Extreme time extrapolation capabilities and thermodynamic consistency of physics-inspired Neural Networks for the 3D microstructure evolution of materials via Cahn-Hilliard flow

Abstract

A Convolutional Recurrent Neural Network (CRNN) is trained to reproduce the evolution of the spinodal decomposition process in three dimensions as described by the Cahn-Hilliard equation. A specialized, physics-inspired architecture is proven to provide close accordance between the predicted evolutions and the ground truth ones obtained via conventional integration schemes. The method can accurately reproduce the evolution of microstructures not represented in the training set at a fraction of the computational costs. Extremely long-time extrapolation capabilities are achieved, up to reaching the theoretically expected equilibrium state of the system, consisting of a layered, phase-separated morphology, despite the training set containing only relatively-short, initial phases of the evolution. Quantitative accordance with the decay rate of the Free energy is also demonstrated up to the late coarsening stages, proving that this class of Machine Learning approaches can become a new and powerful tool for the long timescale and high throughput simulation of materials, while retaining thermodynamic consistency and high-accuracy.

Figures (9)

• Figure 1: (a) Sketch of the NN architecture and specification of the "toFlux" layer converting the hidden state $h^2_{t+\tau}$ to the vector field $\vec{u}_t$. (b) Lossplot obtained training the Convolutional Recurrent Neural Network. No sign of overfitting is present. Differences in initial values for the curves are expected and due to the training procedure (see text for more information). (c) Comparison between the ground truth evolution and the predicted one for a validation set case. In the upper half of the snapshots, only $\varphi \ge 0.5$ is shown to better appreciate the internal structure.
• Figure 2: (a) Spatial generalization test. The domain size is 8 times larger than that used in the training set (twice as large in all directions). One-to-one accordance with the ground truth evolution can still be observed. (b) Evolution on a computational cell with the same size of training for a $\varphi$ initial profile shaped as the Stanford bunny. In the upper half of the snapshots, only $\varphi \ge 0.5$ is shown. MSE values for the whole sequences are reported in the insets.
• Figure 3: Time generalization test. A domain enclosing a region in which $\varphi$ rapidly fluctuates is evolved for $40$ times more steps than those used in training. Despite local variations, long-time behavior still exhibits quantitative correspondence. In the upper half of the snapshots, only $\varphi \ge 0.5$ is shown. MSE loss for the whole sequence is reported in the inset.
• Figure 4: Extreme time generalization test. A domain initialized with Perlin random noise is evolved until a flat configuration is reached, consistently with one of the possible stationary states for the system. A comparison with the evolution obtained using the finite difference integration is provided for $\approx 1/3$ of the sequence. In the upper half of the snapshots, only $\varphi \ge 0.5$ is shown. Close-ups of regions presenting the largest local deviations at time $1.2 \times 10^4 \tau$ are also shown in the insets.
• Figure 5: (a) Representative stages comparing the ground-truth evolution and the corresponding NN prediction. (b) RMSE curve between the predicted and ground truth $\varphi$. (c) $\bar{\delta}$ error (Eq. \ref{['eq::measure']}) as a function of time.