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PMosFM: Preconditioned Manifold Matching for One-Step Physics-Constrained Generation

Zhangyong Liang, Haibin Ling

arXiv:2609.40287v1cs.LG

TL;DR

Physics-constrained generation must match a target distribution while satisfying constraints, but constraint handling can add sampling corrections or training-time residual optimization and unrolling. PMosFM learns one-step transport in feasible coordinates using preconditioning and physical endpoint matching. The paper reports competitive physical and distributional fidelity, lower optimizer-update time than PBFM, and faster one-step sampling than multi-step baselines.

  • Problem

    Constraint enforcement in physics-constrained generation can add iterative sampling corrections or training costs from residual optimization and trajectory unrolling.

    Source: §1 · p. 1

  • Method

    PMosFM encodes constraints in a manifold decoder and learns preconditioned one-step transport in feasible coordinates with velocity supervision and physical endpoint matching.

    Source: §1 · p. 2

  • Results

    Across evaluated PDE settings, PMosFM achieves competitive physical and distributional fidelity with near-zero encoded residuals, reduces optimizer-update time relative to PBFM, and lowers inference time versus multi-step baselines.

    Source: §1 · p. 3, §5 · p. 9

  • Takeaways & Limitations

    The evaluated approach combines feasible-state generation with learned transport while using one neural transport evaluation followed by physical decoding.

    Source: §1 · p. 2

    The guarantees apply on the represented regular feasible region and valid coordinate domain; encoding data outside the represented set yields a physical-metric projection.

    Source: §3.3 · p. 4, §3.1 · p. 4

Abstract

from arXiv · show

Physics-constrained generative models aim to generate physical fields that match a target distribution and satisfy prescribed constraints. However, enforcing these constraints often increases sampling costs through iterative corrections or training costs through residual optimization and trajectory unrolling. To address this issue, we introduce \textbf{P}reconditioned \textbf{M}anifold \textbf{o}ne-\textbf{s}tep \textbf{F}low \textbf{M}atching (\textbf{PMosFM}), a preconditioned manifold matching framework for one-step physics-constrained generation. By encoding constraints in a manifold decoder, PMosFM learns transport in intrinsic coordinates without separate residual losses or terminal residual unrolling. A geometric preconditioner rescales coordinates using the decoder-induced metric, while a regularized covariance transform approximately whitens the interpolation-state inputs. A finite-interval objective couples velocity supervision with consistency between decoded endpoints in physical space. We show that exact parameterization removes residual-induced Gauss--Newton curvature, that geometric and covariance effects separate in a local conditioning bound, and that physical flow-map error bounds endpoint distributional error. Controlled ablations examine conditioning, and experiments evaluate optimizer-update time and memory footprint. At inference, PMosFM uses one neural transport evaluation followed by physical decoding. Experiments across benchmarks show lower training and sampling time than the multi-step baselines at comparable physical and distributional fidelity. Code and datasets will be released publicly.

1 INTRODUCTION

Existing physics-constrained generators often place constraint handling in iterative sampling corrections or training-time residual optimization and unrolling. PMosFM instead learns one-step transport in feasible coordinates, combining preconditioning and physical endpoint matching; the paper reports faster sampling and training than multi-step baselines across five benchmarks.

1 INTRODUCTION

  • Existing approaches place constraint handling in online sampling projections or in training-time residual-gradient computations and terminal unrolling.These different computational bottlenecks motivate unifying constraint satisfaction and transport.Source: §1 · p. 1
  • One-step physical generation must represent feasible states and measure finite-interval errors in an appropriate metric, not merely shorten transport depth.Feasible coordinates expose decoder-induced metric distortion and interpolation-state covariance as distinct conditioning factors.Source: §1 · p. 2
  • PMosFM encodes constraints in a manifold decoder and learns one-step finite-interval transport in feasible coordinates without separate residual penalties or terminal residual unrolling.Its objective couples velocity supervision with decoded-endpoint consistency in physical space.Source: §1 · p. 2
  • Exact residual coordinates remove encoded-constraint Gauss–Newton curvature, while geometric and statistical preconditioning condition the remaining transport.The paper also bounds endpoint distribution error by physical flow-map error and distinguishes constraint satisfaction from matching the target law.Source: §1 · pp. 2–3
  • Across five benchmarks, PMosFM outperforms multi-step baselines in distributional accuracy and sampling speed with one network evaluation followed by physical decoding.The paper also reports reduced training time per update compared with residual unrolling.Source: §1 · p. 3

2 RELATED WORK

Related work spans physics-constrained generation, one-step transport, and manifold-aware modeling with preconditioning. These approaches address constraint handling, transport depth, or geometric and covariance effects in different ways.

