Navigating predictions at nanoscale: a comprehensive study of regression models in magnetic nanoparticle synthesis

Abstract

The applicability of magnetic nanoparticles (MNP) highly depends on their physical properties, especially their size. Synthesizing MNP with a specific size is challenging due to the large number of interdepend parameters during the synthesis that control their properties. In general, synthesis control cannot be described by white box approaches (empirical, simulation or physics based). To handle synthesis control, this study presents machine learning based approaches for predicting the size of MNP during their synthesis. A dataset comprising 17 synthesis parameters and the corresponding MNP sizes were analyzed. Eight regression algorithms (ridge, lasso, elastic net, decision trees, random forest, gradient boosting, support vectors and multilayer perceptron) were evaluated. The model performance was assessed via root mean squared error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE) and standard deviation of residuals. Support vector regression (SVR) exhibited the lowest RMSE values of 3.44 and a standard deviation for the residuals of 5.13. SVR demonstrated a favorable balance between accuracy and consistency among these methods. Qualitative factors like adaptability to online learning and robustness against outliers were additionally considered. Altogether, SVR emerged as the most suitable approach to predict MNP sizes due to its ability to continuously learn from new data and resilience to noise, making it well-suited for real-time applications with varying data quality. In this way, a feasible optimization framework for automated and self-regulated MNP synthesis was implemented. Key challenges included the limited dataset size, potential violations of modeling assumptions, and sensitivity to hyperparameters. Strategies like data regularization, correlation analysis, and grid search for model hyperparameters were employed to mitigate these issues.

Fig. 1. The envisioned workflow of an autonomous MNP synthesis process with machine learning optimization for the production target. This study aims to build the predictive modelling unit of this workflow. The necessary steps are given in the outer extension of the “Predictive Modelling” section.

Fig. 2. (A) A histogram of the MNP size distribution. The mean value is visualized by the red dashed line and the standard deviation by the green dashed lines. (B) The correlation matrix of all variables. Values are calculated by Pearson correlation coefficient. Blue indicates positive correlation; red indicates negative correlation. The magnitude of the correlations is indicated by color intensity and square size.

Fig. 3. (A) Bar chart with prediction results of the trained models for the validation set. The blue bars represent the RMSE values, orange the MAE values, green the MAPE values and red the standard deviation of residuals. (B) Visualizes the distribution of the residuals of the validation set. The dashed line shows the ideal value of zero for residuals, the orange bars and triangles in the boxplots show the median and mean residual, respectively. Circles denote individual outliers.

Fig. 4. 3D Surface plot of principal component analysis (PCA) applied to the support vector regression model. The original dataset, comprising 17 features, was transformed into a 3-dimensional space using PCA, with the first two principal components (PC1 and PC2) plotted on the x and y axes, respectively, and the SVR predictions on the z axis.