A Physics-Enforced Neural Network to Predict Polymer Melt Viscosity

TL;DR

This work addresses rapid prediction of polymer melt viscosity for additive manufacturing by introducing a Physics-Enforced Neural Network (PENN) that embeds empirical η(T, M_w, γ̇) relationships while learning material-specific parameters from polymer fingerprints. PENN is benchmarked against physics-unaware ANN and GPR models and demonstrates superior extrapolation in data-sparse chemical spaces, with improved physical realism and interpretability of the learned parameters. The dataset comprises 1903 observations across 93 repeat units, including homopolymers, copolymers, and blends, with strategies to augment underrepresented Mw regimes. The approach provides a blueprint for physics-informed materials informatics in rheology and offers a pathway to accelerate AM material development by reducing experimental burden.

Abstract

Achieving superior polymeric components through additive manufacturing (AM) relies on precise control of rheology. One key rheological property particularly relevant to AM is melt viscosity ($η$). Melt viscosity is influenced by polymer chemistry, molecular weight ($M_w$), polydispersity, induced shear rate ($\dotγ$), and processing temperature ($T$). The relationship of $η$ with $M_w$, $\dotγ$, and $T$ may be captured by parameterized equations. Several physical experiments are required to fit the parameters, so predicting $η$ of a new polymer material in unexplored physical domains is a laborious process. Here, we develop a Physics-Enforced Neural Network (PENN) model that predicts the empirical parameters and encodes the parametrized equations to calculate $η$ as a function of polymer chemistry, $M_w$, polydispersity, $\dotγ$, and $T$. We benchmark our PENN against physics-unaware Artificial Neural Network (ANN) and Gaussian Process Regression (GPR) models. Finally, we demonstrate that the PENN offers superior values of $η$ when extrapolating to unseen values of $M_w$, $\dotγ$, and $T$ for sparsely seen polymers.

Figures (5)

