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Discrete Action Matching: Learning Stochastic Dynamics from Samples via State Graphs

Mikhail Persiianov, Alexander Korotin

arXiv:2610.05071v1cs.LGcs.AI

TL;DR

Unpaired temporal marginals leave population dynamics underdetermined, motivating structural principles for selecting among compatible motions. DAM adapts Action Matching to finite state graphs by estimating neighboring density ratios and learning an action potential along an empirical interpolation. It reports lower mean marginal errors than graph-gradient-flow methods on synthetic dynamics and mouse gastrulation data, and approximates numerical surface-transport paths from paired samples.

  • Problem

    Unpaired temporal marginals do not reveal individual transitions, and graph kinetic energy depends on unknown endpoint masses, making sample-based learning of the action potential challenging.

    Source: §1 · pp. 1–2

  • Method

    DAM uses positive 1-homogeneity to estimate neighboring density ratios from an empirical interpolation, then learns the action potential and defines a graph-supported Markov sampler.

    Source: §1 · p. 2, §3.2 · p. 7

  • Results

    DAM achieves lower mean marginal errors than graph-gradient-flow methods on controlled synthetic dynamics and mouse gastrulation data, while histogram-based and learned-ratio variants perform comparably.

    Source: §1 · p. 3

  • Takeaways & Limitations

    DAM provides a finite-state minimum-action reconstruction of a prescribed marginal path and also approximates numerical surface-transport paths from paired samples.

    Source: §1 · pp. 2–3

    Temporal marginals generally do not determine a unique microscopic history; DAM selects a canonical current from the prescribed graph geometry and mobility, while that current can admit multiple compatible Markov realizations.

    Source: §6 · p. 9

Abstract

from arXiv · show

Learning population dynamics from unpaired temporal marginals is an ill-posed inverse problem that requires structural assumptions on the underlying dynamics. We introduce $\textit{Discrete Action Matching}$ (DAM), a finite-state counterpart of Action Matching based on discrete Wasserstein geometry. For a prescribed marginal path and transport geometry, we derive an action-minimization objective for its canonical minimum-kinetic-energy current. Our key observation is that the density dependence of the discrete action reduces to neighboring density ratios. Along an empirical interpolation of the snapshots, DAM first estimates these ratios and then learns an action potential. The learned fields also define a graph-supported Markov sampler. Experiments on controlled synthetic dynamics and real mouse gastrulation data evaluate marginal reconstruction and interpolation. Additional experiments approximate numerical surface-transport paths from paired samples.

1 INTRODUCTION

Unpaired population snapshots leave the motion between distributions unobserved, so inferring dynamics requires a principle for choosing among compatible motions. Discrete Action Matching (DAM) adapts minimum-energy Action Matching to finite states by estimating neighboring density ratios, learning an action potential, and defining a graph-supported Markov sampler.

Discrete Action Matching

  • The discrete action’s dependence on unknown endpoint masses can be reduced to neighboring density ratios using positive 1-homogeneity, making the action estimable from marginal samples.Sampling supplies the endpoint probability implicitly, separating density-ratio estimation from learning the action potential.Source: §1 · p. 2
  • For a prescribed marginal path and fixed graph and mobility, DAM selects the unique current with minimum kinetic energy, a principled reconstruction rather than recovery of exact historical trajectories.Unpaired observations generally do not determine unique individual trajectories.Source: §1 · p. 2

Learning and sampling

  • DAM estimates neighboring density ratios on a positive empirical interpolant, then freezes the ratio model and learns the action potential from samples of that same interpolant.The learned fields define graph-supported nonnegative transition rates for a Markov process that reproduces the prescribed marginal path under stated conditions.Source: §1 · p. 2

Experiments

  • DAM achieves lower mean marginal errors than graph-gradient-flow methods on controlled synthetic dynamics and mouse gastrulation data, while its histogram-based variant performs comparably to its learned-ratio variant.It also approximates numerical surface-transport trajectories from paired samples on sphere and hand meshes.Source: §1 · p. 3

2 BACKGROUND

The background frames dynamics inference from unpaired snapshots as underdetermined and reviews minimum-energy selection through continuous Action Matching and its discrete graph-geometric counterpart. In both settings, the prescribed marginal path is central, while finite-state learning faces the additional challenge of unknown endpoint masses in the kinetic energy.

