Divergence-Free Gaussian Reference Measure

• The divergence-free Gaussian measure is a probability measure defined on a subspace of vector fields where the incompressibility condition is inherently satisfied.
• It is constructed via a curl-pushforward of a scalar Gaussian or through divergence-free covariance kernels in bounded domains, ensuring structural consistency.
• This framework underpins generative modeling and interpolation in fluid dynamics by integrating spectral methods and kernel-based Gaussian processes for accurate incompressible flow representation.

A divergence-free Gaussian reference measure is a Gaussian probability measure whose support lies in a divergence-free subspace of vector fields, so that almost every sample path satisfies the incompressibility constraint $\nabla \!\cdot u = 0$. In the recent literature, this object appears in two closely related settings: as a physically admissible prior for generative modeling of incompressible flows on the torus, constructed by pushing a scalar Gaussian through a curl operator (Li et al., 25 Mar 2026), and as the law of a vector-valued Gaussian process on a bounded domain whose covariance kernel is divergence-free by construction (Gia et al., 16 Nov 2025). Across both settings, the central purpose is structural rather than penalized enforcement of incompressibility: the Gaussian reference measure is defined directly on the divergence-free subspace, so admissibility is built into sampling, interpolation, conditioning, and generative transport.

1. Divergence-free structure as a measure-theoretic constraint

In the torus-based formulation, the ambient velocity-field space is

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$

the real Hilbert space of square-integrable, zero-mean, $2$-vector fields with periodic boundary conditions, and the scalar stream-function space is

$X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$

The divergence-free subspace is

$V=\{u\in H:\nabla\cdot u=0\},$

and the orthogonal complement is

$$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$

This yields the Helmholtz-Hodge splitting

$H=V\oplus V^\perp,$

which provides the basic functional-analytic setting for incompressible Gaussian measures (Li et al., 25 Mar 2026).

In the bounded-domain Gaussian-process formulation, one considers a compact domain $D\subset\mathbb{R}^d$ with $d=2,3$ and vector fields in $C(D)^d$ or Sobolev spaces $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$0. A continuously differentiable process $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$1 is called divergence-free if

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$2

Lemma 3.2 in the summarized report states that this is equivalent to the mean $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$3 being divergence-free and, for each fixed $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$4, every column of the matrix-valued map $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$5 being divergence-free (Gia et al., 16 Nov 2025).

These two settings differ in representation—spectral on $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$6, kernel-based on $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$7—but they agree on the defining property of the measure: it is not merely centered on divergence-free data, but supported on a divergence-free function class.

2. Gaussian measures on vector-field spaces

A centered Gaussian measure $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$8 on $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$9 is determined by its covariance operator $2$0, assumed bounded, self-adjoint, nonnegative, and trace-class. Writing

$2$1

means that for any $2$2 the linear functional $2$3 has law $2$4. Under periodic boundary conditions, $2$5 is diagonalized by the Fourier basis $2$6, so that

$2$7

and a draw $2$8 admits the spectral expansion

$2$9

(Li et al., 25 Mar 2026).

In the Gaussian-process setting, a vector-valued process

$X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$0

is characterized by the mean

$X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$1

and covariance kernel

$X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$2

For any finite set $X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$3, the joint random vector $X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$4 is Gaussian with mean $X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$5 and block-matrix covariance $X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$6 (Gia et al., 16 Nov 2025).

The measure-theoretic language is explicit in the GP construction: by Kolmogorov’s extension theorem, the law of $X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$7 is a Borel probability measure

$X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$8

completely determined by its mean function and covariance operator

$X:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}).$9

(Gia et al., 16 Nov 2025).

3. Curl-based pushforward on the torus

The torus construction in "Project and Generate: Divergence-Free Neural Operators for Incompressible Flows" (Li et al., 25 Mar 2026) defines a divergence-free Gaussian reference measure by pushing forward a scalar Gaussian through the stream-function operator

$V=\{u\in H:\nabla\cdot u=0\},$0

By elementary calculus,

$V=\{u\in H:\nabla\cdot u=0\},$1

so $V=\{u\in H:\nabla\cdot u=0\},$2.