• Figure 1: The melt viscosity ($\eta$) learning problem and machine-learning workflow. (A) Depictions of the functions used to describe the behavior of $\eta$ with respect to temperature ($T$), molecular weight ($M_w$), and shear rate ($\dot{\gamma}$). The functions are parametrized by empirical parameters with physical significance, elaborated in Table \ref{['tbl:PENN_const']} and in the Methods section. The $\eta$ dependence on $M_w$ is given by $\log {\eta}_{M_w}$ (Equation \ref{['eq: Mw_smooth']} in the Methods section). Empirical parameters define the slopes of the relationship at low $M_w$ ($\alpha_1$) and high $M_w$ ($\alpha_2$), the critical molecular weight ($M_{cr}$), the y-intercept of $\eta_{M_w}$ ($k_1$), and the rate of transition from low to high $M_w$ regions ($\beta_{M_w}$). The $\eta$ dependence on $T$ and $M_w$ is given by $\log \eta_0(T, M_w)$ (Equation \ref{['eq: T']} in the Methods section), and is parameterized by reference temperature ($T_r$) and empirical fitting parameters ($C_1$ and $C_2$). The effects of $C_1$ and $C_2$ are visualized by comparing the trends with different sampled values. The $\eta$ dependence on $\dot\gamma$ is given by $\log \eta(T, M_w, \dot{\gamma})$ (Equation \ref{['eq: shear_smooth']} 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 $\eta_0$ 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 $\eta$, 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, $T$, $M_w$, $\dot{\gamma}$, and PDI.
• Figure 2: The joint distributions of A) molecular weight ($M_w$), B) shear rate ($\dot\gamma$), C) temperature ($T$), and D) polydispersity index (PDI) with respect to melt viscosity ($\eta$) are presented. The single distributions for the physical conditions are given on the top axes and the distribution of $\eta$ is given on the right-most axis. Each subplot contains all 1903 datapoints from the dataset. A, B, and C have highlighted samples in red that exemplify the dependencies depicted in Figure \ref{['fig:workflow']}A. E) Visual depiction of train-test splitting across chemical space and physical spaces for N monomers in the dataset.
• Figure 3: Parity plots are used to assess the models' overall predictive capabilities in new physical regimes based on the physical variable split for molecular weight ($M_w)$, shear rate ($\dot\gamma$), and temperature $T$. Results are compared between Gaussian Process Regression (GPR), Artificial Neural Network (ANN), and Physics Enforced Neural Network (PENN) models. Each plot compares experimental values for melt viscosity ($\eta$) to the predicted $\eta$ across 3 unique test-train splits for each physical variable. The top row (A-C) contains GPR results for A) the $M_w$ split, B) the $\dot\gamma$ split C) the $T$ split. The middle row (D-F) contains ANN results for D) the $M_w$ split, E) the $\dot\gamma$ split F) the $T$ split. The bottom row (G-I) contains PENN results for D) the $M_w$ split, E) the $\dot\gamma$ split F) the $T$ split. The dotted black lines represent perfect predictions. The coefficient of determination ($R^2$) and Order of Magnitude Error (OME) are reported over these test sets.
• Figure 4: Normalized distributions of empirical parameter values found in the dataset (Ground Truth) are compared to parameter values predicted by Gaussian Process Regression (GPR), Artificial Neural Network (ANN) and Physics Enforced Neural Network (PENN) models. Each column compares a different parameter for the melt viscosity ($\eta$) relationship with molecular weight ($M_w$), shear rate ($\dot\gamma$), and temperature ($T$). The examined parameters include: A) $\alpha_1$, the slope of zero-shear viscosity ($\eta_0$) vs. $M_w$ correlation at low $M_w$ (accepted value of 1 depicted by the red dashed line) B) $\alpha_2$, the slope of $\eta_0$ vs. $M_w$ at high $M_w$ (accepted value of 3.4 depicted by the red dashed line), C) critical molecular weight ($M_{cr}$), D) $n$, the rate of shear thinning (accepted range of 0.2-0.8 depicted by the dashed red lines), E) critical shear rate ($\dot\gamma_{cr}$), F) reference temperature ($T_r$) of a polymer. G and H) show distributions for the $C_1$ and $C_2$ fitting parameters for the $\eta$-$T$ trend. The ground truth distributions represent 41 samples for $M_w$ parameters, 33 samples for $\dot\gamma$ parameters, and 22 samples for $T$ parameters. The Kullback–Leibler (KL) divergence of the model estimation distributions from the ground truth is given in the top left of each histogram. The lowest KL divergence among the three models is bolded for each parameter.
• Figure 5: Examples of accurate (A-C) and inaccurate (D-F) melt viscosity ($\eta$) and zero-shear melt viscosity ($\eta_0$) predictions over wide ranges of molecular weight ($M_w$), shear rate ($\dot\gamma$), temperature ($T$) by the Physics Enforced Neural Network (PENN) models. The extrapolated predictions are compared to those by Gaussian Process Regression (GPR) and Artificial Neural Network (ANN) models given the same training information. A) is a good $\eta_0$-$M_w$ extrapolation for [*]CCCCCCCCCCOC(=O)CCCCC(=O)O[*] at $T=382.15$ K. B) is a good $\eta$-$\dot\gamma$ extrapolation for a copolymer of [*]CC([*])CC(C)C and [*]CC([*])CCCCCCCC (0.968:0.032) ($M_w = 290000$ g/mol, PDI = 7.8) at $T = 543.15$ K. C) is a good $\eta$-$T$ extrapolation for [*]CCOCCOCCOC(=O)CCCCCCCCC(=O)O[*] ($M_w$ = 2000 g/mol, $\dot{\gamma}$ = 60 1/s). D) is an unsuccessful $\eta_0$-$M_w$ extrapolation for [*]C=CCC[*] at $T = 490.15$ K, with possible mispredictions of $M_{cr}$ and $k_1$. E) is an unsuccessful $\eta$-$\dot\gamma$ extrapolation for a copolymer of [*]C[*] and [*]CC([*])OC(C) (0.72:0.28) ($M_w = 60000$ g/mol), with possible misprediction of $\hat{\dot}\gamma_{cr}$ and $\eta_0$. F) is an unsuccessful $\eta$-$T$ extrapolation for [*]CC(O)COc1ccc(C(C)(C)c2ccc(O[*])cc2)cc1 ($M_w$ = 1696 g/mol, $\dot{\gamma}$ = 0.0 1/s) with a possible misprediction of $T_r$.