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Gaussian Flow Dynamics: Simulation-Free Neural SDE Learning Beyond One-Time Marginals

Grigory Bartosh, Christian A. Naesseth

arXiv:2610.04390v1stat.MLcs.LG

TL;DR

Simulation-free latent SDE methods parameterize one-time marginals, but those marginals do not determine the process’s temporal structure and fixed gauges restrict the posterior family. Gaussian flow dynamics expose this freedom for Gauge Matching, which learns the gauge at quadratic cost. It reduces the conjugate-setting gap to within about a nat of Helmholtz-SDE, matches it on nonlinear systems, and applies to state-dependent noise.

  • Problem

    One-time marginals do not determine latent SDE posterior dynamics, while implicit fixed-gauge choices restrict the available posterior family.

    Source: §1 · p. 1, §2.2 · p. 3

  • Method

    Gaussian flow dynamics expose the marginal-preserving gauge, and Gauge Matching learns it using the SDE Matching objective.

    Source: §1 · p. 2, §7 · p. 9

  • Results

    Gauge Matching reduces conjugate-setting gaps to 0.8–3.9 nats versus 14–26 nats for fixed gauges, matches Helmholtz-SDE on nonlinear systems, and extends to state-dependent noise.

    Source: §5.1 · p. 7, §7 · p. 9

  • Takeaways & Limitations

    The gauge can be learned beyond one-time marginals at quadratic cost, including for state-dependent diffusion where Helmholtz-SDE does not apply.

    Source: §1 · p. 2, §7 · p. 9

    At observation times, posterior means and Cholesky factors have kinks and the gauge jumps, so a continuously differentiable time-network approaches but does not attain the zero-infimum KL.

    Source: §4 · p. 6

Abstract

from arXiv · show

Simulation-free training of latent Stochastic Differential Equations (SDEs) relies on a variational posterior process whose one-time marginals are tractable, typically Gaussian. Such marginals, however, do not determine the underlying dynamics: many processes share the same marginals while differing in their temporal structure, and existing parameterizations fix this structure implicitly, which restricts the posterior family and biases the learned model. We introduce Gaussian flow dynamics, which construct stochastic processes directly from smoothly evolving Gaussian marginals while making the marginal-preserving, or gauge, degrees of freedom explicit and parameterizable. The construction admits state-dependent diffusion coefficients and recovers every linear SDE with additive noise and a non-degenerate Gaussian initial distribution. Building on it, we propose Gauge Matching, a simulation-free method for latent SDE learning that combines Gaussian flow dynamics with the SDE Matching objective. Gauge Matching costs at most quadratically in the latent dimension per step, like SDE Matching, but learns the temporal structure of the posterior beyond its one-time marginals. It comes within a nat of Helmholtz-SDE, which computes the gauge from the prior Jacobian at cubic cost, on the linear benchmark where the exact posterior is known, matches it on nonlinear systems, and applies where Helmholtz-SDE does not, to state-dependent noise.

1 INTRODUCTION

Simulation-free latent SDE learning avoids integrating the posterior process at every gradient step by parameterizing its one-time marginals. Because those marginals do not determine temporal dynamics, fixed implicit gauges can restrict the posterior family; Gaussian flow dynamics make that freedom explicit for Gauge Matching.

1 INTRODUCTION

  • Simulation-free training makes gradient-step cost independent of series length, but one-time marginals leave the posterior process’s temporal structure undetermined.Existing parameterizations implicitly choose among processes with the same marginals, potentially restricting the variational family.Source: §1 · p. 1
  • Gaussian flow dynamics expose the marginal-preserving gauge as a parameterized degree of freedom, and Gauge Matching selects it through the SDE Matching objective.The construction supports state-dependent diffusion at quadratic cost and recovers every linear SDE with additive noise and non-degenerate Gaussian initialization.Source: §1 · p. 2
  • Gauge Matching removes most of the fixed-gauge gap in the linear-Gaussian setting, matches Helmholtz-SDE on nonlinear systems at lower cost, and extends to state-dependent noise.Helmholtz-SDE requires the prior-drift Jacobian, has at least cubic cost, and requires state-independent noise.Source: §1 · pp. 1–2

2 BACKGROUND

The background connects evolving distributions to their generating dynamics, then describes variational training of latent SDEs and how simulation-free methods estimate its objective from tractable posterior marginals.