2 RELATED WORK

  • Physics-constrained models incorporate physical information through residual losses, virtual observables, residual-gradient optimization, weak-form fine-tuning, sampling guidance, projections, or hard constraints.The passage groups PBFM with residual optimization and unrolling, and identifies ECI and PCFM as hard-constraint methods.Source: §2 · p. 3
  • One-step transport methods include rectified flows, consistency and shortcut models, distillation, finite-interval velocity learning, and PDE-constrained post-hoc distillation.The passage also describes SoFlow as combining flow matching with solution consistency.Source: §2 · p. 3
  • Manifold methods incorporate non-Euclidean geometry, while preconditioned flow matching addresses anisotropy in interpolation covariance through invertible transformations.Riemannian MeanFlow learns finite-interval transport through tangent-space alignment.Source: §2 · p. 3
  • Table 1 compares PMOSFM with other physics-constrained generative models.The supplied caption identifies the comparison's scope but does not report its outcomes.Source: §2 · p. 3

3 METHODOLOGY

The methodology represents a regular feasible component in intrinsic coordinates, then learns transport between source and data coordinates. PMosFM preconditions those coordinates and inputs, and trains with velocity supervision plus physical-space endpoint matching.

3.1 PRELIMINARIES

  • The method models a regular component of the feasible set, where the residual specifies feasible support and the target law specifies probability mass on that support.Physical error is measured using a symmetric positive-definite quadrature matrix.Source: §3.1 · p. 4

3.2 PHYSICS-BASED FLOW MATCHING

  • Ambient interpolation can leave the feasible set, while residual penalties can make endpoint regression ill-conditioned.Existing approaches impose constraints during training or sampling.Source: §3.2 · p. 4

3.3 PMOSFM: PRECONDITIONED MANIFOLD ONE-STEP FLOW MATCHING

  • PMosFM learns one-step transport on a represented feasible manifold and compares decoded endpoint estimates on that manifold.The schematic contrasts this with PCFM's iterative projection and PBFM's residual-gradient steering with unrolling.Source: §3.3 · p. 4
  • A differentiable local parameterization maps independent coordinates into the physical state space, and decoded states satisfy encoded constraints within the valid coordinate domain.For data outside the represented set, encoding followed by decoding produces a physical-metric projection.Source: §3.3 · p. 4
  • Geometric preconditioning rescales the decoder-induced pullback metric, while a regularized covariance transform rescales network inputs without changing the encoded physical target law.The endpoint loss is evaluated directly in physical space.Source: §3.3 · p. 5
  • The training objective combines ordinary flow-matching velocity supervision with consistency between decoded physical endpoints for finite intervals.The two terms are balanced by γ; in the valid coordinate domain, decoded endpoints satisfy the encoded residual.Source: §3.3 · pp. 5–6
  • For coordinates in the valid domain, the decoder yields zero encoded residual, so that residual contributes no Gauss–Newton curvature.Inference applies one learned transport map and then explicitly decodes its endpoint.Source: §3.3 · p. 6

4 EXPERIMENTS

Across the evaluated benchmarks, PMosFM pairs one-step sampling with competitive physical and distributional accuracy. Results also examine physical fidelity, sparse reconstruction, and the effects of preconditioning and objective choice.

4.1 EXPERIMENTAL SETUP

  • The evaluation includes Darcy flow, dynamic stall, Burgers, turbulence forecasting and reconstruction, Kolmogorov-flow generation and reconstruction, and a constrained-PDE reference suite.Baselines include training-time PBFM and sampling-time ECI, PCFM, D-Flow, CoCoGen, and DiffusionPDE.Source: §4.1 · p. 6, Figure 4 · p. 6, Figure 6 · p. 7

4.2 TRAINING AND SAMPLING EFFICIENCY

  • PMosFM maintains competitive physical and distributional accuracy while using one learned-network evaluation, versus 20–200 for multi-step baselines.Sampling time includes physical decoding; PCFM has higher inference time than PMosFM.Source: §4.2 · p. 7, Table 3 · p. 8

