LeapFrog: Getting the Jump on Multi-Scale Materials Simulations Using Machine Learning

TL;DR

Phase-field simulations of directional solidification across multiple length scales are computationally intensive, especially with AMR. LeapFrog couples Adaptive Mesh Refinement with a Multiscale Convolutional Long Short-Term Memory–based U-Net trained with a dynamic multiscale loss to advance the fields $\phi$ and $C$ over adaptive time steps. The study demonstrates 5–10× wall-time reductions while preserving microstructure statistics such as $(\phi, C)$ fields, concentration profiles, PSDs, and primary arm spacing (PAS). This approach enables efficient exploration of parameter space and rapid generation of statistically representative microstructures for alloy solidification and related first-order phase transitions.

Abstract

The development of novel materials in recent years has been accelerated greatly by the use of computational modelling techniques aimed at elucidating the complex physics controlling microstructure formation in materials, the properties of which control material function. One such technique is the phase field method, a field theoretic approach that couples various thermophysical fields to microscopic order parameter fields that track the phases of microstructure. Phase field models are framed as multiple, non-linear, partial differential equations, which are extremely challenging to compute efficiently. Recent years have seen an explosion of computational algorithms aimed at enhancing the efficiency of phase field simulations. One such technique, adaptive mesh refinement (AMR), dynamically adapts numerical meshes to be highly refined around steep spatial gradients of the PDE fields and coarser where the fields are smooth. This reduces the number of computations per time step significantly, thus reducing the total time of computation. What AMR doesn't do is allow for adaptive time stepping. This work combines AMR with a neural network algorithm that uses a U-Net with a Convolutional Long-Short Term Memory (CLSTM) base to accelerate phase field simulations. Our neural network algorithm is described in detail and tested in on simulations of directional solidification of a dilute binary alloy, a paradigm that is highly practical for its relevance to the solidification of alloys.

Figures (12)

• Figure 1: 5:2 Leapfrog Algorithm Results. Comparison of the time savings and final results between traditional Adaptive Mesh Refinement (AMR) and neural network accelerated AMR phase field simulations, with $N_{ML}/N_{PF}=5/2$. a) System Time vs. Wall Time comparison between a Phase Field (PF) simulation using Adaptive Mesh Refinement (AMR) (black) and the LeapFrog algorithm (composite) evolution of the same system in the dendritic regime. b) Breakdown total Wall Time between PF+AMR, an “Idealized” scenario where the PF+AMR data is always organized in memory (minimizing lookup time), and the LeapFrog Algorithm (broken down by process). c) Comparison of the resulting $\phi$ and d)$C$ system fields after being evolved with the PF+AMR and the LeapFrog algorithm introduced in this work.
• Figure 2: 5:5 LeapFrog Algorithm results. Comparison of the time savings and final results between traditional Adaptive Mesh Refinement (AMR) and neural network accelerated AMR phase field simulations, with $N_{ML}/N_{PF}=1$. a) System Time vs. Wall Time comparison between a Phase Field (PF) simulation using Adaptive Mesh Refinement (AMR) (black) and the LeapFrog algorithm (composite) evolution of the same system in the dendritic regime. b) Breakdown total Wall Time between PF+AMR, an “Idealized” scenario where the PF+AMR data is always organized in memory (minimizing lookup time), and the LeapFrog Algorithm (broken down by process). c) Comparison of the resulting $\phi$ and d) system fields after being evolved with the PF+AMR and the LeapFrog algorithm introduced in this work.
• Figure 3: PF and LeapFrog Concentration Profiles Comparison. Comparison of concentration profiles along a dendrite's core (centerline) for two simulations, one evolved with Adaptive Mesh Refinement (AMR) (right frame, dashed line) and the other with the LeapFrog (LF) algorithm (right frame, continuous line). The left frame is split into two sub-frames, with the left sub-frame showing the centerline along which $C$ is measured from the AMR output, and the right sub-frame showing that of from the LeapFrog outputs. In the right frame. The point $y=0$ corresponds to the top of the centerline in the left sub-frames.
• Figure 4: Generation of mask used to smooth dendrite concentration profiles output from the LeapFrog algorithm a) The concentration field of a system reaching the end of the dendritic growth regime. b) The magnitude of the gradients in said concentration field. c) Constraining the mask to areas where $\phi$ corresponds to the solid phase as well filtering the concentration gradient magnitude through a magnitude threshold. d) Restricting the resulting mask to the portion of the system generated after the initial transient phase of the simulation ($y > V_{p}\frac{65000dt}{dx}$).
• Figure 5: Comparison of concentration profiles along a dendrite's core (centerline) for a simulation evolved with Adaptive Mesh Refinement (AMR) and LeapFrog (LF) at $t^{*}=250000dt$. (red lines in Fig. \ref{['fig:CProfiles']}) with smoothing mask illustrated in Fig. \ref{['fig:CSmoothMask']} applied to the $C$-field output from the LeapFrog algorithm. The core (centerline) concentration profile is shown in a). Also shown are the concentration profiles along the system's isotherms in the liquid (b) and solid (c) phases.