def one_rng(rng):
_, params = dynamics_init(rng, (x_dim, ))
fwd_x_T = ode.odeint(lambda x, _: dynamics(params, x),
jnp.zeros((x_dim, )),
times,
mxstep=1e9)
bwd_x_0 = vmap(
lambda T, x_T: ode.odeint(lambda x, _: -dynamics(params, x),
x_T,
jnp.array([0.0, T]),
mxstep=1e9)[1],
in_axes=(0, 0))(times, fwd_x_T)
return jnp.sum(bwd_x_0**2, axis=-1)
def eval_from_x0(policy_params, x0, total_time):
# Zero is necessary for some reason...
t = jp.array([0.0, total_time])
y0 = jp.concatenate((jp.zeros((1, )), x0, jp.zeros((z_dim, ))))
# odeint_kwargs = {"rtol": 1e-3, "mxstep": 1e6}
odeint_kwargs = {"mxstep": 1e6}
y_fwd = ode.odeint(ofunc, y0, t, policy_params, **odeint_kwargs)
# This is similar but not exactly the same as the place that the rev-mode
# solution since the step sizes can vary when using all the other
# parameters.
y_bwd = ode.odeint(lambda y, t, *args: -ofunc(y, -t, *args),
y_fwd[1], -t[::-1], policy_params,
**odeint_kwargs)
return y_fwd, y_bwd[::-1]
def update(r):
v1_normalized = r * saep * v1
y = np.asarray(
(r_axis + ct * v1_normalized[0] + st * v1_normalized[1],
z_axis + v1_normalized[2]))
sol = odeint(step_partial, y, t_eval)
return sol[:, 0], sol[:, 1]
def test_disable_jit_odeint_with_vmap(self):
# https://github.com/google/jax/issues/2598
with jax.disable_jit():
t = jnp.array([0.0, 1.0])
x0_eval = jnp.zeros((5, 2))
f = lambda x0: odeint(lambda x, _t: x, x0, t)
jax.vmap(f)(x0_eval) # doesn't crash
def manifold_ode_log_prob(params: List, rng: random.PRNGKey, num_samples: int) -> Tuple[jnp.ndarray]:
"""Forward model of the neural manifold ODE. The base distribution is uniform
on the sphere. Computes both the samples from the forward model and the
log-probability of the generated samples.
Args:
params: Parameters of the neural manifold ODE.
rng: Pseudo-random number generator key.
num_samples: Number of samples use to estimate the KL divergence.
Returns:
sph: samples generated from the neural manifold ODE under the forward
model.
log_prob: The log-probability of the generated samples.
"""
rng, rng_x, rng_b = random.split(rng, 3)
x = sample_uniform(rng_x, [num_samples, 4])
b = project_to_sphere(x + 0.01 * random.normal(rng_b, x.shape))
v = log(b, x)
fldj = log_det_jac_exp(b, v)
vector_field = ambient_to_spherical_vector_field(lambda x, t: net(params, stacked(x, t)))
cfunc, divfunc = spherical_to_chart_vector_field(b, vector_field)
init = (v, jnp.zeros(len(v)))
time = jnp.array([0.0, 1.0])
tang, trace = tuple(_[-1] for _ in odeint(divfunc, init, time))
sph = exp(b, tang)
ildj = -log_det_jac_exp(b, tang)
log_prob = uniform_log_density(x) + fldj + ildj + trace
return sph, log_prob
def manifold_reverse_ode_log_prob(params: List, rng: random.PRNGKey, revx: jnp.ndarray) -> jnp.ndarray:
"""Given observations, compute their log-likelihood under the neural manifold
ODE by integrating the dynamics backwards and applying the
change-of-variables formula (computed continuously).
Args:
params: Parameters of the neural manifold ODE.
rng: Pseudo-random number generator key.
revx: Observations whose log-likelihood under the neural ODE model
should be computed.
Returns:
rev_log_prob: The log-probability of the observations.
