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

Vinay Sharma, Olga Fink

arXiv:2609.30344v1cs.LG

TL;DR

Coarse observations make it difficult to infer mechanical response that evolves between observations while interactions propagate across a system. Newmark-β-DGN combines semi-implicit updates with an operator-weighted virtual hub, maintaining coarse-step prediction while learned quantities track independent mechanical references.

  • Problem

    Coarse trajectories often leave forces and mechanical response unobserved, while response can evolve and influence can propagate across the system between observations.

    Source: §1 · p. 1

  • Method

    Newmark-β-DGN combines semi-implicit nodal updates with an operator-weighted virtual hub to approximate system-wide coupling while learning mechanical quantities from kinematics.

    Source: §1 · p. 3, §3 · p. 20

  • Results

    Across coarse-step benchmarks, Newmark-β-DGN maintains bounded rollouts; walking-derived moments track reference hip and knee moments (r = 0.94 and r = 0.87).

    Source: §3 · pp. 20–21

  • Takeaways & Limitations

    The learned forces and response operators remain mechanically interpretable and can be evaluated against independent references despite receiving no direct supervision.

    Source: §3 · p. 21

    On the refined beam mesh, learned operators over-damp by roughly 2.5× and mis-predict frequency by −27%.

    Source: §4.7 · p. 39

Abstract

from arXiv · show

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.

1 Introduction

Newmark-β-DGN targets coarse-step prediction by combining semi-implicit nodal updates with an operator-weighted virtual hub for system-wide coupling. It learns mechanically interpretable quantities from motion without direct supervision of internal forces, moments, or response operators.

Motivation

  • At coarse temporal resolution, response evolves between observations while disturbances can span more of the system, making both stable finite-time integration and system-wide coupling necessary.Motion alone may not identify the underlying mechanics, and accurate trajectory prediction does not establish that a model recovered them.Source: §1 · p. 1

Method

  • The method pairs independent semi-implicit nodal updates with a rank-one-motivated virtual hub that approximates dense implicit coupling without a global matrix solve.The hub is an operator-weighted shared state, connected to all physical nodes through O(N) virtual edges; the augmented graph has diameter two.Source: §1 · pp. 3–4
  • Each observed transition is advanced through recurrent semi-implicit substeps, allowing mechanical response and system-wide coupling to evolve within the interval.At each substep, edge-decoded fluxes and response operators are aggregated into local translational and rotational Newmark updates; positive-definite operators make the nodal systems well posed.Source: §1 · pp. 4, 6

Training and interpretation

  • Training uses observed kinematics and available boundary or load inputs, not internal-force, moment, or response-operator labels; those learned quantities remain accessible as mechanical readouts.The operators are interpreted as relative mechanical indicators, not uniquely identified material or constitutive properties.Source: §1 · p. 6

Results

  • Across beam, human-motion, and protein benchmarks, Newmark-β-DGN maintains bounded coarse-step rollouts; its staged human-motion ablation reaches 0.54% mean whole-body error.The ablation progresses from 3.00% for twelve-substep hub-free explicit DGN to 1.26% with the hub, 0.70% with semi-implicit response, and 0.54% with the full update.Source: §1 · p. 6
  • Joint-moment estimates from walking kinematics track reference hip and knee moments (r = 0.94 and r = 0.87), while beam operators recover the spatial and directional organization of tangent stiffness.The learned beam dynamics also reproduce the fundamental frequency and dominant vibration mode.Source: §1 · p. 6

2 Results

The Results section evaluates Newmark-β-DGN across beam, human-motion, protein, and walking-biomechanics settings, including prediction, rollout, and rotation robustness. It also examines inferred mechanical readouts against reference measurements, momentum-balance errors, and the virtual hub’s effect on protein connectivity cost.

2.1 Overview of experiments

  • The experiments span four systems and use controlled autoregressive comparisons, with models trained on identical trajectories and evaluated over full rollouts using errors across all body nodes.The systems test coarse-step stability, spatial generalization, sparse irregular-graph prediction, and recovery of mechanical quantities; comparisons include six graph-based approaches and beam-specific IGNS.Source: §2.1 · pp. 7–8

2.2 Stable beam elastodynamics under extrapolation

  • Newmark-β-DGN stayed below ~1% beam-length error over 95 test steps and achieved 0.5%–1.2% refined-mesh error, versus 15%–16% DGN, 54%–69% MGN, and 126%–149% IGNS.Across nine held-out beam configurations, mean whole-body rollout error ranged from 0.21% to 3.45%; DGN had lower mean error only on the two longest beams. Figure 2 shows the bounded, non-monotone test rollout, while Figure 4 reports extrapolation errors.Source: Figure 2 · p. 8, Figure 4 · p. 9, §2.2 · pp. 11–12
  • The staged beam ablation reduced mean whole-body error from 3.00% for hub-free DGN to 0.54% for full Newmark-β-DGN, whose rollout also recovered the finite-element fundamental frequency and dominant mode.The dominant mode had a Modal Assurance Criterion of 1.00; explicit DGN instead peaked at a spurious lower frequency. No frequency or mode was supervised.Source: Figure 5 · p. 10, §2.2 · p. 12