2.1 DYNAMICS AND THEIR MARGINALS

  • An ODE flow transports an initial density through a continuity equation, while adding diffusion gives an SDE whose marginal densities satisfy the Fokker–Planck–Kolmogorov equation.The passages introduce the flow map and the role of divergence in the density evolution.Source: §2.1 · p. 2

2.2 SIMULATION-FREE LATENT SDE LEARNING

  • Latent SDE training uses variational inference with a data-dependent posterior drift and the prior’s shared diffusion coefficient.The posterior process approximates the latent trajectory distribution when the marginal likelihood is intractable.Source: §2.2 · p. 2
  • The ELBO includes a path-space KL term that depends on more than the posterior marginals, alongside a term depending on the initial-state distributions.The shared diffusion makes the path-measure KL finite under the stated setup.Source: §2.2 · pp. 2–3
  • When posterior marginals can be sampled at arbitrary times, simulation-free estimation samples a random time and state; SDE Matching constructs them from an affine noise map and converts its flow to an SDE.The conversion uses the prior diffusion while preserving the marginals.Source: §2.2 · p. 3
  • Gaussian posterior marginals still leave many compatible processes, so SDE Matching implicitly selects one drift among them.Making that selection explicit motivates the next section’s treatment of the gauge.Source: §2.2 · p. 3

3 GAUSSIAN FLOW DYNAMICS

Gaussian flow dynamics construct processes from evolving Gaussian marginals and expose the freedom to change their temporal dynamics without changing those marginals. The resulting affine gauge parameterization is complete for the stated linear-SDE family and supports quadratic-cost evaluation.

3 GAUSSIAN FLOW DYNAMICS

  • A Gaussian flow diffusion combines an affine Gaussian flow with a chosen diffusion coefficient while retaining the Gaussian marginals defined by the flow map.With state-independent diffusion the process is Gaussian; with state-dependent diffusion its joint distributions are generally not Gaussian, although its marginals remain Gaussian.Source: §3 · p. 3
  • Every linear SDE with continuous affine drift, state-independent noise, and a non-degenerate Gaussian initial distribution has a Gaussian flow diffusion representation.The representation is unique up to relabeling the initial Gaussian noise.Source: §3 · p. 4
  • Factoring the flow’s square root into a Cholesky factor and an orthogonal factor separates the marginal parameters from the gauge that changes the dynamics.The orthogonal factor enters through a skew-symmetric generator, and affine marginal-preserving fields have exactly this form.Source: §3 · p. 4
  • For fixed Gaussian marginals and diffusion, varying the orthogonal factor spans the compatible processes whose drift differs from the Cholesky-gauge drift by an affine field.For state-independent diffusion, this covers all Gaussian processes with those marginals and diffusion.Source: §3 · p. 4
  • Directly parameterizing the mean, Cholesky factor, and gauge generator evaluates the drift in O(d^2) when the diffusion operations have quadratic cost.This cost also applies to the stated state-dependent diffusion structures.Source: §3 · p. 4

4 GAUGE MATCHING

Gauge Matching leaves the Gaussian posterior’s gauge free and optimizes it with the SDE Matching objective, retaining simulation-free training. The objective selects a unique gauge at each time, with theoretical coverage of the conjugate posterior and a limitation at observation-time discontinuities.

