Differentiable, Hardware Accelerated, Molecular Dynamics

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Molecular dynamics is a workhorse of modern computational condensed matter physics. It is frequently used to simulate materials to observe how small scale interactions can give rise to complex large-scale phenomenology. Most molecular dynamics packages (e.g. HOOMD Blue or LAMMPS) are complicated, specialized pieces of code that are many thousands of lines long. They typically involve significant code duplication to allow for running simulations on CPU and GPU. Additionally, large amounts of code is often devoted to taking derivatives of quantities to compute functions of interest (e.g. gradients of energies to compute forces).

However, recent work in machine learning has led to significant software
developments that might make it possible to write more concise
molecular dynamics simulations that offer a range of benefits. Here we target
JAX, which allows us to write python code that gets compiled to XLA and allows
us to run on CPU, GPU, or TPU. Moreover, JAX allows us to take derivatives of
python code. Thus, not only is this molecular dynamics simulation automatically
hardware accelerated, it is also **end-to-end** differentiable. This should
allow for some interesting experiments that we're excited to explore.

JAX, MD is a research project that is currently under development. Expect sharp edges and possibly some API breaking changes as we continue to support a broader set of simulations. JAX MD is a functional and data driven library. Data is stored in arrays or tuples of arrays and functions transform data from one state to another.

To get started playing around with JAX MD check out the following colab notebooks on Google Cloud without needing to install anything. For a very simple introduction, I would recommend the Minimization example. For an example of a bunch of the features of JAX MD, check out the JAX MD cookbook.

- JAX MD Cookbook
- Minimization
- NVE Simulation
- NVT Simulation
- NVE with Neighbor Lists
- Custom Potentials
- Neural Network Potentials
- Flocking
- Meta Optimization

You can install JAX MD locally with pip,

pip install jax-md --upgradeIf you want to build the latest version then you can grab the most recent version from head,

git clone https://github.com/google/jax-md pip install -e jax-md

We now summarize the main components of the library.

space.py)

In general we must have a way of computing the pairwise distance between atoms. We must also have efficient strategies for moving atoms in some space that may or may not be globally isomorphic to R^N. For example, periodic boundary conditions are commonplace in simulations and must be respected. Spaces are defined as a pair of functions,

(displacement_fn, shift_fn). Given two points

displacement_fn(R_1, R_2)computes the displacement vector between the two points. If you would like to compute displacement vectors between all pairs of points in a given

(N, dim)matrix the function

space.map_productappropriately vectorizes

displacement_fn. It is often useful to define a metric instead of a displcement function in which case you can use the helper function

space.metricto convert a displacement function to a metric function. Given a point and a shift

shift_fn(R, dR)displaces the point

Rby an amount

dR.

The following spaces are currently supported: -

space.free()specifies a space with free boundary conditions. -

space.periodic(box_size)specifies a space with periodic boundary conditions of side length

box_size. -

space.periodic_general(T)specifies a space as a periodic parellelopiped formed by transforming the unit cube by an affine transformation

T. Note that

Tcan be a time dependent function

T(t)that is useful for simulating systems under strain.

Example:

from jax_md import space box_size = 25.0 displacement_fn, shift_fn = space.periodic(box_size)

energy.py)

In the simplest case, molecular dynamics calculations are often based on a pair potential that is defined by a user. This then is used to compute a total energy whose negative gradient gives forces. One of the very nice things about JAX is that we get forces for free! The second part of the code is devoted to computing energies.

We provide the following classical potentials: -

energy.soft_spherea soft sphere whose energy incrases as the overlap of the spheres to some power,

alpha. -

energy.lennard_jonesa standard 12-6 lennard-jones potential. -

energy.morsea morse potential. -

energy.eamembedded atom model potential with ability to load parameters from LAMMPS files.

We also provide the following neural network potentials: -

energy.behler_parrinelloa widely used fixed-feature neural network architecture for molecular systems. -

energy.graph_networka deep graph neural network designed for energy fitting.

For finite-ranged potentials it is often useful to consider only interactions within a certain neighborhood. We include the

_neighbor_listmodifier to the above potentials that uses a list of neighbors (see below) for optimization.

