Rethinking materials simulations: Blending direct numerical simulations with neural operators

Direct numerical simulations (DNS) are accurate but computationally expensive for predicting materials evolution across timescales, due to the complexity of the underlying evolution equations, the nature of multiscale spatio-temporal interactions, and the need to reach long-time integration. We develop a new method that blends numerical solvers with neural operators to accelerate such simulations. This methodology is based on the integration of a community numerical solver with a U-Net neural operator, enhanced by a temporal-conditioning mechanism that enables accurate extrapolation and efficient time-to-solution predictions of the dynamics. We demonstrate the effectiveness of this framework on simulations of microstructure evolution during physical vapor deposition modeled via the phase-field method. Such simulations exhibit high spatial gradients due to the co-evolution of different material phases with simultaneous slow and fast materials dynamics. We establish accurate extrapolation of the coupled solver with up to 16.5$\times$ speed-up compared to DNS. This methodology is generalizable to a broad range of evolutionary models, from solid mechanics, to fluid dynamics, geophysics, climate, and more.

Learning two-phase microstructure evolution using neural operators and autoencoder architectures

Phase-field modeling is an effective mesoscale method for capturing the evolution dynamics of materials, e.g., in spinodal decomposition of a two-phase mixture. However, the accuracy of high-fidelity phase field models comes at a substantial computational cost. Hence, fast and generalizable surrogate models are needed to alleviate the cost in computationally taxing processes such as in optimization and design of materials. The intrinsic discontinuous nature of the physical phenomena incurred by the presence of sharp phase boundaries makes the training of the surrogate model cumbersome. We develop a new framework that integrates a convolutional autoencoder architecture with a deep neural operator (DeepONet) to learn the dynamic evolution of a two-phase mixture. We utilize the convolutional autoencoder to provide a compact representation of the microstructure data in a low-dimensional latent space. DeepONet, which consists of two sub-networks, one for encoding the input function at a fixed number of sensors locations (branch net) and another for encoding the locations for the output functions (trunk net), learns the mesoscale dynamics of the microstructure evolution in the latent space. The decoder part of the convolutional autoencoder can then reconstruct the time-evolved microstructure from the DeepONet predictions. The result is an efficient and accurate accelerated phase-field framework that outperforms other neural-network-based approaches while at the same time being robust to noisy inputs.

Inferring topological transitions in pattern-forming processes with self-supervised learning

The identification and classification of transitions in topological and microstructural regimes in pattern-forming processes is critical for understanding and fabricating microstructurally precise novel materials in many application domains. Unfortunately, relevant microstructure transitions may depend on process parameters in subtle and complex ways that are not captured by the classic theory of phase transition. While supervised machine learning methods may be useful for identifying transition regimes, they need labels which require prior knowledge of order parameters or relevant structures. Motivated by the universality principle for dynamical systems, we instead use a self-supervised approach to solve the inverse problem of predicting process parameters from observed microstructures using neural networks. This approach does not require labeled data about the target task of predicting microstructure transitions. We show that the difficulty of performing this prediction task is related to the goal of discovering microstructure regimes, because qualitative changes in microstructural patterns correspond to changes in uncertainty for our self-supervised prediction problem. We demonstrate the value of our approach by automatically discovering transitions in microstructural regimes in two distinct pattern-forming processes: the spinodal decomposition of a two-phase mixture and the formation of concentration modulations of binary alloys during physical vapor deposition of thin films. This approach opens a promising path forward for discovering and understanding unseen or hard-to-detect transition regimes, and ultimately for controlling complex pattern-forming processes.