2 BACKGROUND

  • The paper distinguishes one-time population marginals from the underlying trajectory law and reviews the observation model and geometric ingredients used to transfer Action Matching to finite states.The detailed derivations are deferred to Appendix A.Source: §2 · p. 3

2.1 LEARNING DYNAMICS FROM UNPAIRED MARGINAL SNAPSHOTS

  • Unpaired snapshots do not determine how probability mass moves between states, and finite observations also leave the intermediate marginal path unknown; the path chosen for training is therefore a modeling assumption.The paper uses a positive empirical interpolant between observed snapshots.Source: §2.1 · p. 3

2.2 ACTION MATCHING ON CONTINUOUS SAMPLE SPACES

  • For a prescribed smooth positive continuous path, Action Matching minimizes an action objective to recover the minimum-energy gradient velocity using marginal samples without observing velocities or evaluating the density.With finitely many observations, intermediate marginals cannot be sampled directly, so this sample-based objective requires a path choice.Source: §2.2 · p. 4

2.3 GRAPH WASSERSTEIN GEOMETRY AND CANONICAL DISCRETE DYNAMICS

  • On a connected weighted graph, discrete Wasserstein geometry represents each marginal tangent by a potential whose gradient is the minimum-kinetic-energy edge field; nonlinear mobility makes this energy depend on unknown endpoint masses.Positive 1-homogeneity allows the dependence to be rewritten using neighboring density ratios, leading to a sample-based action objective.Source: §2.3 · p. 5

3 METHOD

DAM transfers Action Matching to finite-state graph dynamics by learning the canonical minimum-energy motion along a prescribed marginal path. Its sample-based pipeline estimates neighboring density ratios, learns an action potential, and constructs a graph-supported Markov sampler.

3.1 DISCRETE ACTION MATCHING

  • The discrete action objective recovers the canonical minimum-kinetic-energy potential for a prescribed path, up to a time-dependent additive constant.Positive 1-homogeneous mobility lets local density ratios make the action sample-evaluable without reconstructing the full probability vector; arithmetic mobility also admits a ratio-free action estimator.Source: §3.1 · p. 6

3.2 LEARNING LOCAL DENSITY RATIOS

  • DAM estimates neighboring density ratios along a positive empirical interpolant, whose reference mixture provides full support and permits unpaired sampling between snapshots.Under exact population minimization, the ratio field recovers log p plus an additive constant, and thus the neighboring ratios of the chosen interpolant.Source: §3.2 · p. 7
  • The action identity applies interval by interval to the C1, piecewise-smooth interpolated path used for training.The prescribed interpolation is a modeling choice, and the unknown intermediate population evolution is evaluated empirically.Source: §3.2 · p. 7

3.3 TRAINING AND MARKOV REALIZATION

  • After freezing the ratio model, DAM learns an action potential and uses the learned fields to define a graph-supported Markov generator.With exact ratios and a canonical potential, the sampler reproduces the target path when initialized from its initial distribution; approximate fits generally yield a different law.Source: §3.3 · p. 7

4 RELATED WORK

DAM extends the fixed-path Action Matching principle to finite state spaces and complements methods that infer autonomous gradient flows. Its local density-ratio formulation avoids discrete-geodesic preprocessing.

Action Matching

  • DAM transfers continuous fixed-path Action Matching to finite state spaces using mobility-dependent discrete Wasserstein geometry.Action Matching learns continuous dynamics from independent temporal marginals through a sample-based variational objective.Source: §4 · p. 8

Discrete Wasserstein geometry and gradient-flow learning

  • Discrete graph transport supplies finite-state analogues of Wasserstein continuity equations, kinetic energies, and gradient flows.Prior work also uses discrete-Wasserstein geometry to study gradient-flow dynamics.Source: §4 · p. 8
  • Unlike Rancati et al. (2026), which assumes an autonomous free-energy gradient flow and computes discrete-Wasserstein geodesics, DAM learns a canonical potentially time-dependent velocity for a prescribed evolution without that preprocessing.The methods address complementary inverse problems.Source: §4 · p. 8

5 EXPERIMENTS

DAM is evaluated on synthetic graph dynamics and mouse gastrulation for marginal reconstruction and interpolation. Both DAM variants outperform FEGF on the reported comparisons, with harmonic mobility performing best among the tested choices for gastrulation interpolation.