Let $V=\{u\in H:\nabla\cdot u=0\},$3 be any Gaussian measure on $V=\{u\in H:\nabla\cdot u=0\},$4; the divergence-free measure is then

$V=\{u\in H:\nabla\cdot u=0\},$5

meaning that for any measurable $V=\{u\in H:\nabla\cdot u=0\},$6,

$V=\{u\in H:\nabla\cdot u=0\},$7

Because $V=\{u\in H:\nabla\cdot u=0\},$8 is linear and bounded, $V=\{u\in H:\nabla\cdot u=0\},$9 is again Gaussian, with mean zero and covariance

$$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$0

where $$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$1 is the $$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$2-adjoint of $$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$3 (Li et al., 25 Mar 2026).

When $$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$4 in the scalar Fourier basis, the covariance on velocity modes is expressed as

$$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$5

with $$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$6. The resulting samples are exactly divergence-free, since for any $$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$7,

$$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$8

pointwise, and hence up to discretization error on a grid (Li et al., 25 Mar 2026).

This construction is closed-form and intrinsic: the incompressibility constraint is enforced by the map defining the measure itself, not by a subsequent correction step.

4. Spectral Leray projection and subspace geometry

The same torus-based framework pairs the divergence-free Gaussian measure with the spectral Leray projector. The $$V^\perp=\{\nabla\phi:\phi\in H^1_{\mathrm{per}}(\mathbb{T}^2),\ \int \phi=0\}.$$9-orthogonal projection $H=V\oplus V^\perp,$0 is given in Fourier space by

$H=V\oplus V^\perp,$1

or equivalently in physical space by

$H=V\oplus V^\perp,$2

For every mode, one checks

$H=V\oplus V^\perp,$3

which makes $H=V\oplus V^\perp,$4 the canonical linear mechanism for restricting a field to the divergence-free subspace (Li et al., 25 Mar 2026).

Within the paper’s broader framework, the projector is used for deterministic regression and for constrained generative dynamics. The abstract states that learning-based models for fluid dynamics often operate in unconstrained function spaces, leading to physically inadmissible, unstable simulations, while penalty-based methods offer soft regularization but provide no structural guarantees, resulting in spurious divergence and long-term collapse. The Leray projection is introduced as a differentiable spectral projection grounded in the Helmholtz-Hodge decomposition, restricting the regression hypothesis space to physically admissible velocity fields (Li et al., 25 Mar 2026).

A central clarification made in the same work is that simply projecting model outputs is insufficient when the prior is incompatible. This identifies a common misconception in constrained generative modeling: output projection alone does not ensure that the reference measure, intermediate measures, and transported law all remain subspace-consistent. The divergence-free Gaussian reference measure is introduced precisely to resolve this mismatch.

5. Kernel-based divergence-free Gaussian processes on bounded domains

A second construction uses matrix-valued covariance kernels. Let $H=V\oplus V^\perp,$5 be a scalar positive-definite radial kernel of sufficient smoothness, such as a Matérn or Wendland kernel, and define

$H=V\oplus V^\perp,$6

Componentwise,

$H=V\oplus V^\perp,$7

By construction, $H=V\oplus V^\perp,$8, is positive-definite, and for each fixed $H=V\oplus V^\perp,$9 its columns lie in $D\subset\mathbb{R}^d$0 and satisfy $D\subset\mathbb{R}^d$1. Hence

$D\subset\mathbb{R}^d$2

is a centered divergence-free Gaussian process on $D\subset\mathbb{R}^d$3 (Gia et al., 16 Nov 2025).

The induced Gaussian reference measure is supported on

$D\subset\mathbb{R}^d$4

and almost every sample path of $D\subset\mathbb{R}^d$5 lies in this closed subspace (Gia et al., 16 Nov 2025). This kernel formulation is the bounded-domain analogue of the torus pushforward construction in the sense that divergence-freeness is encoded at the covariance level rather than imposed after sampling.

The associated reproducing-kernel Hilbert space is

$D\subset\mathbb{R}^d$6

with reproducing property

$D\subset\mathbb{R}^d$7

By Mercer’s theorem and the Karhunen-Loève expansion, there exist eigenpairs $D\subset\mathbb{R}^d$8 of the integral operator $D\subset\mathbb{R}^d$9 in $d=2,3$0, with $d=2,3$1 divergence-free and

$d=2,3$2

$d=2,3$3

The RKHS is the Cameron-Martin space of $d=2,3$4, with norm

$d=2,3$5

or equivalently

$d=2,3$6

whenever $d=2,3$7 lies in the range of $d=2,3$8 (Gia et al., 16 Nov 2025).

