Accelerating Calculations of Leonard-Jones Potentials using Physics-Informed Neural Network

Computing Leonard-Jones potentials dominates the computational costs in many simulations involving molecular dynamics (MD), for example, in platelet dynamics that plays an essential role in understanding the mechanisms of blood clotting. The extreme computational workload demand of MD simulations often hinders the study of bioprocess with large spatial and temporal scales. In the recent decades, the rise of machine learning (ML) techniques allows for a significant reduction of computational workload while maintaining the accuracy, by utilizing the ML predictions for MD calculations. Our work aims to simplify the computation of Lennard-Jones (LJ) potentials, which is the intermolecular force fields between interacting platelets, by proposing a generalized equation for theoretically similar ellipsoid models.<br/><br/><b>Problem Statement</b>: The study considers the two interacting platelets, B₁ and B₂, to be modeled as ellipsoids, r as the distance between B₁ and B₂, Φ, Ψ, and θ and as the relative rotation angle of B₁ and B₂, respectively. Numerical experiments are conducted to generate training samples. We caculated the LJ potential for each 300 thousand random distrubuted points for each ellipsoid and used Monte Carlo Integration to estimate the double summation over each particle in a fixed form. The length, angle, and distance from center for x, y, z directions are measured to be the independent variables. By using the massive training set with 208,000 samples and 10 independent variables, we train our models to describe the predicted equation that governs the integrated potentials between interacting ellipsoids while significantly reducing the computation time.<br/><br/><b>The Results</b>: With the numerical training set, we designed a physics-informed neural network (PINN) [1] guided by the proposed equation for LJ potentials with variables of relative distance and orientation (λ₂) between platelets and their respective correction parameters, λ₁and λ₃, presented in the equation as LJP= λ₁λ₂[λ₃(σ/r)⁷-(σ/r)]. We used 10 neural network hidden layers each with 20 nerons and employed the Adam optimizer with a learning rate of 0.001.We splited the data to training and testing sets into 1:1 ratio to avoid potential overfitting issues. We then trained the data for 15K iterations with the relative orientation (λ₂) as the target output in the neural net framwor and use the results to calculate the predicted LJ potential value using the aformentioned equation. Finally, we assess the PINN training with a loss function of root mean square error (RMSE) to penalize the difference between predicted LJ potential and the true LJ potential value. In the range of LJ potential value less than 0, we got a RMSE error of 0.01568. Our learned force-field parameters for orientation (λ₁) and distance (λ₃) are -1.526 and -1.915, repectively.<br/><br/>We learned the paramaters for relative orientation and distance for the generalized LJ potential equation to describe the large-scale biophysics systems. The leveraged PINN effectively accelerates the learning process and improve the overall approximation accuracy of the model prediction by embedding the governed physics principles. Our work contributes to thrombosis simulation which could help us better understand the complex risk of blood clotting, providing promising applications to learn the generalized equations of other similar biophysics systems. Future work can investigate other neural network model architectures to improve performance.<br/><br/>Acknowledgment: The project is supported by the Stony Brook Garcia Scholars for Materials Research and funded by Morin Charitable Trust.<br/><br/>References:<br/>[1] Raissi, Maziar, Paris Perdikaris, and George E. Karniadakis. "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations." Journal of Computational Physics 378 (2019): 686-707.