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Markovian Dynamics Enforcer: Feasibility Preserving Correction on Learned Dynamics Manifolds
Kevin Yu, Tao Guo, Constantinos Antoniou, Panagiotis Angeloudis
TL;DR
Neural trajectory predictions can violate physical constraints despite low prediction error, especially when physics is incomplete and controls are unobserved. MaDE corrects proposals through inferred controls and a known-physics-plus-learned-residual completion model; it achieves near-zero known-model residuals on fully specified simulations, while its guarantees remain model-relative.
Problem
Neural trajectory predictions can have low prediction error yet violate dynamics, actuator limits, or state constraints, while existing approaches may require predictor retraining or analytically specified dynamics.
Source: §1 · p. 1Method
MaDE is a stateless post-hoc operator that infers controls from adjacent states, recomputes states through known physics plus a learned residual, and corrects controls against inequality constraints.
Source: §3 · p. 3, §3.1 · p. 3, §3.3 · p. 5Results
On fully specified simulated systems, MaDE drives Dyn.-K below 0.0004 on each and is the only method to do so; on underspecified DB it has the lowest Dyn.-K and Dyn.-T among four methods.
Source: §5.1 · p. 9Takeaways & Limitations
MaDE is intended for post-hoc feasibility checking, but its guarantees are relative to a learned completion model and supplied anchor, not a safety certificate for deployment.
Source: §6 · p. 10, Ethics statement · p. 11Its dynamics consistency is model-relative to the learned completion model and supplied previous-state anchor, and inequality satisfaction is best-effort within the correction iteration budget.
Source: §6 · p. 10, Ethics statement · p. 11
Abstract
from arXiv · showhide
Neural trajectory predictors can reach low prediction error while violating dynamics, actuator limits, or state constraints, especially when controls are unobserved and dynamics are partially specified. We introduce the Markovian Dynamics Enforcer (MaDE), a time-invariant post-hoc operator mapping state-transition proposals onto a learned feasible dynamics manifold, trained on feasible states without ground-truth controls. For each transition it infers a control and recomputes the state through a completion model of known physics plus a learned residual. It then corrects that control by gradient-based inequality reduction, so inequality satisfaction is best-effort within an iteration budget. Since every correction iterate re-enters the completion model, the returned state is dynamically consistent by construction relative to that model and the supplied previous-state anchor. MaDE drives dynamics residuals to essentially zero on fully specified simulated systems, and on an underspecified system leaves a smaller true-dynamics residual than the baselines. Designed to attach to arbitrary predictors, the frozen operator is evaluated downstream of recurrent, structured state-space, and transformer predictors. On recorded vehicle trajectories the one-step residual against a kinematic bicycle model is 0.0071 to 0.0072 for MaDE and 0.1703 to 0.1714 for raw predictors. MaDE raises average displacement error by a factor of 1.57 to 1.83.
1 Introduction
Neural trajectory predictions can be accurate yet violate physical constraints relied on by downstream systems, while existing feasibility approaches may require retraining, specified dynamics, or tight coupling to a predictor. MaDE is a downstream correction operator for partially specified dynamics and unobserved controls, evaluated on simulated systems and recorded vehicle trajectories.
1 Introduction
- Predictions with low error can still violate actuator limits, state constraints, or dynamics, undermining physical interpretation, safety analysis, and control-aware reasoning.Downstream modules may implicitly assume physical admissibility.Source: §1 · p. 1
- Existing feasibility approaches may require predictor retraining, analytically specified dynamics, or tight coupling to an upstream architecture, despite incomplete physics and unobserved controls.Prior approaches include optimisation layers, physics priors, and structured dynamical models.Source: §1 · p. 1
- MaDE is a differentiable post-hoc correction operator for neural trajectory predictions that infers latent controls and recomputes states through known physics augmented by a learned residual.A control-space correction then reduces explicit state and control bound violations while keeping the state on the learned dynamics manifold.Source: §1 · pp. 1–2
- MaDE is evaluated on a simulated dynamics ladder and downstream of three independently trained predictors on recorded vehicle trajectories, with dynamics residuals and inequality violations reported.The vehicle evaluation also quantifies the associated accuracy cost.Source: §1 · p. 2
2 Related Work
MaDE is positioned among feasibility methods as a completion–correction operator that uses a learned dynamics manifold rather than a fully specified analytical equality map. Its distinctions from related approaches include per-transition correction, inference of unobserved controls, and post-hoc use on finished trajectory proposals.
