import jax
import jax.numpy as jnp
import jax_cosmo as jc
from jax_cosmo import Cosmology
from jaxpm.growth import growth_factor, growth_rate, dGfa, growth_factor_second, growth_rate_second, dGf2a
from jaxpm.kernels import PGD_kernel, fftk, gradient_kernel, invlaplace_kernel, longrange_kernel
from jaxpm.painting import cic_paint, cic_read
def pm_forces(positions, mesh_shape, delta=None, r_split=0):
"""
Computes gravitational forces on particles using a PM scheme
"""
if delta is None:
delta_k = jnp.fft.rfftn(cic_paint(jnp.zeros(mesh_shape), positions))
elif jnp.isrealobj(delta):
delta_k = jnp.fft.rfftn(delta)
else:
delta_k = delta
# Computes gravitational potential
kvec = fftk(mesh_shape)
pot_k = delta_k * invlaplace_kernel(kvec) * longrange_kernel(kvec, r_split=r_split)
# Computes gravitational forces
return jnp.stack([cic_read(jnp.fft.irfftn(- gradient_kernel(kvec, i) * pot_k), positions)
for i in range(3)], axis=-1)
def lpt(cosmo:Cosmology, init_mesh, positions, a, order=1):
"""
Computes first and second order LPT displacement and momentum,
e.g. Eq. 2 and 3 [Jenkins2010](https://arxiv.org/pdf/0910.0258)
"""
a = jnp.atleast_1d(a)
E = jnp.sqrt(jc.background.Esqr(cosmo, a))
delta_k = jnp.fft.rfftn(init_mesh) # TODO: pass the modes directly to save one or two fft?
mesh_shape = init_mesh.shape
init_force = pm_forces(positions, mesh_shape, delta=delta_k)
dx = growth_factor(cosmo, a) * init_force
p = a**2 * growth_rate(cosmo, a) * E * dx
f = a**2 * E * dGfa(cosmo, a) * init_force
if order == 2:
kvec = fftk(mesh_shape)
pot_k = delta_k * invlaplace_kernel(kvec)
delta2 = 0
shear_acc = 0
# for i, ki in enumerate(kvec):
for i in range(3):
# Add products of diagonal terms = 0 + s11*s00 + s22*(s11+s00)...
# shear_ii = jnp.fft.irfftn(- ki**2 * pot_k)
nabla_i_nabla_i = gradient_kernel(kvec, i)**2
shear_ii = jnp.fft.irfftn(nabla_i_nabla_i * pot_k)
delta2 += shear_ii * shear_acc
shear_acc += shear_ii
# for kj in kvec[i+1:]:
for j in range(i+1, 3):
# Substract squared strict-up-triangle terms
# delta2 -= jnp.fft.irfftn(- ki * kj * pot_k)**2
nabla_i_nabla_j = gradient_kernel(kvec, i) * gradient_kernel(kvec, j)
delta2 -= jnp.fft.irfftn(nabla_i_nabla_j * pot_k)**2
init_force2 = pm_forces(positions, mesh_shape, delta=jnp.fft.rfftn(delta2))
# NOTE: growth_factor_second is renormalized: - D2 = 3/7 * growth_factor_second
dx2 = 3/7 * growth_factor_second(cosmo, a) * init_force2
p2 = a**2 * growth_rate_second(cosmo, a) * E * dx2
f2 = a**2 * E * dGf2a(cosmo, a) * init_force2
dx += dx2
p += p2
f += f2
return dx, p, f
def linear_field(mesh_shape, box_size, pk, seed):
"""
Generate initial conditions.
"""
kvec = fftk(mesh_shape)
kmesh = sum((kk / box_size[i] * mesh_shape[i])**2
for i, kk in enumerate(kvec))**0.5
pkmesh = pk(kmesh) * (mesh_shape[0] * mesh_shape[1] * mesh_shape[2]) / (
box_size[0] * box_size[1] * box_size[2])
field = jax.random.normal(seed, mesh_shape)
field = jnp.fft.rfftn(field) * pkmesh**0.5
field = jnp.fft.irfftn(field)
return field
def make_ode_fn(mesh_shape):
def nbody_ode(state, a, cosmo):
"""
state is a tuple (position, velocities)
"""
pos, vel = state
forces = pm_forces(pos, mesh_shape=mesh_shape) * 1.5 * cosmo.Omega_m
# Computes the update of position (drift)
dpos = 1. / (a**3 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * vel
# Computes the update of velocity (kick)
dvel = 1. / (a**2 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * forces
return dpos, dvel
return nbody_ode
def get_ode_fn(cosmo:Cosmology, mesh_shape):
def nbody_ode(a, state, args):
"""
State is an array [position, velocities]
Compatible with [Diffrax API](https://docs.kidger.site/diffrax/)
"""
pos, vel = state
forces = pm_forces(pos, mesh_shape) * 1.5 * cosmo.Omega_m
# Computes the update of position (drift)
dpos = 1. / (a**3 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * vel
# Computes the update of velocity (kick)
dvel = 1. / (a**2 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * forces
return jnp.stack([dpos, dvel])
return nbody_ode
def pgd_correction(pos, mesh_shape, params):
"""
improve the short-range interactions of PM-Nbody simulations with potential gradient descent method,
based on https://arxiv.org/abs/1804.00671
args:
pos: particle positions [npart, 3]
params: [alpha, kl, ks] pgd parameters
"""
kvec = fftk(mesh_shape)
delta = cic_paint(jnp.zeros(mesh_shape), pos)
alpha, kl, ks = params
delta_k = jnp.fft.rfftn(delta)
PGD_range=PGD_kernel(kvec, kl, ks)
pot_k_pgd=(delta_k * invlaplace_kernel(kvec))*PGD_range
forces_pgd= jnp.stack([cic_read(jnp.fft.irfftn(- gradient_kernel(kvec, i)*pot_k_pgd), pos)
for i in range(3)],axis=-1)
dpos_pgd = forces_pgd*alpha
return dpos_pgd
def make_neural_ode_fn(model, mesh_shape):
def neural_nbody_ode(state, a, cosmo:Cosmology, params):
"""
state is a tuple (position, velocities)
"""
pos, vel = state
kvec = fftk(mesh_shape)
delta = cic_paint(jnp.zeros(mesh_shape), pos)
delta_k = jnp.fft.rfftn(delta)
# Computes gravitational potential
pot_k = delta_k * invlaplace_kernel(kvec) * longrange_kernel(kvec, r_split=0)
# Apply a correction filter
kk = jnp.sqrt(sum((ki/jnp.pi)**2 for ki in kvec))
pot_k = pot_k *(1. + model.apply(params, kk, jnp.atleast_1d(a)))
# Computes gravitational forces
forces = jnp.stack([cic_read(jnp.fft.irfftn(- gradient_kernel(kvec, i)*pot_k), pos)
for i in range(3)],axis=-1)
forces = forces * 1.5 * cosmo.Omega_m
# Computes the update of position (drift)
dpos = 1. / (a**3 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * vel
# Computes the update of velocity (kick)
dvel = 1. / (a**2 * jnp.sqrt(jc.background.Esqr(cosmo, a))) * forces
return dpos, dvel
return neural_nbody_ode