4 GAUGE MATCHING

  • Gauge Matching jointly learns posterior marginals and their gauge using the unchanged SDE Matching objective and simulation-free estimator.Each iteration uses a sampled time for the path-space KL term and a sampled observation index for reconstruction, with O(d^2) linear algebra.Source: §4 · p. 5
  • The path-space KL term alone selects the process among those with the learned marginals, and its strictly convex quadratic dependence on the skew-symmetric gauge yields a unique optimum.For state-independent diffusion, the optimum is characterized by the equation involving the expected prior-drift Jacobian.Source: §4 · p. 5
  • The selected gauge reproduces the rotational component of prior dynamics as seen from the posterior marginals; for an Ornstein–Uhlenbeck spiral, it makes the posterior process coincide with the prior.The Cholesky gauge is optimal when Σ^-1 times the averaged Jacobian residual is symmetric; with isotropic diffusion, the symmetric gauge is optimal when the expected prior-drift Jacobian is symmetric.Source: §4 · p. 5
  • Helmholtz-SDE’s order-one correction uses the prior Jacobian at the mean and matches Gauge Matching when that Jacobian is affine, but otherwise optimizes the linearized prior process.Gauge Matching instead minimizes the exact objective over the same affine-gauge family, at O(d^2) cost without prior-drift derivatives; it also supports state-dependent diffusion.Source: §4 · pp. 5–6
  • In the conjugate setting, Gauge Matching’s family can represent the true Gaussian posterior between observations, with infimum KL divergence zero.At observation times, kinks in the mean and Cholesky factor and jumps in the gauge mean the infimum is approached rather than attained by a continuously differentiable neural network.Source: §4 · p. 6

5 EXPERIMENTS

Experiments compare Gauge Matching with fixed-gauge and Helmholtz-SDE posterior variants across linear and nonlinear systems. They test whether learning the gauge improves posterior-process fit and how performance scales with latent dimension.

5.1 LINEAR DYNAMICS

  • 14–26 nats of KL gap for fixed gauges falls to 0.8–3.9 with Gauge Matching, versus 0.5–2.8 for exact-gauge Helmholtz-SDE.As rotation speed increases, the fixed-gauge gap rises to 22–23 nats while the learned-gauge gap stays below 0.8.Source: Figure 2 · p. 7, §5.1 · pp. 7–8
  • Without observations, the learned gauge recovers the stationary Ornstein–Uhlenbeck spiral, reducing path-space KL to 0.007 nats over [0, 3], while the fixed gauge shrinks variance from 2.5 to 0.080 at about 18 nats.The rotation angle approaches ωt; the stationary marginals alone carry no information about ω.Source: §5.1 · p. 7, Figure 1 · p. 7

5.2 NONLINEAR DYNAMICS

  • 53 nats per trial for Gauge Matching and Helmholtz-SDE versus 75–78 for fixed gauges on the noisy Lorenz attractor; their prior autocorrelation error is 0.05–0.06 versus 0.11–0.13.The table reports held-out results, and the gains are attributed to the gauge rather than covariance.Source: Table 2 · p. 8, §5.2 · p. 8
  • On the Stuart–Landau oscillator with known nonlinear drift, the learned gauge has 10 times lower gauge excess than the order-one correction and a 1.6 ± 0.3-nat lower negative ELBO on every seed.Source: §5.2 · p. 8
  • In original Lotka–Volterra coordinates, Gauge Matching reduces period error from 22% to 13% and improves all three metrics on every seed; for d = 8, it lowers negative ELBO by 6.6 ± 0.8 or 4.8 ± 0.3 nats, depending on whether the prior is known.Helmholtz-SDE does not support the state-dependent noise in the original coordinates.Source: §5.2 · p. 8
  • On motion capture, Gauge Matching’s test mean squared error is 4.52 ± 0.29, on par with SDE Matching’s 4.50 ± 0.32 and behind the simulation-based latent SDE’s 4.03 ± 0.20.The posterior concentrates on dense, nearly noise-free observations, leaving little for the gauge to add.Source: §5.2 · pp. 8–9
  • Across latent dimensions 2–64, the learned-gauge gap stays within a factor of two of exact Helmholtz-SDE and 4–27 times below fixed gauges; its per-iteration cost is 1.05–1.5 times SDE Matching’s for d ≤128.The learned gauge adds d(d −1)/2 network outputs; Helmholtz-SDE uses d forward-mode passes and an O(d^3) solve.Source: §5.2 · p. 9

6 RELATED WORK

Prior work spans simulation-based latent-SDE inference, simulation-free methods that fix or otherwise restrict the gauge, and related approaches to marginal-preserving dynamics. The paper distinguishes its learned gauge from those choices and from adjacent work in generative modeling.

