A PDE-theory-guided neural operator and the long-time relaxation of the two-dimensional anisotropic Boussinesq system

Abstract

We design a neural operator for the two-dimensional Boussinesq system with anisotropic dissipation on the periodic box, guided by a rigorous analysis of its long-time behavior. The operator maps a small perturbation of hydrostatic equilibrium, of amplitude $\varepsilon$, to the vorticity at a late time. We prove that the flow relaxes to a stratified shear flow, with the horizontally averaged vorticity staying within $O(\varepsilon^2)$ of its initial value $\barω_0$ and the oscillation about it of order $\varepsilon$. Accordingly, the network outputs $\barω_0$ plus these two parts, each scaled to its size, which grows as $\varepsilon^2$ and $\varepsilon$, respectively. Without this guidance the training either does not converge or misses the $\varepsilon^2$ growth. With it, one network reproduces, sample by sample on $2\times10^3$ test runs over a factor of $23$ in amplitude, the quantity bounded by the theorem to a median error of $2.2$ percent. Being fast and differentiable with respect to the initial data, the network is also used to search the space of initial data, and with every candidate checked by the solver the search finds initial data for which this quantity is $4.4$ times the largest value among the test runs, raising the lower bound for the constant of the theorem by that factor.