5 EXPERIMENTS

  • The comparison adapts the synthetic and gastrulation benchmarks of Rancati et al. (2026), matches sampled inputs, and reports mean Hellinger errors with standard deviations across three seeds.Validation is restricted to retained observation times; protocol adaptations include FEGF's time-indexed gastrulation potential and differing transport geometries.Source: §5 · p. 8

5.1 SYNTHETIC GRAPH DYNAMICS

  • The synthetic benchmark retains five unpaired snapshots across 11 graph families and evaluates one continuous rollout against independent test histograms at the other four observation times.Errors are averaged over those four observation times.Source: §5.1 · p. 9

5.1 SYNTHETIC GRAPH DYNAMICS; 5.2 MOUSE GASTRULATION

  • Both DAM variants have lower mean marginal errors than FEGF on all 11 synthetic graph families and lower mean interpolation error on mouse gastrulation; harmonic mobility gives DAM its lowest mean errors.The gastrulation comparison is against omitted, smoothed atlas marginals, and the figure reports lower values as better.Source: §5.1 · p. 9, §5.2 · p. 9, Figure 3 · p. 9

5.2 MOUSE GASTRULATION

  • The gastrulation benchmark uses 34 cell types across nine stages and leaves out each interior stage in turn, fitting on the other eight stages.The smoothed atlas marginal for the omitted stage is used only for evaluation.Source: §5.2 · p. 9

5.1 SYNTHETIC GRAPH DYNAMICS; 5.2 MOUSE GASTRULATION

  • The learned-ratio and histogram-based DAM variants perform comparably on synthetic settings and mouse gastrulation, supporting learned ratios as a scalable alternative when histograms are difficult to estimate.For synthetic results, the variants are comparable when pooled across changes in entropy strength, graph size, and sample budget.Source: §5.2 · p. 9, §5.1 · p. 9

6 DISCUSSION & FUTURE WORK

DAM selects a canonical minimum-action current from the prescribed graph geometry and mobility, but that current need not determine a unique microscopic history. Future work could optimize the marginal interpolation itself and extend the framework to conditional or unbalanced settings.

6 DISCUSSION & FUTURE WORK

  • Temporal marginals do not determine a unique microscopic history; DAM instead selects a minimum-action current conditioned on the prescribed graph geometry and mobility.That signed current can have multiple compatible Markov realizations, including the smooth positive split described here.Source: §6 · pp. 9–10
  • A discrete Wasserstein Lagrangian Flow extension would optimize interpolation between observed marginals using a parameterized path and an outer objective enforcing observation constraints.This would replace the predefined empirical interpolant; conditional and unbalanced extensions are also proposed as future directions.Source: §6 · p. 10
  • Unlike FEGF's single autonomous free-energy assumption, DAM allows a time-dependent potential and therefore covers more general prescribed marginal evolutions.The paper presents local density-ratio estimation as a possible route for transferring further Wasserstein variational methods to discrete population dynamics.Source: §6 · p. 10

ETHICS STATEMENT

The study uses synthetic data, an existing mouse gastrulation atlas, and samples from numerical surface transport; it collects no new biological data. Its geometric background frames discrete transport as a graph-based counterpart to continuous Wasserstein geometry and distinguishes fixed-path motion selection from path optimization.

ETHICS STATEMENT

  • The study uses synthetic data, existing mouse gastrulation atlas data, and samples from numerical surface transport; it recruited no human participants and collected no new biological data.The inferred dynamics are model-dependent descriptions and do not establish biological causation or clinical validity.Source: ETHICS STATEMENT · p. 10

A CONTINUOUS AND DISCRETE WASSERSTEIN GEOMETRY

  • On finite state spaces, graph gradients, mobility-weighted continuity equations, and kinetic energies provide a geometry analogous to continuous Wasserstein geometry on the positive probability simplex.The section presents this as geometric foundation for the paper's later development.Source: A CONTINUOUS AND DISCRETE WASSERSTEIN… · p. 16
  • The paper distinguishes optimizing a probability path globally from selecting a canonical velocity along a prescribed path, making the latter the geometric principle behind Action Matching.The graph construction then develops this fixed-path problem for admissible mobility and characterizes its canonical discrete tangent.Source: A CONTINUOUS AND DISCRETE WASSERSTEIN… · p. 16

A.1 OPTIMAL TRANSPORT AND ACTION MATCHING ON CONTINUOUS SAMPLE SPACES

The appendix develops continuous and discrete dynamical transport as minimum-action geometry. It distinguishes selecting the canonical motion along a fixed marginal path from optimizing the path between endpoints, then formulates the graph analogue using edge mobility.