2.3 Human motion prediction benchmark

  • On unrotated motion trajectories, Newmark-β-DGN achieved the lowest step-five MSE among equivariant models at 1.335, and its error remained unchanged under rotation.GNS achieved 0.5006 unrotated, but its step-five error rose to 2.5097 after a 5° rotation and to 35.6 and 128 times its unrotated value at 15° and 30°, respectively.Source: Figure 6 · p. 13, §2.3 · p. 14

2.4 Protein dynamics prediction benchmark

  • Protein rollout MSE for Newmark-β-DGN rose from 0.221 to a maximum of 0.346 and settled near 0.28 Å2, while DGN climbed to 6.12 and EGHN and EGNO diverged.The hub used 1,710 virtual edges versus roughly 55,610 directed edges for contact-graph baselines. On a six-step visualization rollout, Cα position error was 1.07 Å at the transition midpoint and 0.78 Å at the open state; the mean-displacement baseline was within 4–10% of Newmark-β-DGN, so the benchmark chiefly distinguishes bounded rollout and graph sparsity rather than a large accuracy margin.Source: §2.4 · pp. 14–15, Figure 7 · p. 15, Figure 8 · p. 16

2.5 Internal mechanics inferred from observations

  • The walking benchmark uses kinematics alone for training and compares inferred joint moments with independently measured references from 33 trials comprising 18,631 marker frames.The subject p2 graphs contain 37 marker nodes and 63 anatomical edges, plus a virtual hub; measured joint moments and ground-reaction forces are withheld from training.Source: §2.5 · p. 16
  • Across five evaluation experiments, inferred hip and knee demands correlate with inverse-dynamics measurements at r = 0.943 and r = 0.869, respectively.The knee readout follows measured trends under withheld gait strategies and reproduces a cadence-related minimum in the training/validation sweep; neither moments nor ground-reaction forces enter the objective.Source: Figure 9 · p. 17, §2.5 · pp. 17–18
  • At the withheld 2.0 m s−1 preferred-walking condition, right-knee demand differs from the reference by 5%, versus a 21% under-prediction at 1.8 m s−1 under withheld constant step frequency.The ankle profile is not recovered: pooled correlation is r = 0.492, its amplitude is about one order of magnitude below reference, and one profile is anticorrelated. Removing the angular-momentum channel leaves rollout accuracy essentially unchanged but degrades joint-moment estimates.Source: §2.5 · p. 18
  • Beam response operators match finite-element matrices in dominant direction (cosine similarities approximately 0.85 and 0.92) and nodal-trace organization (correlations approximately 0.63 and 0.94), but not anisotropy or scale.The position-response operator is nearly orthogonal to the decoded-force Jacobian, yet predicts momentum-response changes to velocity perturbations with cosine similarity 0.99 to 1.00; these are relative readouts, not identified material tangents.Source: §2.5 · p. 18

Computational cost.

  • On protein, Newmark-β-DGN uses 3,418 directed edges versus approximately 55,610 for distance-based baselines (16.3 times as many), and 1,067 MB peak memory versus EGHN’s 3,843 MB.The hub’s benefit is specific to dense connectivity: human-skeleton edge budgets are similar (122 versus 130), it does not reduce beam edges, and MGN and EGHN are cheaper per beam training iteration.Source: Figure 10 · p. 19, Computational cost · p. 20

2.6 Conservation, scaling and computational cost

  • The complete update’s synthetic-test relative momentum residual has observed order 0.84, while removing both residual contributions reduces it to approximately 1.7 × 10−8.For bounded masses, states, and response operators, absolute one-substep residual is O(δt^2) and its ratio to non-zero external impulse is O(δt); the residual vanishes in the explicit-response limit. Physical edges balance pairwise and the hub guarantees collective linear balance, but not exact total angular-momentum balance.Source: §2.6 · pp. 18–20

3 Discussion

Newmark-β-DGN combines semi-implicit finite-time response with system-wide coupling, supporting coarse-step prediction and access to learned mechanical readouts. Across beam, human-motion, and protein settings, the paper reports predictive and mechanical results while identifying limits on hub effectiveness, conservation, and absolute mechanical scale.

