50$	$14$ – $20$	$78$ – $123$
$12$	$5$ – $15$	$61$ – $78$	$100$ – $122$
$24$ ( $6\times$ )	$1$ – $7$	$23$ – $25$	$231$ – $338$
Newmark- $\beta$ -DGN ( $S=4$ )	$<1$	$\sim 2$	$0.5$ – $1.5$

Supplementary Table 8: IGNS rollout error under an increasing internal-step budget (whole-body, final step, % of

L

). Four internal steps matches the sub-step budget of Newmark-

\beta

-DGN ; the refined-mesh error does not fall as the budget grows. Newmark-

\beta

-DGN (four sub-steps) is shown for reference.

Supplementary Figure 3: Fundamental frequency and dominant mode recovered from the rollout, on three held-out configurations (top to bottom: a load-amplitude, a load-duration and a cross-section extrapolation, identified in each title). Left, the natural-frequency spectrum of the tip transverse displacement, finite element (black), Newmark-

\beta

-DGN (red), Dynami-CAL GraphNet (blue dashed), with the fundamental frequencies annotated; right, the dominant proper-orthogonal mode shape, finite element against Newmark-

\beta

-DGN , with its MAC. Newmark-

\beta

-DGN matches the finite-element fundamental and mode; Dynami-CAL GraphNet peaks at a spurious lower frequency. No frequency or mode is supervised.

Configuration	Extrapolation axis	FE $\to$ Newmark- $\beta$ -DGN (Hz)	MAC	Modal structure recovered
$L1.0\,W0.5\,D1.0\,F2.0\,T_{c}3.0$	in-distribution	$2.29\to 2.29$	$1.00$	yes
$L1.75\,W0.75\,D0.75\,F3.0\,T_{c}2.5$	load amplitude	$1.25\to 1.25$	$1.00$	yes
$L1.75\,W0.75\,D0.75\,F2.0\,T_{c}1.5$	load duration	$1.25\to 1.25$	$1.00$	yes
$L1.75\,W0.75\,D0.75\,F2.0\,T_{c}4.0$	load duration	$1.25\to 1.25$	$1.00$	yes
$L1.75\,W0.4\,D1.5\,F2.0\,T_{c}2.5$	cross-section	$1.15\to 1.15$	$1.00$	yes
$L1.75\,W1.5\,D0.4\,F2.0\,T_{c}2.5$	cross-section	$1.67\to 1.56$	$1.00$	mode; frequency within one bin

Supplementary Table 9: Natural frequency and dominant mode recovered from the rollout, every configuration. FE is the finite-element fundamental; the Newmark-

\beta

-DGN value is read from its rollout tip spectrum; MAC compares the dominant Newmark-

\beta

-DGN mode shape with the finite-element mode. A frequency counts as recovered when it falls in the same spectral bin (

\approx 0.1\,

Hz at

\texttt{res}=4

). Seed 42.

Supplementary Figure 4: The learned response operators recover part of the finite-element tangent structure, but not its anisotropy or magnitude. Agreement of the decoded per-node position-response operator

\mathbf{K}_{i}

(blue) and velocity-response operator

\mathbf{D}_{i}

(red) with the corresponding finite-element matrices under three scale-separated measures: direction is the Frobenius cosine between the operators; anisotropy is the cosine between their trace-free (deviatoric) parts; spatial pattern is the correlation of their traces across nodes. Each dot is one of three held-out configurations at one of three trajectory stages (loaded, free vibration, rest); bars are the mean and error bars one standard deviation. The operators align in direction (cosine

\approx 0.85

for stiffness,

0.92

for damping) and reproduce where the material is stiff or soft (trace correlation

\approx 0.63

for stiffness,

0.94

for damping), but do not recover the directional anisotropy (deviatoric cosine

\approx 0

) and are off in absolute scale by

2

–

9\times

for stiffness and roughly

10^{3}\times

for damping. They are therefore operational operators, usable as relative readouts, rather than identified material tangents.