A.1 OPTIMAL TRANSPORT AND ACTION MATCHING ON CONTINUOUS SAMPLE SPACES

  • The Benamou–Brenier formulation minimizes kinetic action over both probability paths and velocity fields connecting endpoint distributions.The continuity equation expresses conservation of probability mass along that motion.Source: §A.1 · p. 17
  • For a prescribed marginal path, Action Matching selects its canonical minimum-norm velocity rather than optimizing the path between endpoints.The weighted Helmholtz–Hodge decomposition identifies the gradient field as the unique minimum-norm representative of a tangent direction.Source: §A.1 · pp. 17–18
  • The continuous Action Matching identity expresses its objective through expectations under the path's marginals, enabling learning from marginal samples without observing velocities.The discrete construction follows the fixed-path principle but must additionally handle density-dependent edge mobility.Source: §A.1 · p. 19

A.2 DYNAMICAL OPTIMAL TRANSPORT ON DISCRETE SAMPLE SPACES

  • On a graph, states are vertices and velocities are edge fields; endpoint probabilities determine the edge mobility used in the continuity equation and kinetic energy.Different mobility choices induce different discrete transport geometries.Source: §A.2 · pp. 20–21
  • The discrete dynamical transport distance is distinct from static Kantorovich transport: it specifies infinitesimal graph motion and depends on the chosen mobility.That infinitesimal geometry is the one required for fixed-path Action Matching.Source: §A.2 · p. 21
  • For positive graph marginals, the tangent-space construction gives each tangent direction a unique minimum-energy gradient representative.The paper identifies connectedness and positive conductances as sufficient for this representation, while homogeneity enables ratio factorization.Source: §A.2 · pp. 20–21

A.3 WASSERSTEIN GEOMETRY ON THE PROBABILITY SIMPLEX

On a connected graph, discrete transport equips the positive probability simplex with a mobility-dependent geometry in which each marginal tangent has a unique minimum-energy gradient representation. The action identity then links learning that canonical motion to neighboring density ratios, which are normalization-free and determine the distribution when the graph is connected.

A.3 WASSERSTEIN GEOMETRY ON THE PROBABILITY SIMPLEX

  • The potential gradient is the unique minimum-energy edge velocity inducing a given marginal tangent; divergence-free alternatives leave the tangent unchanged but add kinetic energy.This makes the scalar potential a geometric selection principle, not merely a parameterization choice.Source: §A.3 · p. 23
  • The positive probability simplex carries a smooth Riemannian metric determined by the graph transport mobility, with tangent-space inner product represented by the mobility-weighted graph Laplacian.The metric is distinct from the Fisher–Rao geometry discussed in information geometry.Source: §A.3 · pp. 22–23
  • The discrete action-gap identity shows that minimizing the population objective recovers the canonical current, while positive 1-homogeneity reduces nonlinear mobility dependence to neighboring density ratios.This is the graph analogue of the fixed-path Action Matching identity (Neklyudov et al., 2023, Theorem 2.2).Source: §A.3 · p. 24

B.1 LOCAL DENSITY RATIOS AND DISCRETE SCORES

  • On a connected graph, a complete set of neighboring probability ratios determines the normalized distribution, while valid ratios must also satisfy cycle consistency.Connectedness determines masses relative to a reference state, and normalization fixes their common scale.Source: §B.1 · p. 26

B DISCRETE SCORES AND LOCAL DENSITY RATIOS

  • In DAM, neighboring ratios determine the density-dependent geometry used to learn the action potential, rather than directly parameterizing the population velocity as in discrete score-based diffusion.Related work treats these ratios as normalization-free statistics or discrete score-like quantities (Meng et al., 2022; Lou et al., 2024).Source: §B.1 · p. 27

B.2 DENSITY RATIOS IN DISCRETE ACTION MATCHING

Positive 1-homogeneity lets DAM express edge mobility through a sampled source state and its neighboring log-ratios, avoiding reconstruction of the full probability vector. The section also gives a ratio-free action estimator for arithmetic mobility and establishes consistency of a scalar-field softplus ratio estimator.

