Multi-fidelity surrogate with heterogeneous input spaces for modeling melt pools in laser-directed energy deposition

Multi-fidelity (MF) modeling is a powerful statistical approach that can intelligently blend data from varied fidelity sources. This approach finds a compelling application in predicting melt pool geometry for laser-directed energy deposition (L-DED). One major challenge in using MF surrogates to merge a hierarchy of melt pool models is the variability in input spaces. To address this challenge, this paper introduces a novel approach for constructing an MF surrogate for predicting melt pool geometry by integrating models of varying complexity, that operate on heterogeneous input spaces. The first thermal model incorporates five input parameters i.e., laser power, scan velocity, powder flow rate, carrier gas flow rate, and nozzle height. In contrast, the second thermal model can only handle laser power and scan velocity. A mapping is established between the heterogeneous input spaces so that the five-dimensional space can be morphed into a pseudo two-dimensional space. Predictions are then blended using a Gaussian process-based co-kriging method. The resulting heterogeneous multi-fidelity Gaussian process (Het-MFGP) surrogate not only improves predictive accuracy but also offers computational efficiency by reducing evaluations required from the high-dimensional, high-fidelity thermal model. The tested Het-MFGP yields an $R^2$ of 0.975 for predicting melt pool depth. This surpasses the comparatively modest $R^2$ of 0.592 achieved by a GP trained exclusively on high-dimensional, high-fidelity data. Similarly, in the prediction of melt pool width, the Het-MFGP excels with an $R^2$ of 0.943, outshining the GP's performance, which registers a lower $R^2$ of 0.588. The results underscore the benefits of employing Het-MFGP for modeling melt pool behavior in L-DED. The framework successfully demonstrates how to leverage multimodal data and handle scenarios where certain input parameters may be difficult to model or measure.