Quantifying the Lennard–Jones Potential between Two Hard Ellipsoids using Coarse-Grained Deep Learning Techniques

Equations: Eq. (1) is presented as the 12–6 Lennard–Jones potential: <i>V</i>_LJ = 4<i>ε</i>_LJ[(<i>σ</i>/<i>r</i>)¹² – (<i>σ</i>/<i>r</i>)⁶]. Eq. (2) is presented as the Gay–Berne potential: <i>U</i>_AB = 4<i>ε</i>_AB{[<i>σ</i>₀/(<i>R</i>_AB – <i>σ</i>_AB + <i>σ</i>₀)]¹² – [<i>σ</i>₀/(<i>R</i>_AB – <i>σ</i>_AB + <i>σ</i>₀)]⁶}. Eq. (3) is presented as the double summation of the 12–6 Lennard–Jones potential, <i>V</i>_LJ, over the individual points that subsume the pairwise rigid bodies.<br/><br/>Classical molecular dynamics (MD) is a highly intensive, O(<i>n</i>²), computational method for simulating the kinetics, thermodynamics, and structural properties of a many-body system over time. The initialization of such a method requires the use of multiple energy functions, which, in turn, describe the trajectory and momenta of the particles being simulated. We focus on the Lennard–Jones potential (LJP), which governs the van der Waals (vdW) energetics between, specifically, two interacting particles. We seek to modify the potential’s conventional form, eq. (1), which fails to accurately model the vdW forces between large and anisotropic particles (ellipsoids). Eq. (3), however, overcomes those shortcomings by treating the ellipsoids as a collection of uniformly distributed points and, in turn, applying a double summation over those points to generate the LJP.<br/><br/>We learn the variables of eq. (2) using a physics-informed neural network (PINN), which, unlike our previously proposed feedforward neural network, assumes a physics-principled training process that, in our case, entails the parametrization of eq. (2). Our goal is to replicate the accuracy of eq. (3) by training our PINN to recognize the relationship between the variables of eq. (2) and the imputed parameters—relative orientation, semi-major/-minor axes, and center—of the two ellipsoids. Our PINN then uses those imputed parameters to generate the corresponding <i>ε</i>_AB and <i>σ</i>_AB values, thereby streamlining the complexity involved in the computation of eq. (3).<br/><br/>Our ground truth, serving as the basis of our training data, stemmed from the Monte Carlo approximation of eq. (3) and consisted of 2,080,000 potentials. The parameters of our two ellipsoids—relative orientation, semi-major/-minor axes, and center—were uniquely permuted such that no two potentials were generated using the same parameters.<br/><br/>One of the primary concerns about our data’s accuracy was the extent to which an ellipsoid could be classified as a rigid body. Therefore, we ran three mini-experiments to verify the equations and code used to generate the ground truth. We referenced the three cases shown by Paramonov & Yaliraki and demonstrated (a) that the number of points per ellipsoid, upon exceeding 15,000, minimally impacted the generated potential and (b) that ground truth generation code proved scientifically sound, emulating the well depth and curvature in all three cases shown by Paramonov and Yaliraki. Moreover, in addressing part (a), we randomly sampled a subset of our ground truth (generated using 30,000 points per ellipsoid) and regenerated those LJP values using 5,000, 15,000, 25,000, and 30,000 points. The mean absolute percentage error (MAPE) started at 2.42% for 5,000 points, indicating, on average, a 2.42% error between a value from the dataset to a newly generated value using our code, but upon exceeding 15,000 points, gradually decreased to 0.96% (MAPE_30,000), thus confirming the accuracy of our ground truth.<br/><br/>We implemented two models using this ground truth. The first was a black box model, which used the parameters of two ellipsoids to predict the LJP. The second was a PINN that used the same parameters of the two ellipsoids to find the coefficients of <i>ε</i>_AB and <i>σ</i>_AB through two separate neural networks and tune <i>σ</i>₀ for eq. (2). By employing a combination of Adam and L-BFGS optimization, the PINN was able to achieve a minimum training loss of 0.0004 over 250 epochs. We plan to validate this PINN using a separate testing dataset and further improve upon it by investigating the use of other activation and loss functions.