Variational Inference for Latent SDEs

  • Simulation-based variational inference parameterizes the posterior drift and optimizes the gauge implicitly, using moment ODEs for affine drift or simulation and differentiation for neural drift.The cited approaches include Archambeau et al. (2007a;b), Duncker et al. (2019), and Tzen & Raginsky (2019).Source: §6 · p. 9

Simulation-Free Methods and the Gauge

Marginal-Preserving Fields

  • Related generative-modeling work studies or learns dynamics between shared marginals, whereas this paper treats the divergence-free freedom as a coordinate of the variational family.The passage also relates this freedom to probability-flow ODEs, flow matching, and Helmholtz–Hodge decompositions.Source: §6 · p. 9

7 CONCLUSION

The conclusion presents Gaussian flow dynamics and Gauge Matching as a way to parameterize marginal-preserving freedom explicitly, including for state-dependent diffusion. It reports performance across linear and nonlinear settings and identifies evaluation and non-Gaussian marginals as future work.

7 CONCLUSION

  • Gaussian flow dynamics expose marginal-preserving freedom as a skew-symmetric generator at quadratic cost, including for state-dependent diffusion coefficients.Gauge Matching uses the SDE Matching objective to select the generator.Source: §7 · p. 9
  • On the conjugate benchmark, the learned gauge removes most of the gap of fixed gauges; on nonlinear systems, it matches Helmholtz-SDE at lower cost up to d = 64.It improves on Helmholtz-SDE when the prior drift is known and strongly nonlinear, and extends to state-dependent noise where Helmholtz-SDE does not apply.Source: §7 · p. 9
  • A state-dependent gauge generator extends the construction to general nonlinear marginal-preserving fields beyond Helmholtz-SDE's polynomial gauges and state-independent diffusion.Evaluation of this extension and non-Gaussian marginals are left for future work.Source: §7 · p. 9

A PROOFS FOR SECTION 3

The appendix proves that Gaussian flow dynamics and Gaussian flow diffusion have the prescribed Gaussian marginals. It establishes this through the continuity equation for the flow and the FPK equation for the diffusion.

A PROOFS FOR SECTION 3

  • The appendix proves that Gaussian flow and Gaussian flow diffusion have marginals N(z | m(t), S(t)S(t)⊤).These results are stated as Propositions 5 and 6.Source: A PROOFS FOR SECTION 3 · p. 13

A.1 GAUSSIAN FLOW AND THE CONTINUITY EQUATION

  • The Gaussian flow satisfies the continuity equation for the prescribed Gaussian density because the trace, linear, and quadratic terms cancel.The vector field is affine in z, and its divergence is the trace of its linear part.Source: §A.1 · p. 13

A.2 GAUSSIAN FLOW DIFFUSION AND THE FPK EQUATION

  • For Gaussian flow diffusion, the drift corrections cancel the diffusion term in the FPK equation, reducing it to the continuity equation satisfied by the Gaussian flow.Thus the prescribed Gaussian density satisfies the FPK equation for the diffusion.Source: §A.2 · pp. 13–14

A.3 PROOF OF THEOREM 1

Theorem 1 establishes that, for continuous linear-SDE coefficients and an invertible initial factor, Gaussian flow diffusion uniquely represents the stated linear SDE. The proof constructs the covariance factor using a Cholesky factor and an orthogonal factor, then solves for the mean.

A.3 PROOF OF THEOREM 1

  • For continuous A, u, and g with S0 in GL(d), unique continuously differentiable m and S make Gaussian flow diffusion the linear SDE with flow map m + Sε and diffusion g.The initial values are m(0) = m0 and S(0) = S0.Source: §A.3 · p. 14
  • The proof constructs S from the linear SDE covariance using its Cholesky factor and an orthogonal factor, yielding an invertible S with the required initial value.The orthogonal factor remains orthogonal because the gauge generator is skew-symmetric.Source: §A.3 · pp. 14–15
  • After establishing uniqueness of S, the proof obtains the mean m as the unique continuously differentiable solution of a linear ODE.The cited argument relies on continuity of A and u.Source: §A.3 · p. 15

A.4 PROOF OF PROPOSITION 2

The proof characterizes affine vector fields that are divergence-free with respect to a Gaussian density. Such fields can be added to the drift without changing the marginal laws, for state-dependent or state-independent diffusion.