"""
b = project_to_sphere(revx + 0.01 * random.normal(rng, revx.shape))
vrev = log(b, revx)
revfldj = log_det_jac_exp(b, vrev)
revinit = (vrev, jnp.zeros(len(vrev)))
vector_field = ambient_to_spherical_vector_field(lambda x, t: net(params, stacked(x, t)))
cfunc, divfunc = spherical_to_chart_vector_field(b, vector_field)
revfunc = lambda x, t: tuple(-_ for _ in divfunc(x, 1.0 - t))
time = jnp.array([0.0, 1.0])
revtang, revtrace = tuple(_[-1] for _ in odeint(revfunc, revinit, time))
revsph = exp(b, revtang)
revildj = -log_det_jac_exp(b, revtang)
rev_log_prob = uniform_log_density(revsph) - revfldj - revildj - revtrace
return rev_log_prob
def _run_static(cls, T, x0, theta, rtol=1e-5, atol=1e-3, mxstep=500):
'''
x0 is shape (d,)
theta is shape (nargs,)
'''
t = np.arange(T, dtype='float32')
return odeint(cls.dx_dt, x0, t, *theta)
def integrate(self, x0, n_steps): """ Integrates the equations of motion of the dynamical system. """ t = jnp.asarray([n*self.dt for n in range(n_steps)]) traj = odeint(self.equation_of_motion, x0, t) return traj
def model(N, y=None):
"""
:param int N: number of measurement times
:param numpy.ndarray y: measured populations with shape (N, 2)
"""
# initial population
z_init = numpyro.sample("z_init",
dist.LogNormal(jnp.log(10), 1).expand([2]))
# measurement times
ts = jnp.arange(float(N))
# parameters alpha, beta, gamma, delta of dz_dt
theta = numpyro.sample(
"theta",
dist.TruncatedNormal(
low=0.0,
loc=jnp.array([1.0, 0.05, 1.0, 0.05]),
scale=jnp.array([0.5, 0.05, 0.5, 0.05]),
),
)
# integrate dz/dt, the result will have shape N x 2
z = odeint(dz_dt, z_init, ts, theta, rtol=1e-6, atol=1e-5, mxstep=1000)
# measurement errors
sigma = numpyro.sample("sigma", dist.LogNormal(-1, 1).expand([2]))
# measured populations
numpyro.sample("y", dist.LogNormal(jnp.log(z), sigma), obs=y)
def solve_ode(init_condition, times):
return odeint(f,
init_condition,
t=times,
rtol=1e-10,
atol=1e-10,
**kwargs)
def eval_from_x0(policy_params, x0, total_time):
# Zero is necessary for some reason...
ts = jnp.array([0.0, total_time])
y0 = (jnp.zeros(()), jnp.zeros(()), x0)
odeint_kwargs = {"mxstep": 1e6}
y_fwd = ode.odeint(ofunc, y0, ts, policy_params, **odeint_kwargs)
yT = tree_map(itemgetter(-1), y_fwd)
# This is similar but not exactly the same as the place that the rev-mode
# solution since the step sizes can vary when using all the other
# parameters.
y_bwd = ode.odeint(lambda y, t, *args: tree_map(jnp.negative, ofunc(y, -t, *args)), yT,
-ts[::-1], policy_params, **odeint_kwargs)
y0_bwd = tree_map(itemgetter(-1), y_bwd)
return y0, yT, y0_bwd
def loss(t1, flat_p, omega, U_T):
'''
define the loss function, which is a pure function
'''
t_set = jnp.linspace(0., t1, 5)
D, _, = jnp.shape(U_T)
U_0 = jnp.eye(D, dtype=jnp.complex128)
def func(y, t, *args):
t1, omega, flat_p, = args
return -1.0j * (omega * sz + A(t, flat_p, t1) * sx) @ y
# return -1.0j*( omega* sz)@y
res = odeint(func,
U_0,
t_set,
t1,
omega,
flat_p,
rtol=1.4e-10,
atol=1.4e-10)
U_F = res[-1, :, :]
return (1 - jnp.abs(jnp.trace(U_T.conj().T @ U_F) / D)**2)
def kernel(a):
def integrand(y, t, a):
return dndz(1. / t - t) * cosmo.g(
a, t) # probably not enough, careful about change ofvar
y0 = 0.
