A Simulator-Free Surrogate for Ferroelectric Phase-Fields via an Exact Fourier State-Space Model and Neural Operator

Traditional ferroelectric phase-field simulators are used to predict incubation, nucleation, and saturation trajectories, a task that has been challenging due to their high computational cost. Although surrogates have recently become more popular due to their inherent speed, they struggle to approximate the amplification of incubation noise. As such, we propose a hybrid surrogate with two components: a Fourier state-space model (SSM) that solves the linear pre-nucleation window in closed form, handing off to a next-step Fourier neural operator (FNO) that tracks the nonlinear nucleation/saturation window. We are the first to isolate the incubation window as a stage that, at the Landau parameters of this HfO$_2$ stack, is solvable in closed form. Importantly, our pipeline reproduces the reference solver's (FERROX) full trajectories up to two orders of magnitude faster, while achieving a near-perfect pattern correlation. Additionally, by conditioning the Fourier SSM across the Landau coefficient $\alpha$ without gradient-based retraining, we find a critical $\alpha_c$: for $\vert{}\alpha\vert{} < \vert{}\alpha_c\vert{}$ the device never nucleates.