A.4 PROOF OF PROPOSITION 2

  • An affine field is divergence-free with respect to a Gaussian density exactly when it is LΩL^-1(z − m), with Ω skew-symmetric.Equivalently, its constant term is c = −Mm and L^-1ML is skew-symmetric.Source: §A.4 · pp. 15–16
  • The proof derives this characterization by expressing the divergence-free condition in whitened coordinates and setting each homogeneous part of a quadratic polynomial to zero.The quadratic, linear, and constant parts constrain the linear transformation, affine offset, and trace, respectively.Source: §A.4 · pp. 15–16
  • Adding a Gaussian-density-divergence-free field to the drift leaves the marginal laws unchanged for any diffusion coefficient, including state-dependent diffusion.The condition on the added field does not involve the diffusion coefficient.Source: §A.4 · p. 16

B FIXED GAUGES OF EXISTING METHODS

Existing Gaussian-marginal methods select a fixed gauge, while Gauge Matching optimizes it explicitly. State-dependent skew-symmetric generators extend this framework to nonlinear marginal-preserving fields.

B FIXED GAUGES OF EXISTING METHODS

  • The square-root gauge depends on how the covariance square root is parameterized; with diagonal covariance, it coincides with the Cholesky gauge.Factoring S = LO separates the marginals from the gauge, unlike directly parameterizing a general S.Source: B FIXED GAUGES OF EXISTING METHODS · p. 16
  • The symmetric gauge depends on the diffusion coefficient and does not extend to state-dependent diffusion, because no state-independent gauge makes its drift Jacobian symmetric for all states.Source: B FIXED GAUGES OF EXISTING METHODS · p. 16
  • Gauge Matching treats the gauge as a free function, whereas existing Cholesky, square-root, and symmetric methods choose it deterministically from the marginals and diffusion coefficient.When marginals are stationary and the diffusion and covariance commute, all three fixed gauges give K = 0.Source: B FIXED GAUGES OF EXISTING METHODS · pp. 16–17

B.1 STATE-DEPENDENT GAUGE GENERATORS

  • A state-dependent skew-symmetric generator induces a marginal-preserving field and extends the construction from affine gauges to nonlinear fields.The resulting whitened flow preserves the standard Gaussian, so sampling from the Gaussian marginals remains simulation-free even though joint distributions across times need not be Gaussian.Source: §B.1 · pp. 17–18
  • A single network can represent higher-order marginal-preserving corrections, including settings with state-dependent diffusion that the cited Helmholtz construction excludes.Exact divergence evaluation costs d forward-mode passes; an unbiased directional-derivative estimate uses one Jacobian–vector product, with an upward path-space KL estimator bias that vanishes as the number of directions grows.Source: §B.1 · p. 18

C.1 PROOF OF PROPOSITION 3

The proposition establishes a unique optimal skew-symmetric gauge by showing the objective is a strictly convex quadratic. Its consequences characterize state-dependent diffusion, fixed-gauge optimality, and the relation to the order-one Helmholtz correction.

