Multi-fidelity Bayesian optimization to solve the inverse Stefan problem

Abstract

In this work, we propose an efficient solution of the inverse Stefan problem by multi-fidelity Bayesian optimization. We construct a multi-fidelity Gaussian process surrogate model by combining many low-fidelity estimates of a solidification problem with only a few high-fidelity measurements. To solve the inverse problem, we employ the Gaussian process model in a Bayesian optimization approach based on a multi-fidelity knowledge gradient acquisition function. To account for the specific structure of the target function, we reformulate it as a composite function and thus significantly improve the stability of the optimization procedure. Target values can be switched easily, and previously obtained samples and surrogate models can be reused. The proposed method iteratively improves the recommended solution of the inverse problem. Explicitly adding recommended points of previous iterations to the solution procedure enhances the convergence properties of the algorithm.We demonstrate the applicability of the algorithm by solving the inverse problem for a planar solidification front in a single-fidelity setting. Process parameters are identified for targeted crystal-growth velocities during directional dendritic solidification. The relation between these velocities and process parameters, such as undercooling, thermal diffusivity, capillarity, and capillary anisotropy, defines the thermo-mechanical properties of the solidified part in metal-based additive manufacturing. Material design is based on the corresponding inverse problem. Cost-efficiency of solving the inverse problem is improved by introducing a fidelity hierarchy based on coarse-grid approximations of high-fidelity numerical simulations. The open-source simulation framework ALPACA for multiphase flows allows to generate data at all fidelities. We demonstrate the superior convergence properties of the presented multi-fidelity approach by comparison with an approach solely based on high-fidelity measurements of the tip velocity.