Property	Value
Nodes	31 markers
Physical edges	60 skeletal connections
Hub edges	62 virtual edges to one hub (122 directed in total)
Node features	position, finite-difference velocity
Edge features	scalar attribute ( $-1$ on hub edges)
Edge-local frame	yes ( Dynami-CAL GraphNet )

Supplementary Table 10: Human-motion representation supplied to Newmark-

\beta

-DGN . Baseline-specific features are described in Section 5.2 .

Model	Latent $n_{f}$	Capacity parameter	Batch	Learning rate	Weight decay
Newmark- $\beta$ -DGN	64	$S=4$ substeps	12	$4\times 10^{-4}$	$10^{-6}$
Dynami-CAL GraphNet	64	$S=4$ substeps	12	$4\times 10^{-4}$	$10^{-6}$
Dynami-CAL GraphNet ( global reference frame )	64	$S=4$ substeps	12	$4\times 10^{-4}$	$10^{-6}$
GNS	64	$S=4$ substeps	12	$4\times 10^{-4}$	$10^{-6}$
EGHN	64	5 clusters	12	$4\times 10^{-4}$	$10^{-10}$
EGHNO	64	5 clusters	12	$4\times 10^{-4}$	$10^{-6}$

Supplementary Table 11: Human motion-capture hyperparameters. Capacity parameters have architecture-specific meanings, as explained in Section 2.2 .

Method	1 step	2 steps	3 steps	4 steps	5 steps
Newmark- $\beta$ -DGN (implicit)	$0.0999\pm 0.0069$	$0.2491\pm 0.0153$	$0.4773\pm 0.0246$	$0.8288\pm 0.0282$	$1.3349\pm 0.0195$
GNS	$0.0788\pm 0.0041$	$0.1639\pm 0.0079$	$0.2735\pm 0.0059$	$0.3785\pm 0.0079$	$0.5006\pm 0.0324$
Dynami-CAL GraphNet (global reference frame)	$0.1448\pm 0.0036$	$0.4210\pm 0.0306$	$0.8622\pm 0.0672$	$1.5319\pm 0.1439$	$2.4480\pm 0.3098$
Dynami-CAL GraphNet	$0.2346\pm 0.0086$	$0.7960\pm 0.0288$	$1.7010\pm 0.0459$	$3.2218\pm 0.1868$	$4.7374\pm 0.3951$
EGHN	$0.0872\pm 0.0088$	$0.2577\pm 0.0087$	$0.6128\pm 0.0535$	$1.5396\pm 0.3390$	$4.0918\pm 1.5819$
EGHNO	$0.1208\pm 0.0328$	$0.4931\pm 0.1694$	$1.3513\pm 0.5260$	$2.8605\pm 0.8386$	$5.83\times 10^{6}$

Supplementary Table 12: Per-step rollout error on the human-walk benchmark. Unscaled position mean squared error, mean

\pm

standard deviation over seeds

\{0,42,100\}

n=3

, on the canonical split. Values are not clipped. Every model stays finite across the five-step rollout on this split; EGHNO reaches a large finite value at step five, driven by seed 0 (

1.7\times 10^{7}

) while its other seeds remain near

10

Rotation	1 step	2 steps	3 steps	4 steps	5 steps
$0^{\circ}$	$0.0788\pm 0.0041$	$0.1639\pm 0.0079$	$0.2735\pm 0.0059$	$0.3785\pm 0.0079$	$0.5006\pm 0.0324$
$5^{\circ}$	$0.1566\pm 0.0142$	$0.5243\pm 0.0393$	$1.0978\pm 0.0591$	$1.7665\pm 0.0663$	$2.5097\pm 0.0941$
$15^{\circ}$	$0.8958\pm 0.0655$	$3.2939\pm 0.1031$	$7.3419\pm 0.2866$	$12.3016\pm 0.3948$	$17.8291\pm 0.4667$
$30^{\circ}$	$3.0708\pm 0.0849$	$10.6594\pm 0.1988$	$23.7211\pm 1.4052$	$42.2121\pm 2.0511$	$64.2818\pm 1.8082$

Supplementary Table 13: Degradation of GNS under a rotation of the test set. Per-step rollout error, mean

\pm

standard deviation over seeds

\{0,42,100\}

n=3

. The

0^{\circ}

row reproduces the GNS row of Table 12 . Newmark-

\beta

-DGN is rotation-equivariant by construction and its error is unchanged at every angle, so it is not tabulated per angle.