C.1 PROOF OF PROPOSITION 3

  • Strict convexity of the gauge objective yields a unique minimizer for any positive-definite diffusion coefficient, including state-dependent coefficients.The quadratic part vanishes only at the zero skew-symmetric matrix, making it positive definite on the gauge space.Source: §C.1 · pp. 18–19

C.2 CONSEQUENCES OF PROPOSITION 3

  • For state-dependent diffusion, the optimal gauge remains the solution of a linear system, but its coefficients require expectation estimates because the diffusion inverse depends on state.Gauge Matching sidesteps estimating these coefficients by learning the gauge directly.Source: §C.2 · p. 19
  • A fixed gauge is optimal exactly when its stationarity condition holds; under isotropic diffusion, a symmetric gauge is optimal when the expected prior-drift Jacobian is symmetric.Source: §C.2 · p. 20
  • For the stationary linear example, the optimal gauge recovers the prior process and makes the path-space KL term vanish.Source: §C.2 · p. 20
  • The order-one Helmholtz correction is the optimal gauge for the linearized prior, and it matches the optimal gauge when the expected prior Jacobian equals its value at the mean.The cited higher-order corrections lie outside the constant-gauge family and correspond to state-dependent generators.Source: §C.2 · p. 20

C.3 PROOF OF COROLLARY 4

For a linear prior and linear-Gaussian observations, the true posterior is a Gaussian flow diffusion between observation times. Smoothing its parameters and gauge near observation times gives admissible Gauge Matching processes arbitrarily close in KL.

C.3 PROOF OF COROLLARY 4

  • For a linear prior and linear-Gaussian observation model, the true posterior is a Gaussian flow diffusion between observation times, and Gauge Matching processes approach it in KL.Source: §C.3 · p. 20
  • The true posterior drift is affine and continuous on each closed interval between observations, while its coefficients can jump when the future-observation set changes.Its smoothing marginals remain continuous, and their covariance is positive definite.Source: §C.3 · p. 21
  • Across an observation time, the posterior gauge generally jumps; under isotropic diffusion and isotropic added observation precision, it does not jump.Source: §C.3 · pp. 21–22
  • Smoothing the true posterior marginals and gauge in shrinking neighborhoods of observation times produces admissible processes whose KL divergence tends to zero.The bound is proportional to the neighborhoods’ total length, which shrinks with their width.Source: §C.3 · p. 22

D EXPERIMENTAL DETAILS

The experiments use a shared amortized posterior architecture and common training setup, with baselines distinguished by their gauge parameterization. The appendix defines the reported gap and gauge-excess metrics.

D EXPERIMENTAL DETAILS

  • Gauge Matching’s posterior network outputs the mean, Cholesky factor, and skew-gauge entries, with the gauge produced by a separate head.A GRU processes observations, followed by a softmax-weighted embedding and an MLP.Source: D EXPERIMENTAL DETAILS · p. 23
  • The fixed-gauge baselines omit the gauge head, while Helmholtz-SDE computes a gauge from the prior Jacobian and solves a Sylvester equation at each evaluation.The Jacobian uses d forward-mode passes; the solve uses a Kronecker-sum formula for d ≤ 4 and an eigendecomposition for larger dimensions.Source: D EXPERIMENTAL DETAILS · p. 23
  • Unless otherwise stated, experiments use five seeds and report the mean with twice the standard error.Training uses Adam, gradient clipping at norm 5, and one sampled time, Gaussian noise, and observation index per trial and iteration.Source: D EXPERIMENTAL DETAILS · p. 23
  • The reported gap is KL divergence to the true posterior, while gauge excess is J(Ω) − J(Ω⋆) in nats per unit time and is nonnegative.Gauge excess is zero exactly when the method’s gauge is optimal for its own marginals.Source: D EXPERIMENTAL DETAILS · p. 23

D.1 LINEAR DYNAMICS

The linear benchmarks show that learning the gauge narrows the posterior-process gap left by fixed gauges, while scaling and identifiability affect how the methods compare. The experiments also expose training and observation-model failure modes that can obscure gauge effects.