3 Discussion

  • The framework combines a semi-implicit update inspired by average-acceleration Newmark with an operator-weighted virtual hub for system-wide coupling, making learned mechanical quantities part of the predicted state update.The update uses momentum fluxes, forces, learned inverse mass and inertia, and matrix-valued response operators.Source: §3 · p. 20
  • The ablation reduces mean whole-body error from 3.00% with the operator-weighted hub and explicit update, to 1.26% with the hub, 0.70% with the semi-implicit solve, and 0.54% with the full formulation.The beam results also report low, bounded error and recovery of fundamental vibration frequency and dominant mode without direct supervision.Source: §3 · pp. 20–21
  • Newmark-β-DGN retains rotation-equivariant human-motion prediction, while its hub uses 16.3 times fewer edges than baseline distance-based graphs on the 855-node protein system.The edge saving applies when long-range interactions would otherwise require a dense contact graph, not on a fixed mesh.Source: §3 · p. 21
  • Walking-kinematics-only force estimates track independently derived hip and knee joint-moment trends (r = 0.94 and r = 0.87), while beam operators recover relative response structure but not anisotropy or absolute scale.Removing the angular-momentum channel leaves rollout accuracy essentially unchanged but degrades inferred joint moments.Source: §3 · p. 21
  • A single rank-one hub becomes less effective for the longest beams; independent nodal solves also do not guarantee exact finite-step momentum conservation.The paper identifies higher-rank or multiple hubs and more strongly coupled updates as possible extensions; observed loads or limited calibration could anchor absolute mechanical scales.Source: §3 · p. 21
  • The authors propose that accessible inferred forces and response operators could support virtual sensing for condition monitoring and what-if evaluation under changed loads, configurations, or operating conditions.The stated scope includes demonstrated generalization across unseen loads, operating conditions, and configurations.Source: §3 · p. 21

4 Methods

The Methods section describes Newmark-β-DGN’s learned momentum-flux and response-operator updates, virtual-hub coupling, and training on kinematic transitions. It also specifies the benchmark datasets, autoregressive evaluation protocol, baseline comparisons, and rollout adaptations.

4.1 Graph representation and model inputs

  • The model represents system components and physical connections as a directed graph, augments it with a virtual hub, and carries node positions, velocities, and latent spin.Velocities are obtained by causal finite differencing when not observed; the hub is not an independent degree of freedom.Source: §4.1 · pp. 21–22

4.2 Model update overview

  • Each observation interval is divided into four substeps, each using message passing to decode fluxes, response operators, and node quantities before updating physical states.The hub is recomputed from physical-node states at each substep, and the final positions and velocities define the next-time prediction.Source: §4.2 · p. 22

4.3 Equivariant interaction encoding and processing

  • Equivariant edge frames and, when forces are unobserved, a graph-level global frame encode vector features as invariant scalars for node and edge processing.The edge processor retains recurrent edge-level latent memory across substeps; the global frame is constructed from the physical graph’s first two non-trivial Laplacian eigenvectors.Source: §4.3 · pp. 22–23

4.4.1 Edge-wise Interaction Fluxes; 4.4.2 Edge-wise Response Operators

  • Edge decoders produce linear- and angular-momentum fluxes balanced pairwise on physical connections and collectively zero-sum projected on virtual edges, alongside matrix-valued response operators.The response operators are symmetric positive definite, rotation-equivariant learned update coefficients, not force derivatives or identified material stiffness and damping matrices.Source: §4.4.1 · pp. 23–24, §4.4.2 · p. 24

4.4.3 Node-wise inverse mass, inverse inertia, and external force

  • Node decoders provide positive inverse mass and inverse inertia, and decode external force in a global frame only when force is unobserved; otherwise observed force enters the update directly.The isotropic inverse-mass and inverse-inertia forms preserve rotation equivariance.Source: §4.4.3 · p. 25

4.5 Operator-weighted virtual hub

  • The hub’s position, velocity, and spin are weighted by nodal response operators aggregated from incident edges; identity initialization makes its first-substep state an arithmetic average.The hub carries no independently propagated state, and positive-definite operators make the weighting inverses well-defined.Source: §4.5 · p. 25

4.6 Semi-implicit nodal Newmark update

  • Each substep solves independent 3 × 3 translational and rotational systems locally, holding decoded drives and response operators fixed; finite-step linear-momentum residuals are O(δt^2).The update is semi-implicit because unknown state increments enter the mechanical response; it uses no inner Newton iteration or assembled 3N × 3N system.Source: §4.6 · p. 26

4.7 Training and mechanical readouts; 4.8 Datasets and evaluation protocol

  • Training uses observed kinematic transitions without force, moment, or constitutive supervision, while decoded interactions and response operators remain accessible as mechanical readouts.Walking moments are compared at evaluation with inverse-dynamics references withheld from training; beam trajectories are generated with FEniCS using average-acceleration Newmark parameters γ = 1/2 and β = 1/4. Response operators indicate relative coarse-step response, not identified material tangents.Source: §4.7 · p. 27, §4.8 · p. 27

4.8 Datasets and evaluation protocol

  • The benchmarks use CMU subject 35 walking, adenylate kinase transitions, and treadmill recordings, with autoregressive evaluation and benchmark-specific metrics.Predicted states feed subsequent intervals; errors are evaluated without clipping, and walking joint-moment demand is compared with a withheld inverse-dynamics reference.Source: §4.8 · p. 27

4.9 Baseline implementations

  • Baselines share trajectories, partitions, and observed states, while retaining stated architectures and objectives except where rollout adaptations are required.DGN is the closest controlled comparison; velocity heads are added for EGHN, EGHNO, and EGNO, while IGNS is beam-only and loses stress supervision.Source: §4.9 · pp. 27–28

Supplementary Information Learning coarse-step dynamics and internal mechanical response

The supplementary contents list analyses of residual asymptotic order and the conservative explicit-response limit, as well as the scope of the conservation statements.