2 Related Work
- Figure 1 characterizes MaDE as improving dynamical consistency at a cost in prediction accuracy.The caption gives the trade-off but no magnitude.Source: Figure 1 · p. 2
- MaDE adapts completion–correction methods by using known physics plus a learned residual to supply the equality manifold, rather than requiring an analytical equality map.Optimisation layers embed convex programs, while Donti et al. [2021] reduce inequality violations by gradient steps that remain on an equality manifold.Source: §2 · p. 2
- Unlike sequence models that carry state across whole trajectories, MaDE applies a discrete, stateless correction independently to each transition.Its proposals are projected onto an explicitly constrained learned dynamics manifold.Source: §2 · p. 2
- Imitation-from-observation methods recover unavailable actions for policy learning or transition matching, whereas MaDE infers physically typed controls for its completion step.The described approaches include inverse-dynamics inference [Torabi et al., 2018a], adversarial transition matching [Torabi et al., 2018b], and latent action spaces [Edwards et al., 2019].Source: §2 · pp. 2–3
- Hybrid physical models combine mechanistic dynamics with learned components, while MaDE adds inference of unobserved controls and reduction of inequality violations to the APHYNITY decomposition.APHYNITY [Yin et al., 2021] decomposes dynamics into a known physical component and a minimum-norm learned augmentation.Source: §2 · p. 3
- Vehicle-plausibility methods may assume an analytical motion model or produce trajectories within the predictor, whereas MaDE corrects finished proposals under partially specified dynamics.The comparison covers smoothing, Deep Kinematic Models [Cui et al., 2020], the Realistic Residual Block [Bahari et al., 2021], and differentiable model predictive control [Amos et al., 2018].Source: §2 · p. 3
3 Methodology
MaDE is a stateless, one-step correction operator that maps transition proposals to a state-control pair on a learned dynamics manifold. It infers controls, recomputes states with a completion model, corrects control inequalities, and is trained from state transitions without ground-truth controls.
3 Methodology
- MaDE applies the same stateless, time-invariant one-step map to each transition, allowing use across horizons but trading sequence-level accuracy for local dynamical admissibility.The correction is applied recursively over a horizon.Source: §3 · p. 3
3.1 Problem Setup
- The one-step problem takes a supplied previous-state anchor and raw next-state proposal; the control is unobserved, and feasibility requires dynamics consistency plus analytical inequality constraints.Those inequalities can encode state bounds, actuator bounds, speed limits, or other local domain constraints.Source: §3.1 · p. 3
3.2 The MaDE Cell
- The MaDE cell infers a control, recomputes the next state through its dynamics model, then updates the control to reduce violations and recomputes the state again.Unlike the analogous DC3 procedure, MaDE uses explicit forward numerical integration and re-integration rather than an implicit equality solve or tangent projection.Source: §3.2 · p. 4
3.3 Completion and Inverse Dynamics
- The completion model integrates known physical dynamics plus a learned residual, which is trained with a minimum-norm penalty to capture dynamics the known model cannot explain.Systems without a useful known physics model are outside MaDE’s intended scope.Source: §3.3 · p. 4
- Inverse dynamics combines a system-specific kinematic prior with a learned residual to infer physically typed, bounded controls that explain transitions under the completion model.These are model-relative controls, not estimates of the commands actually applied; the residual corrects discretisation error and prior-data mismatch.Source: §3.3 · pp. 4–5
3.4 Control-Space Inequality Correction
- The corrector optimizes over controls rather than clipping states, re-completing through Tθ after updates so each intermediate state remains on its dynamics manifold.It starts from the inferred control; the correction runs a fixed number of steps during training and an adaptive, tolerance-based number at evaluation.Source: §3.4 · p. 5
3.5 Training Objectives
- MaDE trains from state transitions without ground-truth controls, using forward and inverse cycle consistency with minimum-norm residual penalties.Two-phase training first fits consistency and residual terms, then adds inequality-aware training through the corrector; the inequality signal updates the inverse model, not the forward residual.Source: §3.5 · pp. 5–6
4 Experimentation
MaDE is evaluated first as a one-step operator on simulated systems and then downstream of neural predictors on recorded vehicle trajectories. The experiments measure dynamics consistency, inequality violations, and trajectory accuracy under distinct simulated and real-data conditions.