Property	Value
Nodes	855 backbone atoms
Physical edges	1708 covalent backbone bonds
Hub edges	1710 virtual edges to one hub (3418 directed in total)
Node features	position, causal backward-difference velocity, atom-type one-hot
Edge features	scalar attribute ( $-1$ on hub edges)
Edge-local frame	yes ( Dynami-CAL GraphNet )

Supplementary Table 14: Protein representation supplied to Newmark-

\beta

-DGN . The input and contact graph used by EGHN and EGNO are described in the text.

Model	latent $nf$	capacity parameter	batch	learning rate	weight decay
Newmark- $\beta$ -DGN	64	$S=4$ sub-steps	8	$5\times 10^{-4}$	$10^{-8}$
Dynami-CAL GraphNet	64	$5$ messages	8	$5\times 10^{-4}$	$10^{-8}$
EGHN	128	$20$ clusters	8	$5\times 10^{-4}$	$10^{-8}$
EGNO	128	$4$ layers	1	$5\times 10^{-4}$	$10^{-8}$

Supplementary Table 15: Protein dynamics: hyperparameters. Newmark-

\beta

-DGN and Dynami-CAL GraphNet are trained at

nf=64

, as on every other system; EGHN and EGNO retain their published width of

128

Method	1 step	2 steps	3 steps	4 steps	5 steps
Newmark- $\beta$ -DGN	$0.221$	$0.328$	$0.346$	$0.296$	$0.277$
Dynami-CAL GraphNet	$0.386$	$1.164$	$2.559$	$4.499$	$6.122$
EGHN	$0.214$	$0.366$	$0.679$	$361.4$	NaN
EGNO	$0.228$	$0.417$	$1.018$	$11.03$	NaN
mean-trajectory	$0.231$	$0.341$	$0.369$	$0.330$	$0.307$

Supplementary Table 16: Per-step rollout error on the transition ensemble. Unscaled position mean squared error (Å 2 ). One step is

15

frames, evaluated on the

30

held-out test paths (

241

five-step windows,

855

atoms) from each model’s minimum-validation checkpoint. NaN entries mark a diverged rollout.

Supplementary Figure 5: The single-step benchmark is bounded by a data floor. (a) One backbone atom’s position at every recorded frame, relative to its mean; its root-mean-square fluctuation is $1.3\,\text{\AA}$ . (b) Single-step position error for assume-no-motion, the published and reproduced learned models, and the one-coefficient restoring rule ( $\Delta\mathbf{x}=-0.60(\mathbf{x}-\bar{\mathbf{x}})$ , labelled the spring rule), against the lowest error any predictor of the current state can reach ( $1.692$ , dashed). Every learned model lies above the restoring rule and the floor. Lower is better.

Property	Value
Nodes	37 markers
Physical edges	63 anatomical connections
Hub edges	$2N=74$ virtual edges to one hub
Node features	position, causal backward-difference velocity
Edge features	scalar attribute ( $-1$ on hub edges)
Edge-local frame	yes ( Dynami-CAL GraphNet )

Supplementary Table 17: Graph representation of the biomechanics system.

Model	latent $nf$	capacity parameter	batch	learning rate	weight decay
Newmark- $\beta$ -DGN	64	$S=4$ sub-steps	128	$5\times 10^{-4}$	$10^{-10}$

Supplementary Table 18: Biomechanics: hyperparameters.

Supplementary Figure 6: Inferred demand at all three joints across the cadence sweep. Hip, knee and ankle, measured against inferred, for the conditions of the main-text figure. The hip and knee track the measurement; the ankle does not, for the reason given above. Shaded groups lie within the training and validation data.

Supplementary Figure 7: In-distribution control. The same three joints evaluated on conditions seen during training. Agreement at the knee (

r=0.81

) is as good on these training conditions as on the withheld ones (

r=0.88

). Readout accuracy is therefore limited by the moment model, not by distribution shift, and the inferred profiles are not artefacts of fitting.

