Surrogate-Based Constrained Langevin Sampling With Applications to Optimal Material Configuration Design

We consider the problem of generating configurations that satisfy physical constraints for optimal material nano-pattern design, where multiple (and often conflicting) properties need to be simultaneously satisfied. Consider, for example, the trade-off between thermal resistance, electrical conductivity, and mechanical stability needed to design a nano-porous template with optimal thermoelectric efficiency. To that end, we leverage the posterior regularization framework andshow that this constraint satisfaction problem can be formulated as sampling froma Gibbs distribution. The main challenges come from the black-box nature ofthose physical constraints, since they are obtained via solving highly non-linearPDEs. To overcome those difficulties, we introduce Surrogate-based Constrained Langevin dynamics for black-box sampling. We explore two surrogate approaches. The first approach exploits zero-order approximation of gradients in the Langevin Sampling and we refer to it as Zero-Order Langevin. In practice, this approach can be prohibitive since we still need to often query the expensive PDE solvers. The second approach approximates the gradients in the Langevin dynamics with deep neural networks, allowing us an efficient sampling strategy using the surrogate model. We prove the convergence of those two approaches when the target distribution is log-concave and smooth. We show the effectiveness of both approaches in designing optimal nano-porous material configurations, where the goal is to produce nano-pattern templates with low thermal conductivity and reasonable mechanical stability.