4.1 Evaluation metrics and training protocol
- Evaluation reports inequality violation rate and magnitude, dynamics residuals against known, augmented, or true dynamics, fidelity, ADE, and FDE.Fidelity measures distance to the unperturbed simulator trajectory, whereas ADE and FDE measure distance to recorded positions over the horizon and at its last step, respectively.Source: §4.1 · p. 6
4.2 Single-agent simulated dynamics
- The simulated study evaluates the bare MaDE cell on single-agent systems, recursively correcting perturbed transitions from each preceding corrected state.It uses four systems, five training seeds per cell, and deterministic bound-violation perturbations; the dynamic bicycle condition also uses Gaussian observation noise and underspecified known dynamics.Source: §4.2 · pp. 6–7
4.3 Correction of neural trajectory predictors
- The real-data experiment trains and freezes MaDE on inD state transitions without control labels, then applies it to 15 runs each of recurrent, state-space, and transformer predictors.Three MaDE seeds correct all 15 predictor runs, yielding 45 evaluation cells; the known model is a 2.7 m kinematic bicycle with a learned per-vehicle wheelbase residual.Source: §4.3 · p. 7
- The real-data comparison uses raw forecasts, state-bound clamping, and an EKF/RTS smoother alongside MaDE, with the smoother using the 2.7 m kinematic bicycle model.Predictors use a 10-step history to decode a 15-step horizon and are trained using mean-squared error alone; real-data constraints are hand-set bounds shared by the compared methods.Source: §4.3 · pp. 7–8
5 Results and Discussion
On simulated systems, MaDE achieves strong dynamics consistency, low inequality violations, and trajectory fidelity, including when known physics are underspecified. On recorded trajectories, it reduces kinematic-bicycle residuals and inequality violations, but at a cost in displacement accuracy.
5.1 Simulated experiment
- MaDE is the only method with Dyn.-K below 0.0004 on DI, UNI, and KB; on underspecified DB, it has the lowest Dyn.-K and Dyn.-T among four methods.On DB, its Dyn.-L is below 10^-15 because its returned state is Tθ evaluated at the returned control.Source: §5.1 · p. 9
- With correction, MaDE has the lowest Ineq. mag. on all four simulated systems and the lowest Ineq. rate on DI, KB, and DB.Remaining violations show that the corrector does not always eliminate inequality error within its iteration budget; clamp-only has the lowest UNI rate but the highest magnitude on every system.Source: §5.1 · p. 9
- On DB, FAB has the highest Dyn.-K, Dyn.-T, and Ineq. rate among the four methods, and its seed-to-seed spread is larger than on the other systems for several metrics.Away from DB, FAB is second-best on Ineq. mag.; its baseline projection does not explicitly use physics, controls, or dynamics-manifold completion.Source: §5.1 · p. 9
5.2 Real-data experiment
- Across predictor families, MaDE’s Dyn.-K is 0.00712 to 0.00722, versus 0.1703 to 0.1714 for raw predictors, while its ADE rises by a factor of 1.57 to 1.83.Recordings score 0.04038 on Dyn.-K; MaDE scores below that level, but the real-data comparison is a consistency–accuracy trade-off under an approximate prior, not evidence that corrected trajectories are physically truer.Source: §5.2 · pp. 9–10
- On inD, combined inequality rate orders MaDE, clamp, and raw forecasts from lowest to highest in every family; completion alone raises the rate from 0.0504 to 0.1178, while full MaDE scores 0.0096.Full MaDE is below its paired raw forecast on both inequality metrics in all 45 cells, whereas completion-only exceeds its paired raw forecast in all 45 cells.Source: §5.2 · p. 10
- MaDE averages 2.5127 ms per trajectory at batch size 16 and 4.9297 ms at batch size 1; its corrector reaches the 50-iteration cap above tolerance on 0.964% of calls.The timing is from one run; the iteration statistic pools 6,681 evaluation windows and 45 cells, rather than the single timed window.Source: §5.2 · p. 10
6 Conclusion
The conclusion presents MaDE as a frozen, post-hoc correction operator whose dynamics consistency is relative to its learned completion model and supplied anchor. It reports improved feasibility-related outcomes alongside an accuracy cost on inD, and frames the method as intended for post-hoc checking rather than deployment safety certification.