11.3 Asymptotic Order of the Residual and Conservative Explicit-Response Limit

11.5 Scope of the Conservation Statements

1 Proposed Model: Training and Shared Configuration

Newmark-β-DGN uses a virtual hub and semi-implicit substeps to predict normalized position and velocity increments, training only on trajectory targets. Shared settings use Adam, early stopping, and a four-substep configuration.

1 Proposed Model: Training and Shared Configuration

  • The shared-configuration section specifies the objective and implementation settings used across four benchmarks, with dataset and case-specific details given in the corresponding sections.The shared settings are listed in Supplementary Table 1.Source: §1 · p. 32, Table 1 · p. 32

1.1 Model state and update

  • Each system is represented by physical nodes and edges plus a virtual hub; S semi-implicit substeps decode momentum exchanges and positive-definite response operators for the Newmark update.The operator-weighted hub supplies shared graph state without constructing a dense inverse, and the model learns its mechanical quantities rather than receiving prescribed mass, damping, or stiffness matrices.Source: §1.1 · p. 32

1.2 Learning objective

  • Training predicts standardized position and velocity increments over one prediction span, using physical nodes in the loss and no force, moment, stress, or ground-reaction-force labels.Response operators and the angular-velocity channel are not directly supervised.Source: §1.2 · p. 32

1.3 Optimisation and shared model settings

  • All four implementations use Adam and early stopping, retaining the checkpoint with the lowest validation loss; shared settings include four Newmark substeps and four message-passing rounds.Table 1 lists shared settings, while case tables report differing quantities.Source: §1.3 · p. 32, Table 1 · p. 32

2 Baseline Implementations and Adaptations

The section specifies baseline adaptations needed for a shared autoregressive rollout while otherwise retaining the published architectures and objectives. Position-only baselines receive separate velocity heads so each model supplies the rollout state.

2 Baseline Implementations and Adaptations

  • Baseline objectives are retained except where adaptation is required for the common kinematic task, and capacity counts retain architecture-specific meanings.For example, capacity denotes Newmark substeps, processor depth, temporal or graph layers, or clusters, depending on the architecture.Source: §2 · p. 33, §2.2 · pp. 33–34

2.1 Velocity heads for autoregressive rollout

  • Velocity heads give position-only EGHN, EGNO and EGHNO models the velocities required for autoregressive rollout.The heads read representations already produced by each backbone; the position and velocity heads are trained jointly but do not share weights.Source: §2.1 · p. 33
  • EGHN and EGHNO heads use the same equivariant vectors and invariant scalars as their position heads, while EGNO also reads backbone velocity, displacement and hidden state.The heads follow their position decoders’ equivariant construction.Source: §2.1 · p. 33, Table 2 · p. 33
  • Velocity is decoded independently because it is defined at one-frame spacing, unlike a position increment spanning the prediction interval.Using their ratio would produce chord velocity rather than the one-frame velocity needed by the next rollout step.Source: §2.1 · p. 33

3 Data Processing and Evaluation Protocol

The protocol standardizes data handling, optimization and autoregressive evaluation within each benchmark, while leaving architectures and training objectives distinct. Comparisons use matching trajectories, partitions and observed physical state.

3.1 Data splits and preprocessing

  • Within each benchmark, models use the same trajectories and train, validation and test partitions, with normalization statistics computed from training data only.Inputs and partition construction are specified in each case section.Source: §3.1 · p. 34
  • Velocities are computed from recorded positions using a causal one-frame backward difference.The same position and velocity records are supplied to every model within a case.Source: §3.1 · p. 34

3.2 Optimisation and checkpoint selection

  • Models use Adam and retain the checkpoint with the lowest validation loss under early stopping.Learning rates, weight decay, batch sizes, patience and training budgets are case-specific.Source: §3.2 · p. 34

3.3 Autoregressive rollout

  • Rollouts feed each predicted state into the next call, and errors are evaluated at each rollout step without clipping.A non-finite value at a step is marked rather than averaged across remaining seeds; aggregate errors are reported only when defined for that case.Source: §3.3 · p. 34

3.4 Comparison protocol

  • Comparisons are controlled within benchmarks, not across them, while models retain their stated objectives and update rules.Case sections identify additional inputs, withheld labels, adaptations, seed sets, error definitions and reproduced versus published results.Source: §3.4 · p. 34, §3 · p. 34

4 Elastodynamics of a Clamped Beam

The beam experiments test whether Newmark-β-DGN can predict coarse-step elastodynamics, remain stable on stiff cases, and recover useful mechanical structure without mechanical labels. The results distinguish accurate trajectory and relative-response readouts from calibrated material identification.