4.3 PHYSICS-CONSTRAINED GENERATION

  • Across physics-constrained benchmarks, PMosFM preserves principal solution structure with encoded conservation residuals near numerical precision.Several baselines show localized violations around dominant transition regions.Source: §4.3 · p. 8, Figure 7 · p. 8

4.4 UNDERDETERMINED SPARSE RECONSTRUCTION

  • At 99% missingness, PMosFM better preserves low-wavenumber energy and velocity-gradient PDF tails than the baselines.The comparison tests recovery of spatial and multiscale structure in unobserved regions beyond constraint satisfaction.Source: §4.4 · pp. 8–9, Figure 8 · p. 9

4.5 ABLATION STUDY

  • Preconditioning improves distribution matching, low-variance-mode error, and interpolation covariance conditioning; PMosFM has the lowest energy distance on most benchmarks, while Shortcut is stronger on turbulence reconstruction.The results suggest the preferred one-step objective varies by physical task.Source: Figure 5 · p. 7, §4.5 · p. 9

5 CONCLUSION

PMosFM separates encoded feasibility from learned transport, combining feasible coordinates and preconditioning with finite-interval matching. Across evaluated PDE settings, it achieves competitive fidelity and near-zero encoded residuals while reducing measured update and inference times.

5 CONCLUSION

  • PMosFM separates encoded feasibility from learned transport, conditioning feasible coordinates and interpolation states before matching decoded endpoints in physical space.Across evaluated PDE settings it achieves competitive physical and distributional fidelity with near-zero encoded residuals, faster distributional convergence in controlled preconditioning studies, lower measured optimizer-update time than PBFM, and lower inference time than evaluated multi-step baselines.Source: §5 · p. 9

AI USE STATEMENT

The statement describes the authors’ use and review of generative AI tools. The notation appendix groups symbols by residual manifold, preconditioning, and finite-interval matching.

AI USE STATEMENT

  • Generative AI assisted with manuscript organization, language editing, revision, and figure preparation; the authors reviewed and revised assisted content and take responsibility for the final work.The authors state that experimental measurements came from reported computational pipelines and that derivations, results, and references were independently verified.Source: AI USE STATEMENT · p. 10

A NOTATION

B DETAILED TRAINING AND SAMPLING PROCEDURES

The procedures calibrate preconditioners on training data and hold them fixed, then perform one learned full-interval map evaluation before physical decoding. Appendices specify benchmark protocols and separately account for training, sampling, and decoding costs.

B DETAILED TRAINING AND SAMPLING PROCEDURES

  • Preconditioner statistics are estimated from training data and fixed; training interpolates source and feasible target coordinates and combines velocity and endpoint losses.Calibration and preconditioners remain fixed for validation and test evaluation.Source: B DETAILED TRAINING AND SAMPLING… · p. 14, D ADDITIONAL EXPERIMENTS · p. 16
  • Sampling draws a source coordinate, applies the frozen input and geometric preconditioners, evaluates the full-interval map once, and decodes on the feasible manifold.Solver iterations, when used by a decoder, count as physical decoding rather than learned transport.Source: B DETAILED TRAINING AND SAMPLING… · p. 14

C DATASET DETAILS

  • Benchmark protocols use training-only normalization and physical-unit residual checks, reporting full prescribed residuals when required variables are available and identifiable components otherwise.The suite spans elliptic flow, conservation laws, compressible aerodynamics, turbulence, and inverse problems.Source: C DATASET DETAILS · p. 14, D ADDITIONAL EXPERIMENTS · p. 16, Table 6 · p. 17

C.2 DYNAMIC STALL

  • Dynamic-stall evaluation uses two algebraic residuals, while the Burgers diagnostic uses a finite-difference residual and reports solver results separately.For dynamic stall, the parameterization and evaluator share a sign convention.Source: §C.2 · p. 15, §C.3 · p. 15

C.4 TURBULENCE MASS TRANSFER

  • The turbulence mass-transfer evaluator combines incompressibility, normal velocity near the manifold, signed-distance Eikonal error, and weak pressure smoothness.Kolmogorov-flow evaluation uses periodic divergence; turbulence forecasting checks three-dimensional divergence, and its Fourier Helmholtz projection leaves the fourth channel unchanged.Source: §C.4 · p. 15, §C.5 · p. 15, §C.6 · p. 15

