Coulomb interaction, following an inverse-square force-law, quantifies the amount of force between two stationary and electrically charged particles. The long-range nature of Coulomb interactions poses a major challenge to molecular dynamics simulations, which are major tools for problems at the nano-/micro-scale. Various algorithms are developed to calculate the pairwise Coulomb interactions to a linear scale, but poor scalability limits the size of simulated systems. Here, we use an efficient molecular dynamics algorithm with the random batch Ewald method on all-atom systems where the complete Fourier components in the Coulomb interaction are replaced by randomly selected mini-batches. By simulating the N-body systems up to 10⁸ particles using 10 000 central processing unit cores, we show that this algorithm furnishes O(N) complexity, almost perfect scalability, and an order of magnitude faster computational speed when compared to the existing state-of-the-art algorithms. Further examinations of our algorithm on distinct systems, including pure water, a micro-phase separated electrolyte, and a protein solution, demonstrate that the spatiotemporal information on all time and length scales investigated and thermodynamic quantities derived from our algorithm are in perfect agreement with those obtained from the existing algorithms. Therefore, our algorithm provides a promising solution on scalability of computing the Coulomb interaction. It is particularly useful and cost-effective to simulate ultra-large systems, which is either impossible or very costly to conduct using existing algorithms, and thus will be beneficial to a broad range of problems at nano-/micro-scales.

We develop an accurate, highly efficient, and scalable random batch Ewald (RBE) method to conduct molecular dynamics simulations in the isothermal–isobaric ensemble (the NPT ensemble) for charged particles in a periodic box. After discretizing the Langevin equations of motion derived using suitable Lagrangians, the RBE method builds the mini-batch strategy into the Fourier space in the Ewald summation for the pressure and forces such that the computational cost is reduced to $\mathcal{O}(N)$ per time step. We implement the method in the Large-scale Atomic/Molecular Massively Parallel Simulator package and report accurate simulation results for both dynamical quantities and statistics for equilibrium for typical systems including all-atom bulk water and a semi-isotropic membrane system. Numerical simulations on massive supercomputing cluster are also performed to show promising central processing unit efficiency of the RBE.

Machine-learning interatomic potentials have emerged as a revolutionary class of force-field models in molecular simulations, delivering quantum-mechanical accuracy at a fraction of the computational cost and enabling the simulation of large-scale systems over extended timescales. However, they often focus on modeling local environments, neglecting crucial long-range interactions. We propose a sum-of-Gaussians neural network (SOG-Net), a lightweight and versatile framework for integrating long-range interactions into a machine-learning force field. The SOG-Net employs a latent-variable learning network that seamlessly bridges short-range and long-range components, coupled with an efficient Fourier convolution layer that incorporates long-range effects. By learning sum-of-Gaussians multipliers across different convolution layers, the SOG-Net adaptively captures diverse long-range decay behaviors while maintaining close-to-linear computational complexity during training and simulation via nonuniform fast Fourier transforms. The method is demonstrated effective for a broad range of long-range systems.

We develop an accurate, highly efficient, and scalable random batch sum-of-Gaussians (RBSOG) method for molecular dynamics simulations of systems with Coulomb interactions. The idea of the RBSOG method is based on a sum-of-Gaussians decomposition of the Coulomb kernel, and then a random batch importance sampling on the Fourier space is employed for approximating the Fourier expansion of the Gaussians with large bandwidths (the far components). The importance sampling significantly reduces the computational cost, resulting in a scalable algorithm by avoiding the use of communication-intensive fast Fourier transform. Theoretical analysis is presented to demonstrate the unbiasedness of the approximate force, the controllability of variance, and the convergence of the algorithm. The resultant method has [Formula: see text] complexity with low communication latency. Accurate simulation results on both dynamical and equilibrium properties of benchmark problems are reported to illustrate the attractive performance of the method. Simulations on parallel computing are, also performed to show the high parallel efficiency. The RBSOG method can be straightforwardly extended to more general interactions with long-ranged kernels and thus is promising for constructing fast algorithms for molecular dynamics simulations of various systems.

We have developed an accurate and efficient method for molecular dynamics simulations of charged particles confined by planar dielectric interfaces. The algorithm combines the image-charge method for near field with the harmonic surface mapping, which converts the contribution of infinite far-field charges into a finite number of charges on an auxiliary spherical surface. We approximate the electrostatic potential of far-field charges via spherical harmonic expansion and determine the coefficients by fitting the Dirichlet-to-Neumann boundary condition, which only requires the potential within the simulation cell. Instead of performing the direct evaluation of spherical harmonic series expansion, we use Green’s second identity to transform the series expansion into a spherical integral, which can be accurately represented by discrete charges on the sphere. Therefore, the fast multipole method can be readily employed to sum over all charges within and on the sphere, achieving truly linear $\mathcal{O}\left(N\right)$ complexity. Our algorithm can be applied to a broad range of charged complex fluids under dielectric confinement.