Two-dimensional Navier–Stokes benchmark

In the InverseBench two-dimensional Navier–Stokes case, the spatial domain is the periodic square $(0,2\pi {)}^2$ (the flat torus). In the physical vorticity model, the periodic velocity $u(x,t)\in {\mathbb{R}}^2$ is divergence-free and its scalar curl is $\omega ={\mathrm{\partial }}_1{u}_2-{\mathrm{\partial }}_2{u}_1$. This curl has zero spatial mean. Taking the spatial mean of $u$ to be zero fixes the constant-velocity ambiguity. For viscosity $\nu >0$, forcing $f(x)=-4\cos (4x_2)$, and compatible initial curl ${\omega }_0$, the physical model is

${\mathrm{\partial }}_t\omega +u\cdot \mathrm{\nabla }\omega =\nu \mathrm{\Delta }\omega +f,\mathrm{\nabla }\cdot u=0,\omega (\cdot ,0)={\omega }_0\mathrm{.}$

The released benchmark uses a finite-grid scalar state denoted $w$. Its pinned solver computes velocity by Fourier streamfunction derivatives, which select zero spatial mean velocity and ignore the constant Fourier mode of the scalar state. The nonlinear term uses a dealiased state with zero mode removed, while the time-update formula retains the original state's zero mode; the specified forcing also has zero mean. Thus the solver defines a scalar-state extension even when $\bar{w}\mathrm{\ne }0$. One must not identify such an entire state with the curl of a periodic velocity. In the corresponding continuum mean-split convention, $w=\bar{w}+\omega$, $-\mathrm{\Delta }\psi =\omega$, and $u=({\mathrm{\partial }}_2\psi ,-{\mathrm{\partial }}_1\psi )$; only $\omega =w-\bar{w}$ is the curl, and $\bar{w}$ is a conserved scalar offset. This convention explains the zero-mode distinction; it does not assert that the finite-grid dealiased scheme is an exact continuum solution or that the released dataset has been projected to zero mean.

The reconstruction target is the supplied dataset field $z=w_0$ used as initial state for the inversion solver; the observation is $y=PL(z)+n$, where $L$ is this numerical scalar-state solution map over the specified inversion horizon $T$, $P$ samples the final field, and $n$ is Gaussian measurement noise. The dataset-generation horizon is a separate five time units: §4.1 and Appendix B.5 report independent Gaussian seed fields $\xi \sim \mathcal{N}(0,(-\mathrm{\Delta }+9I{)}^{-4})$, evolved under the stated forcing for five time units to produce the dataset of nontrivial initial fields. Accordingly, $z$ denotes the evolved field, not a claim that the inversion-input distribution remains Gaussian with the seed covariance. On the full periodic space the reported covariance gives the constant eigenmode variance $9^{-4}>0$; the paper does not specify a zero-mode projection for the seed law. The scalar-state implementation permits retaining that offset without asserting the incompatible identity $w=\mathrm{\nabla }\times u$ for arbitrary seeds.

InverseBench uses a pseudo-spectral solver with adaptive time stepping; the reported setup has a $128\times 128$ grid, Reynolds number $200$, $\mathrm{10,000}$ training samples, one validation sample, and ten test samples. The paper reports black-box forward access without a closed-form gradient for this implementation; the one-validation/ten-test split is part of this empirical benchmark, not a population-level guarantee.

The solver, noise, compute budget, and test law are explicit benchmark fields

The Navier–Stokes results are empirical for the reported split

Zheng et al., InverseBench, §3, §4.1, Appendix B.5, Tables 2 and 8

Versioned full-text counterpart, §4.1 and Appendix B.5

Pinned Navier–Stokes scalar-state implementation