Fig. 1: The melt viscosity (η) physical trends, learning problems, and machine-learning workflow.
From: A physics-enforced neural network to predict polymer melt viscosity
A Depictions of the functions used to describe the behavior of η with respect to temperature (T), molecular weight (Mw), and shear rate (\({\dot{\gamma }}\)). The functions are parametrized by empirical parameters with physical significance, elaborated in Table 1 and in the Methods section. The η dependence on Mw is given by \(\log {\eta }_{{M}_{w}}\) (Eq. (8) in the Methods section). Empirical parameters define the slopes of the relationship at low Mw (α1) and high Mw (α2), the critical molecular weight (Mcr), the y-intercept of \({\eta }_{{M}_{w}}\) (k1), and the rate of transition from low to high Mw regions (\({\beta }_{{M}_{w}}\)). The η dependence on T and Mw is given by \(\log {\eta }_{0}(T,{M}_{w})\) (Eq. (5) in the Methods section), and is parameterized by reference temperature (Tr) and empirical fitting parameters (C1 and C2). The effects of C1 and C2 are visualized by comparing the trends with different sampled values. The η dependence on \({\dot{\gamma}}\) is given by \(\log \eta (T,{M}_{w},{\dot{\gamma}})\) (Eq. (4) in the Methods section). The relevant parameters include shear thinning slope (n), the critical shear rate (\({\dot{\gamma }}_{cr}\)), and the rate of transition from η0 to shear thinning (\({\beta }_{\dot{\gamma }}\)). B The Physics-Enforced Neural Network (PENN) architecture starts with an input containing the polymer fingerprint and the PDI. A Multi-Layer Perceptron (MLP) uses the concatenated input to predict the empirical parameters. Next, the computational graph uses the predicted empirical parameters to calculate η, via the encoded \(\log {\eta }_{{M}_{w}}\), \(\log {\eta }_{0}(T,{M}_{w})\), and \(\log \eta (T,{M}_{w},{\dot{\gamma}})\) functions. The physical condition variables \(\log {M}_{w}\), \(\log \dot{\gamma }\) and T are input to their respective functions. C Physics unaware Artificial Neural Network (ANN) and a Gaussian Process Regression (GPR) are baselines to compare with the PENN model. The input features to the ANN and GPR models are the concatenated polymer fingerprint, T, Mw, \({\dot{\gamma}}\), and PDI.