C.7 TURBULENCE FLOW RECONSTRUCTION

  • Turbulence flow reconstruction evaluates measurement consistency and incompressibility separately because overwriting observations can reintroduce divergence after projection.Kolmogorov-flow reconstruction conditions on low-resolution observations and separates reconstruction error from flow diagnostics.Source: §C.7 · p. 16, §C.9 · p. 16

D.1 COMPREHENSIVE TRAINING COST AND MEMORY FOOTPRINT

  • Sampling cost is partitioned into network evaluation, preconditioning, and physical decoding; calibration and optimizer updates are accounted for separately.Profiling used batch size 8, 80 updates with the first ten discarded as warmup, synchronized CUDA timing, and peak allocated memory.Source: §D.1 · p. 17
  • PMosFM has the lowest peak memory on four benchmarks and second-lowest on turbulence reconstruction; its update time is lowest on Burgers and second-lowest on turbulence reconstruction.The benchmark summary distinguishes physical checks of conservation from statistical metrics of coverage.Source: §D.1 · p. 17, Table 7 · p. 18, Table 8 · p. 18

D.3 ZERO-SHOT CONSTRAINED-PDE REFERENCE

Across the zero-shot constrained-PDE suite, PMosFM preserves encoded constraints while maintaining strong field-statistics agreement, though feature-space distances vary by task. Additional studies examine target-law recovery, conditioning, held-out fields, turbulence forecasting, and sparse reconstruction.

D.3 ZERO-SHOT CONSTRAINED-PDE REFERENCE

  • Across five benchmark configurations and ten evaluations, PMosFM residuals are below 10^-5, while penalty-based methods show substantial violations and unconstrained baselines exceed 1.Moment-based metrics indicate strong agreement with reference field statistics, while feature-space distances remain task dependent; Reaction–Diffusion shows complementary behavior across metrics.Source: §D.3 · pp. 18–19, Table 9 · p. 19

D.3.1 BURGERS WITH FIXED INITIAL CONDITION

  • PMosFM more closely follows the reference Burgers mean profiles and standard-deviation pattern than PCFM, although both methods have small mass and boundary residuals.Figures 9 and 10 show residual diagnostics and evolving field moments, respectively.Source: §D.3.1 · p. 19, Figure 9 · p. 20, Figure 10 · p. 20

D.4 FEASIBLE-SET AND TARGET-LAW STUDY

  • With five seeds and 24,000 samples per seed, the co-area law lowers KL from 0.369 to 0.00207 versus the volume law despite near-10^-16 residual RMS for both exact parameterizations.Exact feasibility alone therefore does not recover the target probability law.Source: §D.4 · pp. 20–21, Table 10 · p. 20

D.5 HELD-OUT DIAGNOSTICS

  • For Kolmogorov flow, external RMS is 1.353 × 10^-1 with central differences and 2.422 × 10^-6 with spectral differentiation, while both implementations have relative ℓ2 error near 0.92.The held-out pre-training study separately reports field error and decoder-side and independently evaluated residuals.Source: §D.5 · p. 21

D.6 TANGENT-REGRESSION CONDITIONING

  • Grid refinement worsens raw conditioning, while the ideal residual-space metric keeps active condition numbers near one; exact whitening removes covariance anisotropy and accelerates population-regression convergence.The population study uses 64 eigenmodes and a fixed step rule.Source: §D.6 · p. 21, Table 11 · p. 21

D.7 DYNAMIC STALL FIELD AND ENSEMBLE RESULTS

  • On one held-out Dynamic Stall condition, PMosFM places the shock-associated transition at the reference location across pressure, velocity-gradient, temperature, density, and wall-shear channels.Its conditional ensemble mean preserves the dominant shock structure, but standard-deviation maps show weaker fluctuations than the reference under shared color scales.Source: §D.7 · p. 21

D.8 HELD-OUT ONE-STEP TURBULENCE

  • Among evaluated turbulence methods, PMosFM has the lowest field MSE, divergence RMS, and all three Q–R distribution distances, and follows reference Q–R contours more closely than PCFM, ECI, and PBFM.The full-baseline comparison also reports that FM-OT, FluidFlow, and PCFT retain much of the contour structure.Source: §D.8 · p. 21

D.9 QUALITATIVE TURBULENCE DIAGNOSTICS

  • In turbulence reconstruction, PMosFM has the lowest missing-region RMSE at all three rates and retains a coarse, identifiable large-scale pattern at 99% missingness, although spectra and gradient PDFs differ from the reference.Reconstruction becomes harder as observations become sparser.Source: §D.9 · p. 21, Figure 11 · p. 22, Figure 12 · p. 23

E ADDITIONAL ABLATIONS

The additional ablations examine objective balance, endpoint-gradient supervision, preconditioning, and decoder iteration cost. The accompanying technical sections state assumptions and results concerning residual factorization, one-step transport, and exact endpoint feasibility.