D.1 LINEAR DYNAMICS

  • For α = 0.2 and ω = 2π, the fixed-gauge rate is 6.09 nats per unit time at its minimizing variance, versus 98.7 at stationary variance; the learned gauge’s rate there is zero.The isotropic restriction applies only to the closed-form calculation; numerical minimization confirms its optimum is global even when the trained model can learn general marginals.Source: §D.1 · p. 24
  • At ω = 2π, fixed-gauge family optima are 12–28 nats, versus 0.02–2.2 nats for gauge-corrected families at σ ≥0.3; at σ = 0.05, training falls short of family optima.At ω = 0, the symmetric gauge coincides with Helmholtz-SDE, while the square-root gauge retains a gap when the posterior covariance is anisotropic.Source: §D.1 · p. 25, Figure 3 · p. 25, Figure 4 · p. 26
  • With the generative model known, Gauge Matching’s gap rises from 0.9 nats at d = 2 to 94 at d = 64, while fixed gauges reach 381–496 at d = 64.Its covariance gap is 0.9–32 against 24–393 for fixed gauges, removing 89–97% of their covariance gap at every dimension.Source: §D.1 · pp. 26–27, Figure 5 · p. 27
  • With a learned generative model and partial observations, all five posterior processes reach the same negative ELBO within seed noise for d ≤16.The fitted prior recovers little of the true rotation, leaving it close to a gradient field where every gauge is near-optimal.Source: §D.1 · p. 27
  • GPU iteration costs are 1.05–1.5 times SDE Matching for Gauge Matching, versus 2.2 and 3.2 times for Helmholtz-SDE at d = 64 and 128.On CPU from d = 32, the corresponding overheads are 1.4–1.7 times for Gauge Matching and 6.4–7.3 times for Helmholtz-SDE.Source: §D.1 · p. 27, Figure 5 · p. 27
  • At d = 8 in the identifiable setting, Gauge Matching is 0.5–3 nats above Helmholtz-SDE, while fixed gauges are 3–9 and 4–7 nats above it.The two learned gauges have gauge excess of 0.01–0.02 nats per unit time, versus 0.54–0.80 for fixed gauges.Source: Gauge Matching · p. 28, Figure 6 · p. 29
  • At d = 32 after 6 · 10^4 iterations, SDE Matching is 24–27 nats below Helmholtz-SDE and 9–26 below Gauge Matching on every seed.At d = 16, diagonal Gauge Matching is 8–9 nats below Helmholtz-SDE on every seed; at d = 32, its gauge head diverges near the interval’s end on one seed.Source: Gauge Matching · p. 28
  • When the observation model is learned, all four full-covariance posteriors converge to rank-deficient observation matrices, with reconstruction terms of 650 and 1950 nats at d = 16 and 32.The diagonal posterior’s corresponding reconstruction terms are 218 and 660 nats; this collapse explains the full-covariance lag at d ≥32 and is not a gauge effect.Source: §D.1 · p. 27, Gauge Matching · p. 28

D.2 NONLINEAR DYNAMICS

This section examines Gauge Matching on nonlinear systems and motion-capture data, comparing learned gauges with fixed-gauge baselines across different observation and model settings. It reports reduced gauge excess and improved or similar learned dynamics, with results varying by observation sparsity, posterior covariance, model knowledge, and noise structure.

Noisy Lorenz Attractor

  • At ∆t = 0.5, the Lorenz fixed-gauge deficit falls to −0.5 to +4.8 nats, while all four posterior processes have autocorrelation error 0.23–0.25.With eleven observations per trial, the fitted prior drift no longer recovers within-lobe rotation; the optimal gauge at fixed-gauge marginals shrinks from ∥Ω⋆∥≈12 to 0.3–4.Source: §D.2 · p. 30, Figure 7 · p. 31, Table 3 · p. 31
  • For Lorenz observations with ∆t ≤0.25 and σ ≤0.3, fixed gauges trail Helmholtz-SDE by 20–25 nats, while Gauge Matching is within −0.7 to +1.8 nats.The fixed-gauge deficit comes from a 25–29-nat increase in path-space KL, partly offset by a 1–6-nat reconstruction-term difference. Rare covariance collapse makes the Helmholtz-SDE Sylvester system nearly singular, producing extreme gauges and path-space KL values.Source: §D.2 · p. 30, Table 3 · p. 31, Figure 7 · p. 31