4.1 Case description

  • The test system is a damped, loaded three-dimensional cantilever beam; held-out cases vary load, duration, cross-section, length or mesh resolution.The training set contains 108 configurations at res = 4, while nine configurations are held out entirely; inputs include positions, velocities, mesh, boundary mask and applied load, but no material properties or finite-element matrices.Source: §4.1 · pp. 34–35

4.2 Evaluated baselines; 4.3 Hyperparameters

  • The comparison includes six graph-based baselines, with architecture-specific adaptations and no finite-element mass, damping or stiffness matrices supplied.IGNS is trained without stress targets under the common one-step kinematic objective; MGN results use a corrected output normalization.Source: §4.2 · pp. 35–36, §4.3 · p. 36, Table 4 · p. 36

4.4 Additional results

  • Held-out rollout examples include a refined mesh beyond the training stiffness envelope, a beam three times the training length, and cross-sections that exchange transverse-axis bending stiffness.The rank-one hub becomes inadequate for long-range coupling on the three-times-length beam; the remaining five configurations behave comparably to those shown.Source: Figure 1 · p. 36, Figure 2 · p. 37

4.5 Whole-body error and the role of the semi-implicit solve

  • 1.26% mean whole-body error with the hub, 0.70% after the semi-implicit solve, and 0.54% with the stiffness term, versus 3.00% for Dynami-CAL GraphNet.On refined meshes, Newmark-β-DGN reaches 0.5%–1.2% error versus 15%–16% for Dynami-CAL GraphNet; the first hub comparison also reduces substeps from 12 to 4.Source: §4.5 · p. 37, Table 5 · p. 37

4.6 Accuracy against a temporally converged reference

  • 0.19%–2.06% whole-body error against the temporally converged reference is close to the 0.67% mean coarse-reference error.The reference’s coarse-step discretization error is only 0.09–0.58%, supporting accuracy to the physics rather than just a co-discretized reference.Source: §4.6 · p. 38, Table 6 · p. 38

4.7 Spectral radius and the IGNS baseline

  • At four substeps, IGNS fails on all configurations, while at 24 it reaches 231%–338% error on refined meshes versus 0.5%–1.5% for Newmark-β-DGN.The refined mesh raises the maximum natural frequency beyond the training range; explicit symplectic-Euler amplification there is 455 per substep, while average-acceleration Newmark amplification is unity.Source: §4.7 · pp. 38–39, Table 7 · p. 38, Table 8 · p. 39

4.8 Modal structure recovered from the rollout

  • The dominant mode is recovered on every configuration at MAC ≥0.97, while the fundamental frequency departs from the finite-element value only on length and mesh-refinement extrapolations.Dynami-CAL GraphNet peaks at a spurious 0.3–0.4 Hz on every configuration; the learned nonlinear rollout is not proven to inherit the integrator’s unit amplification.Source: §4.8 · pp. 39, 41, Figure 3 · p. 40, Table 9 · p. 41

4.9 Learned operators and the finite-element tangent

  • Learned operators align in direction and spatial pattern with finite-element matrices, but do not recover anisotropy or absolute scale.Stiffness and damping direction cosines are approximately 0.85 and 0.92, and trace correlations approximately 0.63 and 0.94; the operators predict momentum-response changes but are not constitutive tangents.Source: §4.9 · pp. 41–42

5 Human Motion Capture

The human-motion benchmark tests coarse-span prediction on walking sequences and evaluates rotation sensitivity. Newmark-β-DGN is the most accurate equivariant model, while GNS’s unrotated advantage does not persist under rotation.

5.1 Case description

  • The task predicts marker configuration 30 raw frames ahead from walking sequences, using 31 markers connected by 60 directed skeletal edges.Newmark-β-DGN adds 62 directed hub edges; the baseline graph has two-hop skeletal shortcuts and 130 directed edges.Source: §5.1 · p. 42, Table 10 · p. 43

5.2 Evaluated baselines; 5.3 Hyperparameters

  • The benchmark follows the reference split protocol and evaluates predictions after rotating test trajectories by 5°, 15° or 30° about the vertical walking axis.Models are compared with three seeds; EGHN, EGHNO and EGNO use added velocity heads, and the corrected rollout variant of EGHNO is used.Source: §5.1 · p. 43, §5.2 · p. 43, §5.3 · p. 43

5.4 Additional results

  • On the canonical split, all models remain finite across five rollout steps, but EGHN, EGHNO and EGNO show large unstable growth by the horizon.Their mean squared errors reach roughly four to six; one EGHNO seed drives a large finite step-five value.Source: §5.4 · p. 43, Table 12 · p. 44
  • Newmark-β-DGN is the most accurate equivariant model at every step, while GNS is more accurate unrotated but loses that advantage under rotation.At a 5° rotation, GNS is already less accurate than Newmark-β-DGN at the five-step horizon.Source: §5.4 · p. 44, Table 13 · p. 44

6 Protein Dynamics in Solvent

The adenylate-kinase transition benchmark tests bounded rollout over an enhanced-sampling collective pathway, while the equilibrium benchmark is excluded because its predictable component is bounded by a simple restoring rule. Newmark-β-DGN remains bounded on the transition, but its advantage over a mean-trajectory baseline is modest.