E ADDITIONAL ABLATIONS

  • Balanced velocity and endpoint losses give the lowest field error and physical sliced W2 with near-zero residual on both Kolmogorov and Darcy flow.Moving toward either single-term objective worsens all three metrics, consistent with complementary roles for the two losses.Source: E ADDITIONAL ABLATIONS · p. 23, Table 14 · p. 23
  • Endpoint-gradient supervision slightly reduces physical MSE and lowers Q–R distances; preconditioning improves conditioning and convergence, while higher decoder iteration counts lower errors but increase latency.Physical MSE changes from 5.631×10^-3 to 5.524×10^-3; the decoder latency is separate from one network evaluation.Source: E ADDITIONAL ABLATIONS · pp. 24–25, Table 15 · p. 25, Table 16 · p. 25

F MATHEMATICAL SETTING AND ASSUMPTIONS

  • The mathematical setting assumes a regular residual factorization with constant-rank residual Jacobian and a local embedding whose represented feasible sets cover the target region.It also assumes differentiability, finite interpolation covariance, and C2 maps where second-order expansions are used.Source: F MATHEMATICAL SETTING AND ASSUMPTIONS · pp. 26–27

F.1 BASELINE CONSTRAINT MECHANISMS

  • The baseline mechanisms differ: PBFM coordinates gradients of flow-matching and physical losses, while PCFT, DiffusionPDE, CCFM, ECI, and PCFM use other fine-tuning, guidance, projection, or correction strategies.PCFM applies nonlinear corrections to estimated terminal states during sampling.Source: §F.1 · p. 27

G.1 THREE NOTIONS OF A STEP

  • PMosFM learns a finite-interval map evaluated once at inference; its Euler rollout is used as a conditioning diagnostic, distinct from training-time unrolling and ODE network evaluations.The appendix gives finite-interval endpoint sensitivity results under a continuous velocity Jacobian.Source: §G.1 · p. 27

G.2 RESIDUAL-NORMAL CURVATURE

  • At feasible outputs, the residual-normal term in the exact Hessian decomposition vanishes, and exact residual factorization removes the residual block after generator composition.The composed residual differential's condition number is squared in the stated unconditional result.Source: §G.2 · p. 28

H.1 RESIDUAL-FACTORIZATION DIFFERENTIAL

  • Differentiating the exact residual factorization shows that decoder-Jacobian directions span the residual Jacobian's kernel, identifying tangent directions to the represented manifold.This follows from the constant-rank and local-embedding assumptions.Source: §H.1 · p. 28

H.2 EXACT ENDPOINT FEASIBILITY

  • Within the valid coordinate domain, decoded intrinsic paths remain feasible throughout, and the encoded residual contributes no Gauss–Newton block.The local identity does not control the full endpoint Jacobian or global optimization.Source: §H.2 · pp. 28–29

H.3 INTRINSIC-COORDINATE PENALTY BLOCK

The appendix analyzes penalty curvature, approximate parameterizations, the co-area target law, and representation error. Its results distinguish residual-normal stiffness from feasible-direction conditioning and separate decoder errors from generative error.

H.3 INTRINSIC-COORDINATE PENALTY BLOCK

  • For exact intrinsic coordinates, the residual penalty block scales as Θ(λh^-2p) under the stated operator and weight assumptions, while tangent preconditioning bounds active intrinsic eigenvalues.The ambient condition-number bound is restricted to a declared subspace and does not remove generator-dependent tangential curvature.Source: §H.3 · p. 29

H.4 APPROXIMATE PARAMETERIZATIONS AND ERROR BOUNDS

  • Approximate decoders require separate residual-value and pulled-back residual-differential errors; a small residual value alone need not imply a small differential error.Exact parameterization sets both errors to zero under differentiability.Source: §H.4 · p. 30

