I want to use numpyro to infer the parameters of an ODE system. The complication is that I want to solve the same ODE system for 100s/1000s of different objects (different initial conditions) within a single iteration of free parameter values. In principle I could use vmap for this but I want to use pmap (or shard_map) to take advantage of the embarrassingly parallel nature of this problem to assign each ODE system to a different CPU core on a single node. (vmap over my ODE system takes forever to compile, and while the thousands of cores on a single GPU may help, I might need multiple GPUs anyway due to their limited memory.)

I have adapted the official predator-prey numpyro example (Example: Predator-Prey Model — NumPyro documentation) to illustrate how I imagine using pmap within the numpyro model function. The following code works for this simple example but I get a warning: `UserWarning: The jitted function scan includes a pmap. Using jit-of-pmap can lead to inefficient data movement, as the outer jit does not preserve sharded data representations and instead collects input and output arrays onto a single device. Consider removing the outer jit unless you know what you're doing. See https://github.com/google/jax/issues/2926.`

For my actual more complicated ODE system where pmap takes longer to finish (~30 sec), I get that same warning plus MANY more print statements that look like

```
E0324 17:04:20.251938 815459 collective_ops_utils.h:269] This thread has been waiting for 5000ms for and may be stuck: participant AllReduceParticipantData{rank=0, element_count=280, type=F64, rendezvous_key=RendezvousKey{run_id=RunId: 248276, global_devices=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39], num_local_participants=40, collective_op_kind=cross_replica, op_id=220}} waiting for all participants to arrive at rendezvous RendezvousKey{run_id=RunId: 248276, global_devices=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39], num_local_participants=40, collective_op_kind=cross_replica, op_id=220}
```

**Is there a recommended best practice for using pmap/shard_map within a numpyro model where, within a single iteration of MCMC (single realization of free parameters), we need to evaluate an expensive but embarrassingly parallel function (ODE solver in this case) over 1000s of other fixed parameters?**

```
import os
os.environ["XLA_FLAGS"] = "--xla_force_host_platform_device_count=8" # assume we have 8 CPU cores for pmap
from jax import config
config.update("jax_enable_x64", True)
from jax.random import PRNGKey, uniform, lognormal
import jax.numpy as jnp
from jax import jit, pmap
import numpyro
import numpyro.distributions as dist
from numpyro.examples.datasets import LYNXHARE, load_dataset
from numpyro.infer import MCMC, NUTS, Predictive
from diffrax import diffeqsolve, ODETerm, SaveAt, Tsit5
### download pre-packaged numpyro data for Lotka-Volterra problem
_, fetch = load_dataset(LYNXHARE, shuffle=False)
year, data = fetch() # data is in hare -> lynx order
### Lotka-Volterra ODE function
def vector_field(t, y, args):
prey, predator = y
alpha, beta, gamma, delta = args
dprey_dt = alpha * prey - beta * prey * predator
dpredator_dt = -gamma * predator + delta * prey * predator
return dprey_dt, dpredator_dt
### setup for diffrax
term = ODETerm(vector_field)
solver = Tsit5()
t0 = jnp.asarray(year[0],dtype=jnp.float64)
t1 = jnp.asarray(year[-1],dtype=jnp.float64)
dt0 = 0.1
saveat = SaveAt(ts=jnp.asarray(year,dtype=jnp.float64)) # evaluate at same times as observations
"""
pretend we have 8 different objects aka 8 different ICs we want to pmap over
yes this could be done with vmap, but for my actual problem, I need pmap/shard_map
since I have 1000s of objects/ICs (embarassingly parallel problem) and vmap is slow to compile
"""
# generate 8 random combinations of (alpha,beta,gamma,delta,init_prey,init_predator)
rand_params = uniform(PRNGKey(1),shape=(8,4),minval=0.01,maxval=1.0) # (alpha,beta,gamma,delta) ~ U(0.01,1)
rand_ICs = lognormal(PRNGKey(1),jnp.log(10),(8,2)) # init_prey, init_predator ~ LogNormal(sigma=log(10))
rand_args = jnp.hstack((rand_params,rand_ICs)) # each row is (alpha,beta,gamma,delta,init_prey,init_predator) -- 8 rows for 8 fixed ICs
# function that solves ODE problem for a given array of (alpha,beta,gamma,delta,init_prey,init_predator)
def solve_diffrax(args):
# returning transposed solution so it has same dims as observed data: 91x2
return jnp.asarray(diffeqsolve(term, solver, t0, t1, dt0, tuple(args[4:]), args=args[:4], saveat=saveat).ys).T
# create a pmapped version of that ODE solver function -- this will operate over each row of rand_args in parallel (8 ICs for 8 devices)
pmap_solve_diffrax = pmap(solve_diffrax)
# run it once to compile
ptest = pmap_solve_diffrax(rand_args)
### numpyro model where the 8 ICs will be solved in parallel for different (alpha,beta,gamma,delta)
### obs_data is the observed time series of (prey,predator) vs time and we will fix uncertainty obs_sigma=0.8 instead of fitting for it
def model(rand_args, obs_data, obs_sigma=0.8):
# sample our free parameters (alpha,beta,gamma,delta)
alpha = numpyro.sample('alpha',dist.TruncatedNormal(low=0.0,loc=1.0,scale=0.5))
beta = numpyro.sample('beta',dist.TruncatedNormal(low=0.0,loc=0.05,scale=0.05))
gamma = numpyro.sample('gamma',dist.TruncatedNormal(low=0.0,loc=1.0,scale=0.5))
delta = numpyro.sample('delta',dist.TruncatedNormal(low=0.0,loc=0.05,scale=0.05))
# update rand_args with the random samples of (alpha,beta,gamma,delta)
rand_args = rand_args.at[:,0].set(alpha)
rand_args = rand_args.at[:,1].set(beta)
rand_args = rand_args.at[:,2].set(gamma)
rand_args = rand_args.at[:,3].set(delta)
# use pmap to solve ODEs for all 8 ICs in parallel for current set of (alpha,beta,gamma,delta)
psol = pmap_solve_diffrax(rand_args)
# simple placeholder: compute mean solution of (prey,predator) vs time averaging over the 8 ICs
mean_sol = jnp.mean(psol,axis=0)
# likelihood similar to official example https://num.pyro.ai/en/stable/examples/ode.html
numpyro.sample('obs', dist.LogNormal(jnp.log(mean_sol), obs_sigma), obs=obs_data)
### Finally, prior predictive check to test
### This works but raises "jit-of-pmap" warning
prior_predictive = Predictive(model, num_samples=100)
prior_predictions = prior_predictive(PRNGKey(5), rand_args=rand_args, obs_data=None, obs_sigma=0.8)['obs']
```