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Dissipative particle dynamics (DPD) is a relatively new technique which has proved successful in the simulation of complex fluids. We caution that for the equilibrium achieved by the DPD simulation of a simple fluid the temperature depends…
Gradient-based optimization methods are commonly used to identify local optima in high-dimensional spaces. When derivatives cannot be evaluated directly, stochastic estimators can provide approximate gradients. However, these estimators'…
In this paper we will study some interesting properties of modifications of the Euler-Poincar\'e equations when we add a special type of dissipative force, so that the equations of motion can be described using the metriplectic formalism.…
We propose new continuous-time formulations for first-order stochastic optimization algorithms such as mini-batch gradient descent and variance-reduced methods. We exploit these continuous-time models, together with simple Lyapunov analysis…
An implicit scheme for steady state solutions of diatomic gas flow is presented. The method solves the Rykov model equation in the finite volume discrete velocity method (DVM) framework, in which the translational and rotational degrees of…
Diffusive molecular dynamics is a novel model for materials with atomistic resolution that can reach diffusive time scales. The main ideas of diffusive molecular dynamics are to first minimize an approximate variational Gaussian free energy…
We introduce a scheme for molecular simulations, the Deep Potential Molecular Dynamics (DeePMD) method, based on a many-body potential and interatomic forces generated by a carefully crafted deep neural network trained with ab initio data.…
The gradient discretisation method (GDM) is a generic framework for designing and analysing numerical schemes for diffusion models. In this paper, we study the GDM for the porous medium equation, including fast diffusion and slow diffusion…
Most of numerical methods for deterministic simulations of rarefied gas flows use the discrete velocity (or discrete ordinate) approximation. In this approach, the kinetic equation is approximated with a global velocity grid. The grid must…
Maintaining conservation laws in the fully discrete setting is critical for accurate long-time behavior of numerical simulations and requires accounting for discrete conservation properties in both space and time. This paper derives…
We propose new numerical approach to non-conservative dynamical systems. Our method being of low order, enhances qualitative performance of standard discrete gradient algorithm, thank to new concept of a reservoir. Paper is of explanatory…
Structure-preserving particle methods have recently been proposed for a class of nonlinear continuity equations, including aggregation-diffusion equation in [J. Carrillo, K. Craig, F. Patacchini, Calc. Var., 58 (2019), pp. 53] and the…
In this article, we consider discrete schemes for a fractional diffusion equation involving a tempered fractional derivative in time. We present a semi-discrete scheme by using the local discontinuous Galerkin (LDG) discretization in the…
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…
High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as nodal…
Granular impact -- the dynamic intrusion of solid objects into granular media -- is widespread across scientific and engineering applications including geotechnics. Existing approaches for simulating granular impact dynamics have relied on…
We prove closed-form equations for the exact high-dimensional asymptotics of a family of first order gradient-based methods, learning an estimator (e.g. M-estimator, shallow neural network, ...) from observations on Gaussian data with…
The next generation of force fields for molecular dynamics will be developed using a wealth of data. Training systematically with experimental data remains a challenge, however, especially for machine learning potentials. Differentiable…
We compare the performance of several discretizations of the simple pendulum equation in a series of numerical experiments. The stress is put on the long-time behaviour. We choose for the comparison numerical schemes which preserve the…
Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in…