Ablation	Effect	Established in
Remove the operator-weighted hub
(& 12 substeps instead of 4)	beam error $0.54\%\to 3.00\%$	Table 5
Set the implicit stiffness term $\beta=0$	beam error $0.54\%\to 0.70\%$	Table 5
Per-node update made explicit ( $\mathbf{A}_{i}=\mathbf{I}$ )	beam error $0.54\%\to 1.26\%$	Table 5, Section 10.7
Remove the angular-momentum channel	rollout tied; inferred joint moments degrade	Section 8.1

Supplementary Table 19: Ablations of Newmark-

\beta

-DGN ’s components. Each row removes or replaces one ingredient of the update. The first four are established in the sections cited; the last is treated in Section 8.1. Beam errors are whole-body, mean over the reported configurations, seed 42.

Supplementary Figure 8: The angular-momentum channel is needed for the inferred joint moments, not for the rollout. Inferred against measured hip, knee and ankle demand across the cadence sweep, for the full model (left) and the variant with the angular-momentum channel removed (right). Measured demand is black and grey; inferred is red. With the channel the inferred hip and knee demand follows the measured trend; without it the inferred demand flattens toward unity and loses the trend. The moment construction

\mathbf{M}=-\sum_{i}\mathbf{r}_{i}\times\mathbf{F}_{i}

is identical in both, so the difference is in the decoded force. The ankle is not recovered by either model, for the reason given in Section 7.4 .

	Configuration	$\delta t=0.2$	$\delta t=0.0125$	Observed $p$
A	Original nodal update	$4.24\times 10^{-2}$	$4.10\times 10^{-3}$	$0.84$
B	Common coefficient matrix	$3.23\times 10^{-2}$	$2.21\times 10^{-3}$	$0.97$
D	Known-velocity response term suppressed	$2.83\times 10^{-2}$	$2.97\times 10^{-3}$	$0.81$
C	Both controlled modifications	$1.70\times 10^{-8}$	$1.67\times 10^{-8}$	$\mathrm{n/a}$

Supplementary Table 20: Linear-momentum residual of the independent nodal solve. A common coefficient matrix removes the first term in Equation ( 88 ). Suppressing the known-velocity position-response term removes the second.

Physics-Informed GNNs

Physics-informed graph neural networks encode equations, conservation laws, and boundary conditions into graph models for scientific prediction and simulation.

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Momentum-Conserving Graph Neural Networks for Deformable Objects
Apr 28, 2026Jiahong Wang, Logan Numerow, Stelian Coros +3Graph Neural Networks Physics-Informed ML
Max Planck Institute for Informatics · ETH Zurich · University of Bonn +1
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Oct 16, 2026George Tsialiamanis, Keith WordenNonlinear System Identification Time Series Modeling
Dynamics Research Group, School of Mechanical, Aerospace, and Civil Engineering, University of Sheffield, Sheffield, Mappin St, Sheffield, United Kingdom
Learning Equivariant Neural-Augmented Object Dynamics From Few Interactions
May 4, 2026Sergio Orozco, Tushar Kusnur, Brandon May +2Deformable Object Manipulation Robotic Manipulation
Brown University, Providence, RI · Robotics and AI (RAI) Institute, Cambridge, MA · General Motors

Apr 28, 2026J/K move · Enter open · S save

Momentum-Conserving Graph Neural Networks for Deformable Objects

Jiahong Wang, Logan Numerow, Stelian Coros, Christian Theobalt+2

Max Planck Institute for Informatics · ETH Zurich · University of Bonn+1

Graph Neural Networks Physics-Informed ML Physics-Based Simulation Deformable Object Simulation

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Graph neural networks (GNNs) have emerged as a versatile and efficient option for modeling the dynamic behavior of deformable materials. While GNNs generalize readily to arbitrary shapes, mesh topologies, and material parameters, existing architectures struggle to correctly predict the temporal evolution of key physical quantities such as linear and angular momentum. In this work, we propose MomentumGNN -- a novel architecture designed to accurately track momentum by construction. Unlike existing GNNs that output unconstrained nodal accelerations, our model predicts per-edge stretching and bending impulses which guarantee the preservation of linear and angular momentum. We train our network in an unsupervised fashion using a physics-based loss, and we show that our method outperforms baselines in a number of common scenarios where momentum plays a pivotal role.

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Physics-Informed GNNs

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