Predicting the relative shear storage modulus of highly filled bidisperse particulate suspensions by machine learning and constrained symbolic regression

Abstract

Highly filled suspensions of rigid spheres in a yield stress fluid have a relative shear storage modulus that is
governed not only by the solid volume fraction but also by the grading of the particles. Classical homogenization schemes contain no maximum packing fraction and therefore cannot reproduce the rapid growth of the modulus that is observed as the solid skeleton approaches its densest state, whereas conventional machine learning delivers accurate predictions in the form of a trained model rather than an explicit expression that a designer can use. This paper compares six machine learning models and two symbolic regression variants on a published set of 70 data points covering seven mixtures of spherical particles in an inverse emulsion, one monodisperse and six bidisperse, with size ratios from 1 to 8.25 and fine fractions from 0 to 0.5; no new measurements were made. The set comprises 63 measurements of G′(φ)/G′(0) and seven normalization points at φ = 0, where the ratio equals unity by definition, and every model is scored on the 63 measurements. The
maximum packing fraction, computed from a compressible packing model, is used as an input feature, and predictive ability is assessed with a leave-one-mixture-out protocol that never places a mixture in both the training and the test set. Over the six bidisperse mixtures the Gaussian process model gives a macro-average mean squared error in logarithmic space of 0.0205 against 0.1164 for the one-parameter power law, and its pooled error rises by a factor of 6.4 when the maximum packing fraction is removed from the feature set. The constrained symbolic regression variant returns a power law with no additional numerical constant fitted in the final exponent and reaches a log-space coefficient of determination of 0.950 over the 63 measurements; under the same leave-one-mixture-out protocol, however, the search procedure that produces it is, for the seed-selection rules tested, less accurate than the one-parameter power law. On 16 measurements of Table 3 of [1] at eight gradings that entered no fit, the expression gives a mean squared error of 0.0308 against 0.0081 for the fixed exponent √2, so the fixed exponent is the better closed form for these suspensions. Constraining the functional form is therefore necessary but not sufficient for generalization to new gradings.