Towards Spatio-Temporal Extrapolation of Phase-Field Simulations with Convolution-Only Neural Networks

TL;DR

This work addresses the computational bottleneck of phase-field simulations for liquid metal dealloying by introducing a convolution-only, conditionally parameterized U-Net surrogate capable of extrapolating in space and time. It integrates convolutional self-attention, physics-informed padding, and a flood-fill corrector, and leverages diffusion models to generate synthetic initial conditions, eliminating solver-based initialization. The approach achieves accurate reproduction of key QoIs and microstructure statistics with training errors well below 5% and extrapolation errors generally under 15–20%, while offering speed-ups up to around 10^4–10^5× over traditional solvers. This combination of space–time extrapolation, domain-size independence, and generative initialization enables practical, large-scale, long-horizon LMD simulations, with significant implications for uncertainty quantification and optimization in materials design.

Abstract

Phase-field simulations of liquid metal dealloying (LMD) can capture complex microstructural evolutions but can be prohibitively expensive for large domains and long time horizons. In this paper, we introduce a fully convolutional, conditionally parameterized U-Net surrogate designed to extrapolate far beyond its training data in both space and time. The architecture integrates convolutional self-attention, physically informed padding, and a flood-fill corrector method to maintain accuracy under extreme extrapolation, while conditioning on simulation parameters allows for flexible time-step skipping and adaptation to varying alloy compositions. To remove the need for costly solver-based initialization, we couple the surrogate with a conditional diffusion model that generates synthetic, physically consistent initial conditions. We train our surrogate on simulations generated over small domain sizes and short time spans, but, by taking advantage of the convolutional nature of U-Nets, we are able to run and extrapolate surrogate simulations for longer time horizons than what would be achievable with classic numerical solvers. Across multiple alloy compositions, the framework is able to reproduce the LMD physics accurately. It predicts key quantities of interest and spatial statistics with relative errors typically below 5% in the training regime and under 15% during large-scale, long time-horizon extrapolations. Our framework can also deliver speed-ups of up to 36,000 times, bringing the time to run weeks-long simulations down to a few seconds. This work is a first stepping stone towards high-fidelity extrapolation in both space and time of phase-field simulation for LMD.

Figures (21)

• Figure 1: U-Net Architecture - The model takes as input the field at time $t$ and outputs the field at a future time step $t+\Delta\tau$, where $\Delta\tau$ is a conditional parameter included in $\theta$. The conditional network outputs vectors that scale each channels of the residual connections during reconstruction. The U-Net color scheme is analogous to the one employed in the original U-Net paper ronneberger2015u.
• Figure 2: Padding in each U-Net convolutional layers. The width of the padded tensors is exaggerated for illustration purposes. In practice, since the convolution kernels in the U-Net are $3\times3$, the padding width required is only 1.
• Figure 3: Flood-fill algorithm for field cleaning. The approach first identifies where the un-corroded portions of the species fields stop. It subsequently generates corresponding masks to replace the un-corroded (yet potentially corrupted) nodes with the known ideal species value.
• Figure 4: Example of initial conditions sampled from the diffusion model. The diffusion model is conditioned on $c_A$, and can thus be used to generate fields for arbitrary values of $c_A$ not contained in the training dataset (e.g. $c_A=0.225$). In this figure, samples for $c_A$ ranging from 0.2 to 0.4, with a 0.025 increment are shown (and corresponding $c_B=1-c_A$).
• Figure 5: QoI marginal densities for different values of $c_A$. Results in red () correspond to diffusion model-generated samples, and blue lines () are for DNS-based (solver-generated) samples. Empirically, the distributions match reasonably well, except for the total mass QoI (as seen in table \ref{['tab:pval']}). However, unlike other QoIs, the total mass has an extremely reduced range (typically between 0.165 and 0.18), so the diffusion model still generates samples that are on average close to the true distribution. The distributions are estimated using kernel density estimation (KDE) chen2017tutorialkerneldensityestimation.