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.

Parsimonious neural networks learn classical mechanics and can teach it

We combine neural networks with genetic algorithms to find parsimonious models that describe the time evolution of a point particle subjected to an external potential. The genetic algorithm is designed to find the simplest, most interpretable network compatible with the training data. The parsimonious neural network (PNN) can numerically integrate classical equations of motion with negligible energy drifts and good time reversibility, significantly outperforming a generic feed-forward neural network. Our PNN is immediately interpretable as the position Verlet algorithm, a non-trivial integrator whose justification originates from Trotter's theorem.