y = odeint(integrand, y0, np.array([cosmo._amin, a]), a)
return y[1]
def g(x):
# Two initial values for the ODE
y0_arr = jnp.array([[x, 0.1], [x, 0.2]])
# Run ODE twice
t = jnp.array([0., 5.])
y = jax.vmap(lambda y0: odeint(dx_dt, y0, t))(y0_arr)
return y[:, -1].sum()
def advance(x0, theta):
x1 = odeint(cls.dx_dt,
x0,
t_one_step,
*theta,
rtol=rtol,
atol=atol,
mxstep=mxstep)[1]
return x1, x1
def solve_model_rk45(inputs):
y = odeint(rhs_ode,
y0,
t_eval,
inputs,
rtol=self.rtol,
atol=self.atol,
**self.extra_options)
return jnp.transpose(y)
def batch_warmup(self, x0: Array, total_steps: int) -> Array: """ Integrates the model and returns just the last step. This function is used to spin-up the model to statistically stataionary regime. """ t = jnp.asarray([0, total_steps * self.dt]) traj = odeint(self.equation_of_motion, x0, t) return traj[-1, ...]
def evally(policy_params, rng, x0, gamma):
# policy_params if first since that's what we want gradients wrt.
t = random.exponential(rng, (num_keypoints, )) / -jp.log(gamma)
t = jp.concatenate((jp.zeros((1, )), jp.sort(t)))
print(f"t = {t}")
x_t = ode.odeint(ofunc, x0, t, policy_params, rtol=1e-3, mxstep=1e6)
print(f"x_t = {x_t}")
costs = vmap(lambda x: cost_fn(x, policy(policy_params, x)))(x_t)
print(f"costs = {costs}")
return jp.mean(costs)
def _forward_dormandprince(state, ts, params, diffusivity, stimuli, dt, dx):
return ode.odeint(
step,
state,
ts,
params,
diffusivity,
stimuli,
dx,
)
def check_against_scipy(self, fun, y0, tspace, *args, tol=1e-1):
y0, tspace = np.array(y0), np.array(tspace)
np_fun = partial(fun, np)
scipy_result = jnp.asarray(osp_integrate.odeint(np_fun, y0, tspace, args))
y0, tspace = jnp.array(y0), jnp.array(tspace)
jax_fun = partial(fun, jnp)
jax_result = odeint(jax_fun, y0, tspace, *args)
self.assertAllClose(jax_result, scipy_result, check_dtypes=False, atol=tol, rtol=tol)
def make_sim(key):
x0 = random.normal(key, (n, total_dim)) # (n, dim * 2 + params)
#x0 = index_update(x0, s_[..., -1], np.exp(x0[..., -1])) #all masses
x0 = index_update(
x0, s_[..., -1],
1) # mass set to 1, [ x, y, x', y', random_value, mass]
x_times = odeint(odefunc,
x0.reshape(packed_shape),
times,
mxstep=2000).reshape(-1, *unpacked_shape)
return x_times
def ode_forward(rng: random.PRNGKey, net_params: List[jnp.ndarray],
num_samples: int, num_dims: int) -> Tuple[jnp.ndarray]:
x = random.normal(rng, [num_samples, num_dims])
log_prob_prior = jspst.norm.logpdf(x).sum(axis=-1)
vector_field = lambda x, t: net(net_params, stacked(x, t))
divfunc = primal_to_augmented(vector_field)
init = (x, jnp.zeros(len(x)))
time = jnp.array([0.0, 1.0])
xfwd, trace = tuple(_[-1] for _ in odeint(divfunc, init, time))
log_prob = log_prob_prior + trace
xsph = project(xfwd)
return xfwd, xsph, log_prob
def ode_reverse(net_params: List[jnp.ndarray],
xrev: jnp.ndarray) -> Tuple[jnp.ndarray]:
num_dims = xrev.shape[-1]
vector_field = lambda x, t: net(net_params, stacked(x, t))
divfunc = primal_to_augmented(vector_field)
revfunc = lambda x, t: tuple(-_ for _ in divfunc(x, 1.0 - t))