B.2 DENSITY RATIOS IN DISCRETE ACTION MATCHING

  • Positive 1-homogeneity factors edge mobility into the source probability and a function of one neighboring log-ratio, making the kinetic energy sample-computable.The directed neighborhood sum uses both orientations of each undirected edge.Source: §B.2 · p. 28
  • For arithmetic mobility, the action can be estimated from marginal samples without density ratios because endpoint sampling recovers each edge weight pi + pj.The Markov-rate construction still uses density ratios, so this action simplification does not remove density dependence from the rates.Source: §B.2 · p. 28
  • On a connected graph, the population softplus ratio loss is minimized by log-probabilities up to an additive constant, and the objective needs neither masses nor their normalization constant.This is a population consistency result, not a finite-data recovery guarantee.Source: §B.2 · p. 29
  • A scalar log-density field represents neighboring log-ratios consistently, preventing arbitrary model parameters from violating positivity, reciprocity, or cycle consistency.Only differences of the field matter, so an additive time-dependent constant is immaterial.Source: §B.2 · pp. 29–30

B.3 EMPIRICAL TRAINING AND MARKOV REALIZATION

Empirical training declares an interpolant between snapshots and estimates density ratios before fitting the action under frozen conductances. The resulting fields define graph-supported Markov dynamics, with exact path-law matching requiring exact ratios and the canonical potential.

B.3 EMPIRICAL TRAINING AND MARKOV REALIZATION

  • The declared interpolant supplies the intermediate-time distributions needed for the population action; it interpolates distributions, not matched individual trajectories.The training construction samples time, then a reference or snapshot mixture, then a state.Source: §B.3 · p. 30
  • Snapshot-only ratio training guarantees recovery only at observed knots under positive weights, simultaneous realizability, and global population minimization; reference mixing restores support but changes the target.Zero-count empirical states make unrestricted unsmoothed edge-ratio estimation problematic.Source: §B.3 · p. 31
  • Freezing the fitted ratio field fixes the effective conductances during action optimization, so the potential is learned under the geometry determined by those estimates.The conditional estimator identity does not establish unbiasedness of an optimized network or identification of the unknown intermediate population path.Source: §B.3 · pp. 31–32
  • The graph-supported generator has nonnegative off-diagonal rates only between neighbors and defines a valid law evolution through the forward equation.Trajectory simulation uses a time-discretization approximation when rates vary continuously.Source: §B.3 · p. 32
  • If ratios are exact and the potential is canonical, the simulated law matches the prescribed path; approximate fits do not guarantee that match.The result applies intervalwise to the piecewise-smooth construction.Source: §B.3 · p. 32

C.1 EXTENDED RELATED WORK

DAM is distinguished from methods that optimize transport paths, target distributions, or a shared energy: it selects a canonical graph velocity for a prescribed marginal path. The section also outlines cortical-modeling boundaries and the synthetic benchmark protocol.

C.1 EXTENDED RELATED WORK

  • Unlike graph inverse-JKO learning, which seeks one free-energy functional, DAM selects the minimum-action tangent of a prescribed path, potentially with explicit time dependence.The related work also distinguishes DAM from continuous-space inverse Wasserstein-flow approaches.Source: §C.1 · p. 33
  • DAM's neighboring density ratios determine marginal-dependent mobility, not velocity or reverse-diffusion dynamics; a separate potential represents the population dynamics.This contrasts with discrete score-based models that use such ratios for reverse-time dynamics.Source: §C.1 · p. 33
  • Related graph-transport and gradient-flow work studies distances, paths, or dynamics driven by a supplied objective, unlike DAM's inference of velocity from observed temporal marginals.The cited work includes graph-surface transport and samplers for prescribed targets.Source: §C.1 · pp. 33–34

C.2 CORTICAL PREDICTION FROM UNPAIRED OBSERVATIONS

  • Personalized cortical prediction is not established by unpaired state samples; it requires a justified conditional population model and registered common state labels.The proposed unpaired objective could complement paired supervision, but whether it improves next-visit accuracy remains untested.Source: §C.2 · p. 35
  • Cortical biomarker extensions would need unbalanced transport for changing total burden and evaluation on held-out longitudinal subjects.Normalizing PET maps discards absolute burden, and fitting observed times does not establish extrapolation or clinical benefit.Source: §C.2 · p. 35

D ADDITIONAL EXPERIMENTAL DETAILS

  • The additional experiments cover synthetic reconstruction and gastrulation interpolation, alongside surface transport on sphere and hand meshes.The synthetic benchmark uses five sampled marginals across eleven graph families, with fitting and selection based on sampled observations.Source: D ADDITIONAL EXPERIMENTAL DETAILS · p. 35, §D.1 · pp. 35–36

D.2 TRAINING AND BASELINE SPECIFICATION

The training specifications give DAM's shared fitting procedure and distinguish the synthetic and gastrulation FEGF baselines. Figure 4 summarizes one-factor synthetic sweeps and their variability.