Example:

import jax.numpy as np from jax import random from jax_md import energy, quantity N = 1000 spatial_dimension = 2 key = random.PRNGKey(0) R = random.uniform(key, (N, spatial_dimension), minval=0.0, maxval=1.0) energy_fn = energy.lennard_jones_pair(displacement_fn) print('E = {}'.format(energy_fn(R))) force_fn = quantity.force(energy_fn) print('Total Squared Force = {}'.format(np.sum(force_fn(R) ** 2)))

simulate.py,

minimize.py)

Given an energy function and a system, there are a number of dynamics are useful to simulate. The simulation code is based on the structure of the optimizers found in JAX. In particular, each simulation function returns an initialization function and an update function. The initialization function takes a set of positions and creates the necessary dynamical state variables. The update function does a single step of dynamics to the dynamical state variables and returns an updated state.

We include a several different kinds of dynamics. However, there is certainly room to add more for e.g. constaint strain simulations.

It is often desirable to find an energy minimum of the system. We provide two methods to do this. We provide simple gradient descent minimization. This is mostly for pedagogical purposes, since it often performs poorly. We additionally include the FIRE algorithm which often sees significantly faster convergence. Moreover a common experiment to run in the context of molecular dynamics is to simulate a system with a fixed volume and temperature.

We provide the following dynamics: -

simulate.nveConstant energy simulation; numerically integrates Newton's laws directly. -

simulate.nvt_nose_hooverUses Nose-Hoover chain to simulate a constant temperature system. -

simulate.nvt_langevinSimulates a system by numerically integrating the Langevin stochistic differential equation. -

simulate.brownianSimulates brownian motion. -

minimize.gradient_descentMimimizes a system using gradient descent. -

minimize.fire_descentMinimizes a system using the fast inertial relaxation engine.

Example:

from jax_md import simulate temperature = 1.0 dt = 1e-3 init, update = simulate.nvt_nose_hoover(energy_fn, shift_fn, dt, temperature) state = init(key, R) for _ in range(100): state = update(state) R = state.position

partition.py)

In many applications, it is useful to construct spatial partitions of particles / objects in a simulation.

We provide the following methods: -

partition.cell_listPartitions objects (and metadata) into a grid of cells. -

partition.neighbor_listConstructs a set of neighbors within some cutoff distance for each object in a simulation.

Cell List Example: ```python from jax_md import partition

cell*size = 5.0
capacity = 10
cell*list*fn = partition.cell*list(box*size, cell*size, capacity)
cell*list*data = cell*list*fn(R)
```

Neighbor List Example: ```python from jax_md import partition

neighbor*list*fn = partition.neighbor*list(displacement*fn, box*size, cell*size)
neighbors = neighbor*list*fn(R) # Create a new neighbor list.

neighbors = neighbor*list*fn(R, neighbors) # Update the neighbor list without resizing.
if neighbors.did*buffer*overflow: # Couldn't fit all the neighbors into the list.
neighbors = neighbor*list*fn(R) # So create a new neighbor list.

# DevelopmentJAX MD is under active development. We have very limited development resources and so we typically focus on adding features that will have high impact to researchers using JAX MD (including us). Please don't hesitate to open feature requests to help us guide development. We more than welcome contributions!

## Technical gotchas

## GPU

You must follow JAX's GPU installation instructions to enable GPU support.

## 64-bit precision

To enable 64-bit precision, set the respective JAX flag

beforeimporting`jax_md`

(see the JAX guide), for example:```python from jax.config import config config.update("jax_enable_x64", True)

If you use the code in a publication, please cite the repo using the .bib,

@misc{jaxmd2019, title={JAX M.D.: End-to-End Differentiable, Hardware Accelerated, Molecular Dynamics in Pure Python}, author={Samuel S. Schoenholz and Ekin D. Cubuk}, year={2019}, eprint={1912.04232}, archivePrefix={arXiv}, primaryClass={stat.ML}, howpublished={\url{https://github.com/google/jax-md}, \url{https://arxiv.org/abs/1912.04232}}, }