revinit = (xrev, jnp.zeros(len(xrev)))
time = jnp.array([0.0, 1.0])
yrev, revtrace = tuple(_[-1] for _ in odeint(revfunc, revinit, time))
log_prob_prior = jspst.norm.logpdf(yrev).sum(axis=-1)
rev_log_prob = log_prob_prior - revtrace
return rev_log_prob
def integrate_fieldline(startpoint, objective_dict):
closed_l = r(objective_dict['p'], theta)
ll = closed_l[:, :-1, :]
dl = closed_l[:, :-1, :] - closed_l[:, 1:, :]
jitfield = jit(
lambda r_eval, t: biot_savart(r_eval, dl, ll, objective_dict['I_arr']))
startpos = r_surf[0, 0, :]
times = np.linspace(0, 10, 10000)
fieldline = odeint(jitfield, startpos, times)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot(fieldline[:, 0], fieldline[:, 1], fieldline[:, 2])
def test_swoop(self):
def swoop(y, t, arg1, arg2):
return np.array(y - np.sin(t) - np.cos(t) * arg1 + arg2)
ts = np.array([0.1, 0.2])
tol = 1e-1 if num_float_bits(onp.float64) == 32 else 1e-3
y0 = np.linspace(0.1, 0.9, 10)
integrate = lambda y0, ts: odeint(swoop, y0, ts, 0.1, 0.2)
jtu.check_grads(integrate, (y0, ts),
modes=["rev"],
order=2,
rtol=tol,
atol=tol)
big_y0 = np.linspace(1.1, 10.9, 10)
integrate = lambda y0, ts: odeint(swoop, big_y0, ts, 0.1, 0.3)
jtu.check_grads(integrate, (y0, ts),
modes=["rev"],
order=2,
rtol=tol,
atol=tol)
def loss(t1,flat_p,psi_init,psi0):
'''
define the loss function, which is a pure function
'''
t_set = jnp.linspace(0., t1, 5)
def func(y, t, *args):
t1, flat_p, = args
return -1.0j*Hmat(t, flat_p, t1)@y
res = odeint(func, psi_init, t_set, t1, flat_p, rtol=1.4e-10, atol=1.4e-10)
psi_final = res[-1, :]
return (1 - jnp.abs(jnp.dot(jnp.conjugate(psi_final), psi0))**2)
def state_and_costate_trajectories(initial_costate_and_final_time):
"""Propagates the ODE that defines the shooting method.
Args:
initial_costate_and_final_time: An array of shape (4,) containing (p_x(0), p_y(0), p_θ(0), t_f).
Returns:
A tuple of arrays (times, (states, costates)) where
times: An array of shape (N,) containing a sequence of time points spanning [0, t_f].
states: An array of shape (N, 3) containing the states at `times`.
controls: An array of shape (N, 3) containing the controls at `times`.
"""
initial_costate = initial_costate_and_final_time[:-1]
final_time = initial_costate_and_final_time[-1]
times = jnp.linspace(0, final_time, 20)
return times, odeint(shooting_ode, (initial_state, initial_costate), times)
def model(self, t, X):
noise = sample('noise', dist.LogNormal(0.0, 1.0), sample_shape=(self.D,))
hyp = sample('hyp', dist.Gamma(1.0, 0.5), sample_shape=(self.D,))
W = sample('W', dist.LogNormal(0.0, 1.0), sample_shape=(self.D,))
J0 = sample('J0', dist.Uniform(1.0, 10.0)) # 2.5
k1 = sample('k1', dist.Uniform(80., 120.0)) # 100.
k2 = sample('k2', dist.Uniform(1., 10.0)) # 6.
k3 = sample('k3', dist.Uniform(2., 20.0)) # 16.
k4 = sample('k4', dist.Uniform(80., 120.0)) # 100.
k5 = sample('k5', dist.Uniform(0.1, 2.0)) # 1.28
k6 = sample('k6', dist.Uniform(2., 20.0)) # 12.
k = sample('k', dist.Uniform(0.1, 2.0)) # 1.8
ka = sample('ka', dist.Uniform(2., 20.0)) # 13.
q = sample('q', dist.Uniform(1., 10.0)) # 4.