D.2 TRAINING AND BASELINE SPECIFICATION

  • Both DAM variants use the same smoothstep interpolant with uniform-reference mixing and four sampled neighbors per state; histogram and learned-ratio variants differ in how they obtain log-density information.They share action architectures, initial weights, training streams, update budgets, and selection rules.Source: §D.2 · p. 36
  • Figure 4 varies one factor at a time from the base synthetic setting, reporting mean Hellinger errors across eleven graph families and sample SD across three seeds.Smaller sample budgets reuse prefixes of the same draws.Source: Figure 4 · p. 37
  • Synthetic FEGF learns edge differences and an entropy coefficient, whereas gastrulation FEGF learns a time-indexed state potential without a trained entropy coefficient.Gastrulation FEGF interpolates its potential in time; both variants select checkpoints using observed-time validation TV.Source: §D.2 · p. 37
  • FEGF uses stationary-relative logarithmic mobility, while logarithmic DAM uses probability-coordinate mobility; these geometries differ when the stationary distribution is nonuniform.The comparison evaluates marginal prediction under the respective geometries, and matched optimizer budgets do not imply equal prior tuning effort.Source: §D.2 · p. 37

D.3 SYNTHETIC RESULTS AND ABLATIONS

DAM outperforms FEGF in the reported synthetic and gastrulation marginal-error comparisons, while surface experiments test supplied transport paths and reveal estimator and runtime trade-offs. These results are bounded by the protocols and settings reported for each experiment.

D.3 SYNTHETIC RESULTS AND ABLATIONS

  • Both DAM variants have lower mean Hellinger errors than FEGF across all base-setting graph families and reported synthetic sweep averages.Learned-ratio and histogram variants perform comparably, with no consistent learned-ratio advantage across metrics.Source: §D.3 · p. 37
  • Harmonic mobility gives DAM its lowest mean gastrulation errors among the four tested choices, a descriptive comparison without independent mobility selection.The study separately refits each variant for each mobility while holding data, smoothing, interpolation, architecture, and optimizer budgets fixed.Source: §D.3 · p. 37, Table 3 · p. 38

D.4 MOUSE GASTRULATION

  • With harmonic mobility, both DAM variants achieve lower mean Hellinger and TV errors than FEGF and persistence on omitted gastrulation stages.Differences between learned ratios and histograms depend on the stage; the folds retain later observations, and error bars reflect sampling and optimization seeds rather than biological replicates.Source: §D.4 · p. 39, Figure 6 · p. 39

D.5 SPHERE AND HAND SURFACE TRANSPORT

  • On the two supplied sphere and hand paths, both DAM variants achieve lower omitted-time TV than tuned FEGF, with histogram DAM competitive or better.The experiments use paired vertex trajectories and one seed; transfer to new endpoint pairs or unseen meshes remains untested.Source: §D.5 · p. 39, Table 4 · p. 39

D.6 SURFACE TEACHER, SAMPLING AND LEARNING PROTOCOL

  • Surface DAM trains its potential from paired integration-by-parts samples; histogram DAM accesses stored interpolated histograms at inference, while learned-ratio DAM combines local and global-pair losses.Surface FEGF and DAM use different validation information, geometries, and tuning budgets.Source: §D.6 · p. 40

D.7 MESH ESTIMATOR AND INTEGRATION DIAGNOSTICS

  • In the cumulative sphere study, global-pair ratio estimation improves density and endpoint errors but worsens mean trajectory TV, so better density estimates do not uniformly improve rollouts.For the combined model, midpoint freezing reduces substeps needed to meet the refinement tolerance from 128 to 32, with little accuracy change.Source: §D.7 · p. 40

D.8 ACCURACY VERSUS COMPUTATIONAL COST

  • The reported CPU timings do not establish equal-accuracy or end-to-end speedup on new transport problems.The methods have unequal errors, histogram DAM accesses stored intermediate distributions, and tuned FEGF training time was not measured.Source: §D.8 · p. 43, Figure 9 · p. 43
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