Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbone

TL;DR

This work tackles the computational bottleneck of phase-field simulations for liquid-metal dealloying (LMD), where stiff coupled PDEs force extremely small time steps. It introduces U-AFNO, a hybrid surrogate that combines a U-Net backbone with an Adaptive Fourier Neural Operator to map the field at time t to a future time t+Δτ, ensuring outputs remain within physical bounds via a sigmoid. The authors demonstrate that U-AFNO accurately reproduces invariant field statistics and key quantities of interest (curvature, perimeter, penetration depth, mass loss) while delivering large speed-ups (~10^4×) over high-fidelity solvers in fully auto-regressive roll-outs, and provide a nuanced analysis of hybrid surrogate/high-fidelity stepping. They also show that the gains from relaxation steps are architecture- and problem-dependent, with U-AFNO maintaining robustness where purely data-driven surrogates often fail. Overall, the results support U-AFNO as a promising tool for fast, physics-informed surrogates in phase-field simulations, enabling efficient uncertainty quantification and design exploration in LMD contexts.

Abstract

Prolonged contact between a corrosive liquid and metal alloys can cause progressive dealloying. For such liquid-metal dealloying (LMD) process, phase field models have been developed. However, the governing equations often involve coupled non-linear partial differential equations (PDE), which are challenging to solve numerically. In particular, stiffness in the PDEs requires an extremely small time steps (e.g. $10^{-12}$ or smaller). This computational bottleneck is especially problematic when running LMD simulation until a late time horizon is required. This motivates the development of surrogate models capable of leaping forward in time, by skipping several consecutive time steps at-once. In this paper, we propose U-Shaped Adaptive Fourier Neural Operators (U-AFNO), a machine learning (ML) model inspired by recent advances in neural operator learning. U-AFNO employs U-Nets for extracting and reconstructing local features within the physical fields, and passes the latent space through a vision transformer (ViT) implemented in the Fourier space (AFNO). We use U-AFNOs to learn the dynamics mapping the field at a current time step into a later time step. We also identify global quantities of interest (QoI) describing the corrosion process (e.g. the deformation of the liquid-metal interface) and show that our proposed U-AFNO model is able to accurately predict the field dynamics, in-spite of the chaotic nature of LMD. Our model reproduces the key micro-structure statistics and QoIs with a level of accuracy on-par with the high-fidelity numerical solver. We also investigate the opportunity of using hybrid simulations, in which we alternate forward leap in time using the U-AFNO with high-fidelity time stepping. We demonstrate that while advantageous for some surrogate model design choices, our proposed U-AFNO model in fully auto-regressive settings consistently outperforms hybrid schemes.

Figures (10)

• Figure 1: Liquid metal dealloying with two sets of initial conditions. The initial species fields ($c_A$ and $c_B$ at $t=0$) are contaminated with low-amplitude random white noise (left fields). Given the chaotic nature of the dealloying process, the initial noise perturbation eventually leads to widely different solid phase fields at late time (e.g. right fields, at $t=6\,\mu$s, after 6 million time steps).
• Figure 2: U-AFNO Architecture - The model takes as input the field at time $t$ and outputs the field at a future time step $t+\Delta\tau$. The U-Net color scheme is analogous to the one employed in the original U-Net paper ronneberger2015unet. The abbreviations in the AFNO layers are the following: FFT: Fast Fourier Transform, IFFT: Inverse Fast Fourier Transform, MHA: Mutli-Head Attention, FC: Fully Connected.
• Figure 3: Curvature QoIs - At each time $t$, the parametric curve $\gamma(s,t)$ delineating the interface between the two phases is computed, from which we compute the curvature $k(s,t)$. Histograms of the curvature distribution at two different times ($t=2.5~\mu$s and $t=3.5~\mu$s) are shown, from which means and standard deviations are estimated and computed over time (blue and orange plots).
• Figure 4: U-AFNO-B/1 field predictions in the fully auto-regressive case. The predicted and ground truth fields are shown at $t=(0, 1, 2, 3, 4, 5, 6)~\mu$s. Since the initial $10^6$ time steps are computed with the high-fidelity solver, time $t=0-1~\mu$s for the U-AFNO and the ground truth are identical.
• Figure 5: U-AFNO field auto-correlation maps in the fully auto-regressive case. The auto-correlation maps correspond to the fields shown in figure \ref{['fig:field_far']}. The error maps are the absolute error between the predicted and ground truth auto-correlation.