KI = sample('KI', dist.Uniform(0.1, 2.0)) # 0.52
phi = sample('phi', dist.Uniform(0.05, 1.0)) # 0.1
Np = sample('Np', dist.Uniform(0.1, 2.0)) # 1.
A = sample('A', dist.Uniform(1., 10.0)) #4.
IC = sample('IC', dist.Uniform(0, 1))
# compute kernel
K_11 = W[0]*self.RBF(self.t_t[0], self.t_t[0], hyp[0]) + np.eye(self.N[0])*(noise[0] + self.jitter)
K_22 = W[1]*self.RBF(self.t_t[1], self.t_t[1], hyp[1]) + np.eye(self.N[1])*(noise[1] + self.jitter)
K_33 = W[2]*self.RBF(self.t_t[2], self.t_t[2], hyp[2]) + np.eye(self.N[2])*(noise[2] + self.jitter)
K = np.concatenate([np.concatenate([K_11, np.zeros((self.N[0], self.N[1])), np.zeros((self.N[0], self.N[2]))], axis = 1),
np.concatenate([np.zeros((self.N[1], self.N[0])), K_22, np.zeros((self.N[1], self.N[2]))], axis = 1),
np.concatenate([np.zeros((self.N[2], self.N[0])), np.zeros((self.N[2], self.N[1])), K_33], axis = 1)], axis = 0)
# compute mean
x0 = np.array([0.5, 1.9, 0.18, 0.15, IC, 0.1, 0.064])
mut = odeint(self.dxdt, x0, self.t.flatten(), J0, k1, k2, k3, k4, k5, k6, k, ka, q, KI, phi, Np, A)
mu1 = mut[self.i_t[0],ind[0]] / self.max_X[0]
mu2 = mut[self.i_t[1],ind[1]] / self.max_X[1]
mu3 = mut[self.i_t[2],ind[2]] / self.max_X[2]
mu = np.concatenate((mu1,mu2,mu3),axis=0)
# Concat data
mu = mu.flatten('F')
X = np.concatenate((self.X[0],self.X[1],self.X[2]),axis=0)
X = X.flatten('F')
# sample X according to the standard gaussian process formula
sample("X", dist.MultivariateNormal(loc=mu, covariance_matrix=K), obs=X)
def bwd_spline_segment(ta, tb, args, Q, y_old, aug_tb):
"""Run the backwards RK on just one segment of the spline."""
def adj_dynamics(aug, t, args, Q, y_old):
_, y_bar, _ = aug
y = unravel(eval_spline(ta, tb, Q, y_old, -t))
_, vjpfun = vjp(fun, -t, y, args)
return vjpfun(y_bar)
adj_path = ode.odeint(adj_dynamics,
aug_tb,
jnp.array([-tb, -ta]),
args,
Q,
y_old,
rtol=1e-3,
atol=1e-3)
return tree_map(itemgetter(-1), adj_path)
def make_sim(key):
if sim in ['string', 'string_ball']:
x0 = random.normal(key, (n, total_dim))
x0 = index_update(x0, s_[..., -1], 1); #const mass
x0 = index_update(x0, s_[..., 0], np.arange(n)+x0[...,0]*0.5)
x0 = index_update(x0, s_[..., 2:3], 0.0)
else:
x0 = random.normal(key, (n, total_dim))
x0 = index_update(x0, s_[..., -1], np.exp(x0[..., -1])); #all masses set to positive
if sim in ['charge', 'superposition']:
x0 = index_update(x0, s_[..., -2], np.sign(x0[..., -2])); #charge is 1 or -1
x_times = odeint(
odefunc,
x0.reshape(packed_shape),
times, mxstep=2000).reshape(-1, *unpacked_shape)
return x_times