Known Nonlinear Drift: Stuart–Landau Oscillator

  • At κ = 3, Gauge Matching’s negative ELBO is 1.6 ± 0.3 nats lower than Helmholtz-SDE’s, and its gauge excess is 0.014 ± 0.003 versus 0.135±0.006 for the order-one correction.At κ = 2, the gauge excesses are 0.028 ± 0.003 for Gauge Matching and 0.052 ± 0.004 for the order-one correction, while the ELBO difference is 0.1 ± 0.8 nats.Source: §D.2 · p. 32, Table 4 · p. 33, Figure 9 · p. 33

State-Dependent Noise: Stochastic Lotka–Volterra

  • For Lotka–Volterra in original coordinates, where Helmholtz-SDE does not apply, Gauge Matching improves all three metrics over the fixed gauge on every seed and lowers gauge excess to 0.003 from 0.081 nats per unit time.In log coordinates, gauge excess is 0.215±0.006 for the fixed gauge, 0.000 for Helmholtz-SDE, and 0.004 ± 0.001 for Gauge Matching.Source: §D.2 · pp. 33–34, Table 5 · p. 34

State-Dependent Noise in Higher Dimension

  • With the d = 8 rotating Ornstein–Uhlenbeck generative model known, Gauge Matching gains 6.6 ± 0.8 nats over SDE Matching and 6.9 ± 0.6 over the Cholesky gauge.Gauge excess is 1.29–1.40 nats per unit time for fixed gauges versus 0.041–0.049 for Gauge Matching, whose gauge norm matches the optimal gauge’s at its marginals.Source: Table 6 · p. 36, §D.2 · p. 36
  • With the d = 8 generative model learned, Gauge Matching gains 4.8±0.3 nats over SDE Matching and 5.1 ± 0.5 over the Cholesky gauge, while autocorrelation error drops from 0.072 to 0.040.Gauge excess is 0.85–0.88 for fixed gauges versus 0.040–0.046 for learned gauges; the learned diffusion has relative root-mean-square error 0.15 for all four posterior processes.Source: Table 6 · p. 36, §D.2 · p. 36

Ablation: Gauge versus Covariance

  • The ablation separates learned gauge from covariance shape by adding Gauge Matching with diagonal Lϕ and learned skew-symmetric Ωϕ to the design alongside SDE Matching and the full-covariance Cholesky gauge.The diagonal-Lϕ variant uses marginal means and positive standard deviations, while its gauge outputs come from the same posterior network.Source: §D.2 · p. 37

Results

  • On Lorenz, diagonal-covariance learned-gauge matching reaches 55.3 ± 1.9 nats versus 77.9 (SDE Matching), 75.5 (Cholesky), and 53.2 (Gauge Matching), with the gauge explaining 22.5 of 24.7 nats.The covariance-shape contribution is 2.2 ± 1.8 nats, entirely in the path-space KL term; gauge excess is 0.022 ± 0.001.Source: §D.2 · p. 37
  • On Stuart–Landau, the variant is within 0.4 ± 0.2 nats of Gauge Matching, while on state-dependent-noise Lotka–Volterra it matches Gauge Matching and beats Cholesky on every seed.For Lotka–Volterra, paired differences are within 0.02 on all three metrics over 10 seeds; Stuart–Landau gauge excess is 0.012 ± 0.003, while Cholesky remains with fixed gauges at 64.3.Source: §D.2 · p. 37
  • In the conjugate setting, the variant’s gaps versus Gauge Matching are 2.4, 2.6, 2.9, and 8.6 nats at N = 5, 10, 20, and 50, versus 19–26 nats for SDE Matching.The learned-gauge models have the same mean-part gap, and the difference is in the covariance part; diagonal covariance cannot represent the rotating process’s anisotropic covariance, especially at large N.Source: §D.2 · p. 37

Motion Capture Data

  • On motion capture, Gauge Matching’s test mean squared error is 4.52 ± 0.29, compared with 4.50 ± 0.32 for SDE Matching and 4.03 ± 0.20 for Li et al. (2020).With 300 nearly noise-free observations per sequence, the posterior concentrates on the trajectory and path-space KL is small for every gauge, so the experiment tests the additional 15 gauge outputs without sacrificing accuracy.Source: §D.2 · p. 37

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