6.1 Case description

  • The 200-path enhanced-sampling ensemble follows adenylate kinase’s closed–open transition; one step is a fraction of that transition, not physical time.The evaluation predicts 15 frames ahead over five autoregressive steps, spanning 75 frames and most of the transition.Source: §6.1 · pp. 44–45

6.2 Evaluated baselines

  • The hub graph uses 1710 virtual edges, compared with approximately 55,610 directed contact edges for EGHN and EGNO.The comparison describes the cost of those contact graphs, not a measured performance advantage by itself.Source: §6.2 · p. 45

6.3 Hyperparameters

  • The protein models are trained with early stopping, a 15-frame prediction span, and seed 42; Newmark-β-DGN and Dynami-CAL GraphNet use width 64, while EGHN and EGNO retain width 128.The reference protocol’s widths are retained for EGHN and EGNO.Source: §6.3 · p. 46, Table 15 · p. 46

6.4 Additional results

  • 0.221 to a maximum of 0.346 Å^2: Newmark-β-DGN remains bounded over five rollout steps, while Dynami-CAL GraphNet reaches 6.12 and EGHN reaches 361 before EGNO becomes non-finite.The mean-trajectory baseline is only 4 to 10% worse than Newmark-β-DGN over this horizon.Source: §6.1 · p. 45, Table 16 · p. 46, §6.4 · p. 46

6.5 Why we do not use the equilibrium benchmark

  • The equilibrium trajectory is not used to rank dynamical models because its single-step error has a floor that no learned model beats.At the 3.6 ns horizon, only statistical relaxation survives; even shortening the horizon does not turn the benchmark into a dynamics test.Source: §6.5 · pp. 46–47
  • 1.7252 Å^2 for the one-coefficient restoring rule is already above the 1.6915 Å^2 floor, and every learned model lies above both.Supplementary Figure 5 compares the one-coefficient rule, learned models, and the lowest error any predictor of the current state can reach.Source: Figure 5 · p. 47, §6.5 · p. 47

7 Joint-Moment Inference in Human Walking

The walking study tests whether forces decoded from marker kinematics yield joint-moment profiles that agree with an independent inverse-dynamics reference. Agreement is strong at the hip and knee but weak at the ankle, with limitations in readout scope and gait asymmetry.

7.1 Case description

  • The model receives marker positions and causal velocities but no joint moments or ground-reaction forces; measured moments are reserved for evaluation.The evaluation uses one subject and a split by gait condition rather than time.Source: §7.1 · p. 48
  • The joint-moment readout sums decoded internal forces over markers distal to each joint, with no loss term constraining the spin channel.The readout uses 13 markers at the hip, seven at the knee, and two at the ankle.Source: §7.1 · p. 48

7.2 Evaluated baselines

  • No forward-simulator baseline is evaluated here; the purpose is comparison with an independent inverse-dynamics reference.The prediction span is 30 frames.Source: §7.2 · p. 49, §7.3 · p. 49

7.4 Additional results

  • Pearson correlations are 0.943 at the hip, 0.869 at the knee, and 0.492 at the ankle; normalized RMS errors are 0.171, 0.213, and 0.887, respectively.The ankle is not recovered and is anticorrelated with measurement in one condition (r = −0.322).Source: §7.4 · p. 49, Figure 6 · p. 50
  • Inferred gait imbalance remains 0.048 ± 0.017 while measured imbalance ranges from 0.006 to 0.170; error rises with true asymmetry (r = +0.76).Mean error reaches 17.3% in the condition with the greatest asymmetry.Source: §7.4 · p. 49
  • The readout omits measured ground-reaction contributions, particularly constraining the ankle where only two distal markers contribute.Source: §7.4 · p. 49

8 Ablations

Ablations distinguish the angular-momentum channel’s contribution to inferred mechanics from its effect on rollout accuracy. Removing it does not worsen rollout error, but the inferred joint-moment trend degrades; the proposed explanation remains a hypothesis.

