Rapid Neural Network Prediction of Linear Block Copolymer Free Energies

Abstract

Free energies are fundamental quantities governing phase behavior and thermodynamic stability in polymer systems, yet their accurate computation often requires extensive simulations and post-processing techniques such as the Bennett Acceptance Ratio (BAR). While BAR provides reliable estimates when applied between closely related thermodynamic states, evaluating free energies across large changes in interaction strength typically requires a sequence of intermediate simulations to maintain sufficient phase-space overlap, substantially increasing computational cost. In this work we develop a machine learning framework for rapidly predicting excess free energies of linear diblock copolymer systems from simulation-derived energetic descriptors. Using dissipative particle dynamics simulations of freely-jointed chain polymers, we construct a dataset of per-chain energetic statistics, including heterogeneous interaction energies, homogeneous interaction energies, and bonded spring energies, and train feed-forward neural networks to learn the relationship between these descriptors and free energies computed using a stratified BAR procedure. The resulting models accurately reproduce the reference free energies across a range of chain lengths, compositions, and densities, including polymer architectures held out from training. In regimes where direct, brute-force BAR estimates become unreliable due to poor phase-space overlap, the neural network predictions remain consistent with the reference values. These results demonstrate that physically informed machine learning models can serve as efficient surrogates for expensive free-energy calculations and provide a promising approach for accelerating thermodynamic analysis of polymer systems.

Figures (7)

• Figure 1: dBCP simulations are run across a spectrum of $\chi$ parameters, and a feed-forward neural network is trained on selected parameters to predict computed free energies.
• Figure 2: Comparison between reference excess free energies $\Delta F$ (dashed line) and values estimated using the Bennett acceptance ratio (BAR) method (red crosses) for an $A_9B_9$ diblock copolymer as a function of $\chi N$. Results are shown for densities (a) $\rho=3$ and (b) $\rho=5$. While BAR accurately reproduces the free energy for small $\chi N$, deviations grow as $\chi N$ increases, reflecting the breakdown of phase-space overlap between sampled states.
• Figure 3: Training and testing output parity plots for (a) the $\alpha$ only model and (b) the $\alpha$, $\mathbb{E}\left[ E_{AB} \right]$ model. The black $y=x$ line represents perfect parity. The discrepancy between training and testing MSE is a consequence of the restricted testing dataset, which is not observed in later models.
• Figure 4: Training and testing output parity plots for (a) the $\alpha$, $\mathbb{E}\left[ E_{AB} \right]$, $\mathbb{E}\left[ E_{\text{hom}} \right]$ model and (b) the $\alpha$, $\mathbb{E}\left[ E_{AB} \right]$, $\mathbb{E}\left[ E_{\text{hom}} \right]$, $\mathbb{E}\left[ E_{K} \right]$ model. The black $y=x$ line represents perfect parity.
• Figure 5: Training and testing output parity plots for (a) the $\alpha$, $\mathbb{E}\left[ \cdot \right]$, $\sigma\left[ \cdot \right]$ model and (b) the $\alpha$, $\mathbb{E}\left[ \cdot \right]$, $\sigma\left[ \cdot \right]$, $\mathrm{Cov}\!\left[\cdot,\cdot\right]$ model, both of which incorporate all three energy terms outlined previously. The black $y=x$ line represents perfect parity.