6 Conclusion
- MaDE combines inverse dynamics, known physics plus a learned residual, and a control-space inequality corrector into a differentiable, time-invariant one-step map trained without ground-truth controls.It is designed as a frozen downstream layer for arbitrary predictors; its dynamics consistency is relative to the learned completion operator and supplied previous-state anchor, while inequality satisfaction is limited by the iteration budget.Source: §6 · p. 10
- On the inD experiment, MaDE has the lowest one-step residual against the 2.7 m kinematic bicycle and the lowest inequality violation rate in every predictor family, while ADE rises by a factor of 1.57 to 1.83.Across the simulated ladder, it achieves better trajectory fidelity than learned-projection and direct-mapping baselines.Source: §6 · p. 10
Ethics statement
- The paper states that MaDE is intended for post-hoc feasibility checking, and its guarantees are model-relative rather than guarantees against the true system.The inD recordings are drone-recorded public-road traffic, used under provider terms for non-commercial research and not redistributed; no personally identifying information or human subjects are involved.Source: Ethics statement · p. 11
Reproducibility statement
- The reproducibility statement points readers to the sections and appendices documenting metrics, data splits, model and training details, baselines, ablations, and inference timing.It identifies Appendix E as the source for hardware, training compute, and inference-timing protocol and results.Source: Reproducibility statement · p. 11
A Scope and Failure Modes
MaDE’s guarantees are conditional on its learned model, supplied anchor, and local one-step scope. The paper identifies limits involving state sufficiency, known-physics assumptions, recursive anchors, and finite-budget correction.
A Scope and Failure Modes
- Every correction iterate recomputes the state through Tθ, so dynamics consistency holds relative to the learned model and supplied anchor, not necessarily the true system or a feasible anchor.Inequality satisfaction follows only if the corrector reaches J(u)=0.Source: A Scope and Failure Modes · p. 13
- One-step locality excludes trajectory-level constraints, and the state must contain every variable needed to evaluate one-step dynamics and constraints.A Markov, time-invariant operator cannot recover hidden multi-step modes.Source: A Scope and Failure Modes · p. 13
- The APHYNITY interpretation presumes known physics explains a substantial part of the dynamics; on DB, the kinematic prior substitutes for omitted terms and requires a non-small residual on affected channels.The paper treats transfer of the decomposition argument to controlled systems as motivation rather than a re-proof.Source: A Scope and Failure Modes · p. 13
- Robustness evidence comes from prescribed experimental perturbations, not a distribution-free claim about deployed-predictor errors.Architecture-ablation ordering was tested in simulation only, so whether it holds on real data remains untested.Source: A Scope and Failure Modes · p. 14
- MaDE never corrects its supplied anchor, and an infeasible anchor outside the controlled-invariant subset can leave no feasible next state reachable in one step.Recursive rollouts can therefore carry unresolved violations forward; conditions ensuring invariant anchors are left for future work.Source: A Scope and Failure Modes · p. 14
B Experimental Details
Algorithm 1 trains MaDE in two phases: first for forward and inverse consistency, then optionally with inequality-aware correction in the loop.
Algorithm 1 MaDE training (Phases 1 and 2)
- The procedure samples minibatches of feasible transitions and returns the trained MaDE cell.Algorithm 1 requires a control sampler and constraint map as well as the feasible transitions.Source: B Experimental Details · p. 14
- Phase 1 updates the inverse model for forward and inverse consistency and the dynamics model for forward consistency plus its auxiliary penalty.The inverse-side update also includes the inverse-residual penalty; the dynamics-side update uses sampled controls and freezes inverse-model gradients.Source: B Experimental Details · p. 14
- When inequality-aware training is enabled, Phase 2 runs the inverse model, dynamics model, and fixed differentiable corrector, adding an inequality loss to the inverse-side update.The dynamics-side update retains its auxiliary penalty and receives no gradient from the inequality loss.Source: B Experimental Details · p. 14
B.1 System Specifications
The experimental details specify simulated and inD settings, system-specific integration and inverse priors, model architectures, training and correction procedures, baselines, and default evaluation limits.