8.1 The angular-momentum channel

  • The ablation removes spin state and angular flux while retaining linear momentum and translational response operators.The remaining update is unchanged.Source: §8 · p. 51, §8.1 · p. 52
  • Removing the angular-momentum channel leaves rollout accuracy unchanged or slightly better, but inferred joint demand flattens and loses the measured trend.The linear-only variant’s motion-capture rollout error is 6 to 13% lower across five steps, while the biomechanics errors are 0.477 versus 0.480 mm.Source: §8.1 · p. 52
  • The authors hypothesize that removing the channel’s antisymmetric, zero-sum angular-flux constraint explains the poorer force-based moment readout, but do not establish this mechanism.They have not measured either rollout’s angular-momentum residual.Source: §8.1 · p. 52

9 From implicit coupling to an operator-weighted hub

The operator-weighted hub approximates non-local coupling from an implicit mechanical solve without eliminating physical nodes or assembling a global Schur complement. The derivation links this hub to dense effective response and force transfer arising from a sparse system.

9 From implicit coupling to an operator-weighted hub

  • The hub retains a star topology and uses learned matrix-valued operators to approximate non-local coupling without eliminating physical nodes or assembling a global Schur complement.The derivation describes the star as arising from a rank-one approximation to dense effective interaction after static condensation.Source: §9 · p. 52

9.1 Implicit time stepping produces non-local coupling

  • With the angular-momentum channel, inferred hip and knee demand follows measured trends; without it, inferred demand flattens, while rollout accuracy is not the target of this ablation.The ankle is not recovered by either model.Source: Figure 8 · p. 53, §9.1 · p. 53
  • Although the assembled system is sparse, eliminating degrees of freedom generally creates dense effective coupling and dense force transfer; the implicit-response term contains inter-node coupling.Source: §9.1 · p. 53

9.2 Static condensation of the implicit update

Static condensation partitions a Newmark system into retained and internal degrees of freedom, then eliminates the internal increments algebraically. This thought experiment shows how local interactions can induce non-local coupling and transfer internal force drives to retained nodes.

9.2 Static condensation of the implicit update

  • Static condensation solves internal increments from the partitioned equations and substitutes them into the retained equations, modifying both the retained operator and drive.The internal-node effects remain represented after elimination; the thought experiment does not remove physical nodes in Newmark-β-DGN.Source: §9.2 · p. 54

9.2.1 Left-hand side: Schur-complement coupling and the rank-one virtual-hub topology

  • Eliminating internal increments produces an exact Schur complement that can couple retained nodes that were not neighbours in the original graph.The exact condensed operator can be dense even when the original interaction operator is sparse; the rank-one star is an approximation, not an assertion that the exact operator is rank one.Source: §9.2 · p. 54, §9.2.1 · pp. 55–56

9.2.2 Right-hand side: Transfer of the internal force drive

  • The condensed right-hand side transfers the internal force drive to retained nodes, potentially allowing a force at one internal node to affect many retained nodes.In the rank-one star derivation, the internal drive is distributed among retained nodes according to their hub couplings, preserving the complete internal drive.Source: §9.2.2 · pp. 56–57

9.3 Implementation of the Virtual Hub in Newmark-β-DGN: A Rank-One-Motivated Approximation of Schur-Complement-Induced Non-Local Coupling

The virtual hub implements a rank-one-motivated approximation of non-local coupling without explicitly condensing physical nodes or constructing the exact condensed system. Its operator-weighted state relays interactions, while balanced virtual-edge messages transfer internal drives.

9.3 Implementation of the Virtual Hub in Newmark-β-DGN: A Rank-One-Motivated Approximation of Schur-Complement-Induced Non-Local Coupling

  • Newmark-β-DGN implements the condensation-derived coupling and force-transfer ideas directly, without eliminating physical nodes or computing the exact condensed operator or drive.The implementation replaces scalar star couplings with learned stiffness and damping operators and uses virtual-edge force and angular-flux messages.Source: §9.3 · p. 57

9.3.1 Left-hand-side implementation: Operator-weighted hub coordinates

  • The hub state is an operator-weighted mechanical state, not a geometric centroid; it reduces to the arithmetic centroid only when all node stiffness operators are equal.Its position, velocity and spin are recomputed from equilibrium rather than advanced as an independent state.Source: §9.3.1 · pp. 57–58

9.3.2 Right-hand-side implementation: Balanced virtual-edge interactions

  • Projected virtual-edge force and angular-flux messages sum to zero, so the unloaded hub redistributes interactions without adding a net resultant.Zero net force does not alone establish an energy balance, which would also require a condition on total power.Source: §9.3.2 · p. 58

9.3.3 Computational cost

  • The hub adds 2N directed virtual edges and three 3 × 3 operator-weighted solves per substep, with additional cost linear in the number of physical nodes.The augmented graph has diameter two; physical-edge message passing and independent nodal solves remain unchanged.Source: §9.3.3 · p. 58

10 Virtual-Hub Coupling and the Semi-Implicit Nodal Newmark Update in Newmark-β-DGN

Newmark-β-DGN advances each observed interval through semi-implicit nodal updates informed by physical and virtual messages. Its Newmark-like form does not constitute a classical global implicit solve, and classical stability results do not establish stability of the trained rollout.

