Data-driven machine learning approaches are being increasingly used to solve partial differential equations (PDEs).

Yet a fundamental gap remains: while these methods depend critically on the choice of prior covariance kernel, most kernels are selected for computational convenience (e.g., Gaussian/RBF kernels) or generic smoothness assumptions (e.g., Mat ern) rather than being rigorously grounded in the system's long-time statistical structure. Recent breakthroughs in stochastic PDE theory now make it possible to close this gap, constructing priors directly from the invariant-measure geometry of the underlying dynamics. Recent work of Coe, Hairer, and Tolomeo [7] establishes a remarkable geometric property of the two-dimensional stochastic Navier-Stokes (2D SNS) equations: although the dynamics are highly nonlinear, their unique invariant measure is equivalent-in the sense of mutual absolute continuity-to the Gaussian invariant measure of the linearized Ornstein-Uhlenbeck (OU) process. Equivalence means the two measures share the same support, null sets, and typical events, differing only by a positive Radon-Nikodym derivative. This reveals that the equilibrium statistical geometry is Gaussian, even when individual realizations are not.

Neural PDE solvers used for scientific simulation often violate governing equation constraints. While linear constraints can be projected cheaply, many constraints are nonlinear, complicating projection onto the feasible set. Dynamical PDEs are especially difficult because constraints induce long-range dependencies in time. In this work, we evaluate two training-free, post hoc projections of approximate solutions: a nonlinear optimization-based projection, and a local linearization-based projection using Jacobian-vector and vector-Jacobian products. We analyze constraints across representative PDEs and find that both projections substantially reduce violations and improve accuracy over physics-informed baselines.

Extensive empirical evaluations on a diverse set of PDEs demonstrate that our method is highly data-efficient, more gener-alizable, and even outperforms conventional vision-pretrained models.

We train all the networks for 500 epochs with Adam optimizer. The batch size is set to 32. For the backward pass, we use phantom gradients [ Geng et al., 2021 ] which are For the S-FNO-DEQ used in Table 1, we use Broyden's method [ Broyden, 1965 ] to solve for the The width of an FNO layer set to 32 across all the networks. Additionally, we retain only 12 Fourier modes in FNO layer, and truncate higher Fourier modes. As mentioned in Sec. 5 we use the dataset provided by Li et al. [ 2020a ] for our experiments with All the models are trained on 1024 data samples and tested on 500 samples.