B.1 System Specifications
- The simulated ladder spans double-integrator, unicycle, kinematic-bicycle, and dynamic-bicycle systems, including an underspecified dynamic-bicycle condition using kinematic known physics.The real-data inD setting bounds heading, speed, steering, and acceleration, but leaves position unbounded because recorded positions do not support a reliable bound.Source: §B.1 · pp. 14–15
- InD evaluation uses filtered vehicle tracks, 10-step histories, 15-step forecasts, and recurrent, structured state-space, and transformer predictors.The predictors receive vehicle metadata; they are trained with mean-squared forecast error and validation-based early stopping.Source: §B.1 · p. 16
B.2 ODE Integration
- Controls are held constant and integrated with one Heun step (0.1 s simulated; 0.2 s inD); float32 solvers were unstable near low-vx dynamic-bicycle states.Dyn.-K and Dyn.-T use the pure-physics vector field, while Dyn.-L uses augmented dynamics under the same fixed-step convention.Source: §B.2 · p. 16
B.3 Inverse Priors
- System-specific analytical inverse priors provide a structural baseline; the dynamic-bicycle prior uses staged Newton iterations and matches the Heun simulator to roughly 10^-12 on tested maneuvers.The underspecified condition uses the kinematic prior wrapped to the dynamic state vector, not a promoted dynamic prior.Source: §B.3 · p. 17
B.4 Network Architectures
- The inverse and residual vector-field networks are two-hidden-layer ReLU MLPs initialized with zero-scaled final layers, so MaDE starts from the analytical prior and known physics.The inD wheelbase encoder uses vehicle metadata and location embedding to supply the wheelbase to the dynamics model.Source: §B.4 · p. 17
B.5 Optimisation
- Training combines cycle-consistency Phase 1 with inequality-aware Phase 2, whose corrected-output inequality loss updates the inverse model but is stopped from the dynamics-side loss.At evaluation, correction runs until maximum violation is at most 10^-6 or 50 iterations; phase training also uses validation- and physics-based stopping.Source: §B.5 · p. 18
B.6 Baselines
- Real-data deterministic baselines aggregate 15 predictor-family-by-seed cells; the state-space accuracy-tuned smoother reports ADE 0.7063 ± 0.0180 m and Dyn.-K 0.0291 ± 0.0007.The smoother uses the full forecast and tunes a covariance scale; accuracy- and residual-based criteria select the same scale for recurrent and transformer families but differ by half a decade for state-space.Source: §B.6 · pp. 18–19
B.7 Default Hyperparameters
- The corrector uses one setting across all four simulated systems and ablations; 0.964% of inD calls hit the evaluation cap, which then sets correction depth.The cap bounds worst-case compute but also becomes an accuracy setting for proposals that reach it first.Source: §B.7 · p. 19
C.1 MaDE Ablations
The ablations separate the roles of the learned dynamics residual, corrector, and inverse prior. The prior-only comparison and control-recovery results also show that inferred controls are model-relative, especially under underspecified dynamics.
C.1 MaDE Ablations
- MaDE’s Dyn.-L is below 10^-15 on all four systems, while baseline Dyn.-L is 0.2397 to 0.8547 on DI, UNI, and KB.For DI, UNI, and KB, the known model matches the data-generating model; each baseline Dyn.-L mean differs from its Dyn.-K by at most 1.5 × 10^-5.Source: §C.1 · p. 20
- On DB, full MaDE has the lowest fidelity error among the five variants, although w/o Fa and fixed-I each have lower true-dynamics residuals.The w/o Fa residual is zero for known physics by definition, not by learning.Source: §C.1 · p. 20
- On DB, w/o C and supervised-I produce unstable seeds on reported metrics, unlike w/o Fa and fixed-I.The supervised-I variant replaces cycle consistency with simulator-supervised controls and is available only in simulation.Source: §C.1 · p. 20
C.2 Prior-Only Comparator
- 0.3030 ± 0.0007 vs 0.3736 ± 0.0647 true-dynamics residual: the prior-only comparator beats full MaDE, while full MaDE retains the best fidelity.The comparator also beats full MaDE on both inequality metrics and has 12 bookkeeping floats versus 139,796 for full MaDE.Source: §C.2 · p. 21
C.3 Control Recovery
- On DI and UNI, all control-recovery variants reach at most 2.2 × 10^-5 nRMSE; on underspecified DB, none recovers true controls and all retain negative acceleration bias.On KB, the analytic prior recovers steering at 5.0 × 10^-15 nRMSE, versus 0.0010 ± 0.0005 for the learned inverse.Source: §C.3 · p. 21, Table 6 · p. 21
C.4 Gradient-Depth Probe
The probe tests how recursive correction depth affects training gradients and clarifies how real-data inequality violations are aggregated. The inverse-model gradient decreases modestly with depth, while the forward-model gradient is depth-independent under the probe’s loss setup.