10.1 One learned mechanical substep

  • Each substep decodes forces and response operators from the current graph state, then advances each free physical node by an independent 3 × 3 solve.Virtual-edge operators enter the response on the left-hand side, while virtual-edge forces contribute to the drive on the right-hand side.Source: §10.1 · p. 59

10.2 Connection to average-acceleration Newmark integration and the meaning of semi-implicit

  • The update follows the average-acceleration Newmark coefficient structure, but learned response operators are not identified with physical stiffness or damping matrices.It performs one learned solve per substep without an inner Newton iteration; message passing and the hub provide non-local information instead of a global solve.Source: §10.2 · p. 60

10.3 Response-operator construction and nodal solvability

  • The edge and nodal response operators are rotation-equivariant, and positive-definite response construction keeps each individual nodal system invertible.This solvability guarantee does not establish stability of the complete learned rollout.Source: §10.3 · pp. 60–61

10.4 Classical stability correspondence and its scope

  • The classical undamped linear update is unconditionally stable and second-order accurate, but can accumulate phase error at large steps.These classical properties do not establish unconditional stability for Newmark-β-DGN, whose state-dependent learned updates are assessed empirically.Source: §10.4 · p. 61

10.5 Angular update

  • The angular state uses the same algebraic update, with learned inverse inertia, torque, and rotational response operators.The angular state is latent and receives no direct supervision.Source: §10.5 · p. 61

10.6 External forcing and constrained nodes

  • Beam loads are observed, whereas forcing for motion capture, protein dynamics, and walking is decoded from the predicted state without future forcing input.The nodal update is applied only to free physical nodes; clamped degrees of freedom are prescribed, and the hub is recomputed rather than integrated.Source: §10.6 · pp. 61–62

10.7 Explicit limit

  • In the explicit-response limit, decoded forces are integrated explicitly while retaining the trapezoidal position rule.Its momentum properties depend on whether contributing fluxes come from antisymmetric physical-edge exchanges or projected virtual-hub interactions.Source: §10.7 · p. 62

11 Momentum Balance of the Decoded Interactions and the Residual of the Independent Nodal Update

Decoded physical-edge and virtual-hub interactions are balanced before integration, but independent nodal response solves introduce a finite-substep linear-momentum residual. Under boundedness assumptions, that residual is second order in the substep size; the conservation claims are limited in scope.

11 Momentum Balance of the Decoded Interactions and the Residual of the Independent Nodal Update

  • The independent nodal update introduces a finite-substep linear-momentum residual after balanced decoded drives are processed by different nodal response operators.The residual is not an additional force from the decoder or hub; the analysis makes no energy-conservation claim.Source: §11 · p. 62

11.1.1 Pairwise Balance of Physical-Edge Interactions

  • Pairwise physical-edge forces have zero resultant, and each edge exchanges angular as well as linear momentum under the stated shared-application-point construction.The angular balance includes both orbital force contributions and intrinsic spin torque.Source: §11.1.1 · pp. 62–63

11 Momentum Balance of the Decoded Interactions and the Residual of the Independent Nodal Update

  • Physical edges exchange linear and angular momentum without creating either, while the virtual hub imposes collective balance across its incident edges.The hub's total moment about a common spatial origin additionally depends on learned application points, which have not been measured.Source: §11.1.1 · p. 63, §11.1.2 · p. 63

11.1.2 Collective Balance of Virtual-Hub Interactions

  • Virtual-hub force and angular-flux channels are collectively balanced, but exact total angular-momentum balance also depends on the learned application points.Their geometric contribution to total moment has not been quantified.Source: §11.1.2 · p. 63

11.2 Linear-Momentum Residual of the Independent Nodal Update

  • Equal-and-opposite decoded forces need not produce cancelling momentum increments when different nodes use different response matrices.The independent solves also include a known-velocity position-response contribution; both terms arise from the independent nodal updates.Source: §11.2 · p. 64

11.3 Asymptotic Order of the Residual and Conservative Explicit-Response Limit

  • For bounded masses, states, and response operators, the absolute one-substep linear-momentum residual is O(δt^2), or first order relative to a non-zero impulse.This is a local asymptotic result, not a bound on accumulated conservation drift; the residual vanishes in the explicit-response limit.Source: §11.3 · p. 65

11.4 Numerical Verification

  • The 12-node numerical checks show approximately first-order relative residuals with either residual contribution retained, while removing both reduces the residual to single-precision round-off.These checks verify algebra for prescribed response operators, not accumulated conservation drift in trained rollouts.Source: §11.4 · p. 66

11.5 Scope of the Conservation Statements

  • The analysis establishes a one-substep linear-momentum result, not a finite-substep result for the complete angular update or a measured conservation drift in trained rollouts.Quantifying system drift would require accounting for external impulses and boundary reactions in the walking, beam, and protein settings.Source: §11.5 · p. 66
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