Combined Machine Learning and CALPHAD Approach for Discovering Processing-Structure Relationships in Soft Magnetic Alloys

We aim to investigate relationships between select processing parameters or inputs (composition, temperature, annealing time) and two structural parameters, specifically, the mean radius and volume fraction of the Fe$_3$Si nanocrystals. To this end, we have deviced a combined CALPHAD and machine learning approach that led to well-calibrated metamodels able to predict structural parameters quickly and accurately for any desired inputs. In order to generate data for the mean radius and volume fraction of Fe$_3$Si nanocrystals, we have used a precipitation model based in the software Thermocalc to perform annealing simulations at a set of temperatures (490-550~\degree C) and for varying Fe and Si concentrations (Fe$_{72.89 +x}$Si$_{16.21-x}$B$_{6.90}$Nb$_{3}$Cu$_{1}$, $-3\leq x \leq 3$ atomic \%). Thereafter, we used the data to develop metamodels for the mean radius and volume fraction via the \emph{k}-Nearest Neighbour algorithm. The metamodels are shown to reproduce closely the trends obtained from the precipitation model over the entire annealing timescale. Our further analysis via parallel coordinate charts shows the effect of composition, temperature, and annealing time, and helps identify combinations thereof that lead to the desired mean radius and volume fraction for the nanocrystalline phase. This approach utilizes experimental (thermodynamic and kinetic) databases from the CALPHAD approach so as to capture the physics of nucleation and growth, while the machine learning algorithm provides the robustness needed to analyze the effects of processing parameters for this complex precipitation problem. This work contributes to understanding the linkages between processing parameters and desired microstructural characteristics (crystal size and volume fraction) responsible for achieving targeted properties, and illustrates ways to reduce the time from alloy discovery to deployment.