C.4 Gradient-Depth Probe
- The gradient norm on Iϕ falls 4.18% from depth 0 to 1 and another 0.038% by depth 8, without diverging or collapsing.From depth 1 onward, one added iterate lowers the norm by 6.280 × 10^-4 to 6.307 × 10^-4 on average across probed intervals.Source: §C.4 · p. 21, Table 7 · p. 21
- The Tθ gradient is unchanged across correction depths because the probe removes the inequality term from the dynamics-side gradient.The remaining dynamics-side loss terms contain no corrector.Source: Table 7 · p. 21, §C.4 · p. 22
D.1 Inequality Components
- The combined violation rate counts a step violating both state and control bounds once, so it lies between the larger component rate and their sum.Its shortfall from the sum is the fraction of steps violating both; the bounds cover vehicle-state limits and steering-angle and acceleration limits.Source: §D.1 · p. 22, Table 8 · p. 22
- The combined violation magnitude is the mean of per-step Euclidean norms, so it need not equal the norm of the two component means.In 104 of 105 real-data cells, it lies between that norm and the sum of component means.Source: §D.1 · p. 22
D.2 Per-Model Breakdown
Across real-data evaluation cells, the full operator reduces inequality violations relative to both raw forecasts and completion alone, while pooled dynamics residuals are lower than for raw forecasts. Results vary across predictor runs, and correction adds displacement error.
D.2 Per-Model Breakdown
- Pooled Dyn.-K is 0.00718 for MaDE and 0.17095 for raw forecasts, while MaDE’s pooled mean ADE is 1.74 times the raw-predictor mean.The Dyn.-K means pool 45 MaDE cells and 15 raw-predictor runs.Source: §D.2 · p. 22
- The seed-1 MaDE model has the lowest mean ADE and FDE, 1.0646 m and 2.8443 m; the seed-0 model has the highest, 1.1944 m and 3.1763 m.Seed 0 is highest on both metrics in all 15 runs; seed 1 is below seed 2 on ADE in 14 runs and on FDE in all 15.Source: §D.2 · p. 23
- Across predictor runs, raw inequality violation rates span 0.0228 to 0.0687, while corrected rates range from 0.00036 to 0.02135 across the three MaDE models.The seed-0 MaDE model has the lowest mean rate, 0.00397, and the seed-1 model the highest, 0.01600.Source: §D.2 · p. 23
D.3 Completion-Only Variant
- 0.0032 pooled violation magnitude for full MaDE, versus 0.0340 for raw forecasts and 0.0992 for completion only on evaluation windows.Across all 54,465 test windows, the corresponding magnitudes are 0.0154, 0.0662, and 0.3832, and violation rates are 0.0778, 0.1377, and 0.3207; full MaDE is below both alternatives in all 45 cells.Source: §D.3 · pp. 23–24, Table 10 · p. 23
E Runtime
Runtime measurements cover a specific float64, GPU-based evaluation protocol, with correction cost depending on the proposal and, for batched use, the slowest trajectory. The stated deployment scope excludes real-time onboard closed-loop use.
E Runtime
- The measured protocol uses float64 on one RTX 3090 for a 15-step, 3-second horizon; each correction iteration adds a violation-objective gradient and Heun re-completion.Costs are not claimed for other precisions or devices; the loop early-stops at tolerance 10^-6 and is capped at 50 iterations.Source: E Runtime · p. 24
- Across 6,681 evaluation windows and 45 cells, 0.964% of corrector calls stop at the 50-iteration cap above tolerance.The pooled call count has median 0, 95th percentile 1, mean 0.614, and maximum 50 iterations.Source: E Runtime · p. 24
- MaDE’s batch-size-1 correction cost has a 2.4116 ms median and 4.9297 ms mean across 45 cells, with seven slower cells at 9.0582 to 35.0782 ms.The seven slower cells are exactly those with corrector calls hitting the 50-iteration cap on the timed window.Source: E Runtime · p. 24
- The amortised correction cost averages 2.5127 ms and ranges from 0.6144 to 16.0995 ms across 45 cells.At each timestep, batched correction runs as many iterations as the slowest trajectory in the batch needs.Source: E Runtime · p. 25
- MaDE is scoped to reconstruction, simulation, batched forecasting, and offline certification; real-time onboard closed-loop use is not claimed.Amortised cost is throughput rather than latency; single-trajectory use pays the batch-size-1 cost.Source: E Runtime · p. 25
- Real-data MaDE training took 47.92 to 71.05 minutes per seed and 3.13 GPU-hours over three seeds; simulated-system training compute is unreported.The 15 predictors used 1.34 GPU-hours total, at 2.99 to 6.93 minutes per run.Source: E Runtime · p. 25