6. Conditioning, approximation, and generative transport

The bounded-domain GP formalism provides explicit conditioning formulae. For noiseless observations $d=2,3$9, the posterior process is

$C(D)^d$0

with

$C(D)^d$1

$C(D)^d$2

For noisy observations $C(D)^d$3 with $C(D)^d$4 i.i.d., the predictive mean and covariance become

$C(D)^d$5

$C(D)^d$6

(Gia et al., 16 Nov 2025).

The same report states error estimates in Sobolev norms under assumptions linking the prior RKHS to $C(D)^d$7 and using the fill distance

$C(D)^d$8

For interpolation without noise, Theorem 5.1 gives rates depending on smoothness, target norm, and fill distance; for approximation with noise and Tikhonov regularization, Theorem 5.4 yields bounds containing terms of the form

$C(D)^d$9

plus terms involving the noise norm $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$00, and in the correctly specified Gaussian noise case $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$01 the expectation over the noise yields rates of the same order plus an additional term $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$02 (Gia et al., 16 Nov 2025).

In the torus generative-model setting, the divergence-free Gaussian reference measure is used as the initial law in a flow-matching or normalizing-flow-based model. If $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$03 is the data distribution on $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$04, $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$05, and $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$06, then one sets

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$07

so each $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$08. A time-dependent vector field $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$09 is learned by minimizing

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$10

Because $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$11 itself lies in $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$12 and the support of $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$13 is $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$14, the generative ODE

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$15

remains exactly in $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$16 for all $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$17 and produces incompressible velocity fields (Li et al., 25 Mar 2026).

A plausible implication is that the two lines of work address complementary operations on divergence-free measures: one emphasizes posterior inference and approximation under pointwise observations, while the other emphasizes transport between a divergence-free reference law and a target law under learned dynamics.

7. Support, regularity, and computation

For the curl-pushforward construction, $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$18 on $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$19, and its support is the closure of $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$20 in $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$21. If one chooses $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$22 with eigenvalues $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$23, then draws $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$24 lie almost surely in the Sobolev space $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$25 with $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$26, and hence

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$27

(Li et al., 25 Mar 2026).

The same source gives an FFT-based spectral discretization. On a uniform $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$28 grid, one implements $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$29 and $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$30 by FFTs in $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$31 operations. Sampling $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$32 amounts to three steps: draw scalar Fourier modes $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$33, set $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$34, and apply the inverse FFT to obtain $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$35. Each projection or curl-pushforward requires two forward and two inverse FFTs, and the reported practical remark is that this overhead is minor compared to the cost of evaluating a large neural-operator network (Li et al., 25 Mar 2026).

For the kernel-based GP construction, a Karhunen-Loève truncation is

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$36

where $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$37 are the leading eigenpairs of the integral operator $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$38. Numerically, one approximates the eigendecomposition of the matrix $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$39 and extends to new points by

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$40

where $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$41 is the rank-$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$42 eigendecomposition of $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$43 (Gia et al., 16 Nov 2025).

Kernel evaluation requires computation of

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$44

For a radial kernel $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$45, the report gives

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$46

and

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$47

The algorithmic summary given there is: form the Gram matrix $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$48; solve $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$49 in the noiseless case or $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$50 otherwise; predict by

$H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$51

and, for uncertainty quantification, compute the predictive covariance using the standard Schur-complement formula already stated (Gia et al., 16 Nov 2025).

Across both implementations, the defining computational theme is subspace consistency. In the torus setting, both the reference $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$52 and the learned dynamics lie in $H:=L^2_{\mathrm{per}}(\mathbb{T}^2;\mathbb{R}^2),$53, so the entire generative flow is well-defined on the divergence-free subspace without further corrections or penalties (Li et al., 25 Mar 2026). In the bounded-domain GP setting, the covariance kernel itself has divergence-free columns, so interpolation, regression, and posterior sampling remain inside the divergence-free class (Gia et al., 16 Nov 2025).