H.5 CO-AREA LAW

  • The co-area target density weights tangent volume by the residual-normal co-area Jacobian, and reparameterization changes coordinate density while preserving the induced physical measure.The density is normalized by Zc.Source: §H.5 · p. 30

H.6 ENCODING-INDUCED REPRESENTATION ERROR

  • Encoding data outside the parameterization range induces a feasible projection whose coupling bounds representation error separately from generative error.The projection is defined using the physical metric and the encoder's selected coordinates.Source: §H.6 · p. 31

I SUPPORT, MEASURE, AND TRANSPORT ERRORS

The section separates support, target measure, and transport error, then develops preconditioning and feasible-coordinate tools for conditioning and endpoint matching. Exact constraint parameterizations, local manifold patches, and physical-space diagnostics clarify what is enforced and what remains approximate.

I SUPPORT, MEASURE, AND TRANSPORT ERRORS

  • Eq. 42 separates transport, target-law, and representation error; the latter two vanish for exactly feasible sample-defined targets.Density-defined targets instead use the co-area law in Eq. 38.Source: I SUPPORT, MEASURE, AND TRANSPORT ERRORS · p. 31

J.1 REGULARIZED INPUT COVARIANCE

  • Regularized whitening conditions interpolation-state network inputs while preserving the specified physical target law.The local conditioning analysis treats input covariance separately from the decoder-induced physical metric.Source: §J.1 · p. 31, J PHYSICAL AND STATISTICAL… · p. 31

J.2 LOCAL PHYSICAL-SPACE CONDITIONING

  • The decoded-endpoint loss has a Gauss–Newton matrix whose conditioning is bounded using the decoder metric and the covariance of preconditioned inputs.The stated bound assumes lower and upper metric bounds in the evaluated region and positive-definite input covariance.Source: §J.2 · p. 32

K ONE-STEP PULLBACK GEOMETRY

  • The pullback metric is a local approximation, while decoded endpoint error directly measures finite physical discrepancy.Its finite-displacement error is analyzed under a segment-in-domain assumption and regularity bound.Source: K ONE-STEP PULLBACK GEOMETRY · p. 32

L.1 LINEAR NULL-SPACE PARAMETERIZATION

  • For compatible affine constraints, a null-space parameterization gives an exact global constraint-satisfying representation.It covers discrete conservation, boundary, and observation constraints; numerical quality still depends on the null-space basis representation.Source: §L.1 · pp. 32–33

L.2 DISCRETE STREAMFUNCTION PARAMETERIZATION

  • Compatible discrete derivative operators make the streamfunction parameterization exactly incompressible, while its pullback metric makes gradient and small-scale importance explicit.On a periodic grid, the pullback matrix's Fourier symbol scales with squared discrete wave number.Source: §L.2 · p. 33

M PHYSICAL-SPACE ENDPOINT MATCHING

  • Velocity supervision trains the instantaneous field, while finite-interval consistency compares estimates of the same decoded physical endpoint and links flow-map defects to endpoint-law error.The bound assumes a smooth target path, a valid coordinate domain, and finite second moments.Source: M PHYSICAL-SPACE ENDPOINT MATCHING · pp. 33–34

N.1 LOCAL RESIDUAL-MANIFOLD PARAMETERIZATIONS

  • Regular residual manifolds admit local feasible parameterizations, and patchwise bounds extend conditioning and endpoint-error analyses without globally flattening the manifold.A near-identity metric bound for each coordinate transform requires separate verification.Source: §N.1 · p. 34

O.2 DECODER TOLERANCE AND COST

The section distinguishes finite-iteration decoder accuracy from learned transport cost. It specifies residual and differential checks and separates transport timing from full physical decoding.

O.2 DECODER TOLERANCE AND COST

  • Finite-iteration decoders require separate derivative-accuracy checks.The decoder diagnostics measure both physical residual and its differential in represented coordinates.Source: §O.2 · p. 36
  • For an exact decoder, the encoded residual-normal Gauss–Newton block vanishes; the stated bound covers the block retained by an iterative decoder.Source: §O.2 · p. 36
  • End-to-end timing reports learned transport separately from physical decoding, and residual diagnostics quantify the realized endpoint's accuracy.The decoder iteration count indexes the physical decoder, not the learned transport map.Source: §O.2 · pp. 35–36
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