def _log_abs_determinant(self):
log_det = tfp_math.reduce_kahan_sum(tf.math.log(tf.math.abs(
self._diag)),
axis=[-1]).total
if dtype_util.is_complex(self.dtype):
log_det = tf.cast(log_det, dtype=self.dtype)
return log_det
def error_if_not_real_or_complex(tensor, identifier):
"""Raise a `TypeError` if the `Tensor` is neither real nor complex."""
if not (dtype_util.is_floating(tensor.dtype) or
dtype_util.is_complex(tensor.dtype)):
raise TypeError(
'`{}` must have a floating point or complex floating point dtype.'
.format(identifier))
def abs_square(x):
"""Returns the squared value of `tf.abs(x)` for real and complex dtypes."""
if dtype_util.is_complex(x.dtype):
return tf.math.square(tf.math.real(x)) + tf.math.square(
tf.math.imag(x))
else:
return tf.math.square(x)
def _one_part(tensor):
tensor = tf.convert_to_tensor(tensor)
if dtype_util.is_floating(tensor.dtype) or dtype_util.is_complex(
tensor.dtype):
return tf.stop_gradient(tensor)
else:
return tensor
def _inverse(self, y):
x = y
if self.shift is not None:
x = x - self.shift
if self._is_only_identity_multiplier:
s = (tf.math.conj(self._scale) if self.adjoint
and dtype_util.is_complex(self._scale.dtype) else self._scale)
return x / s
# Solve fails if the op is singular so we may safely skip this assertion.
x = self.scale.solvevec(x, adjoint=self.adjoint)
return x
def _forward(self, x):
y = x
if self._is_only_identity_multiplier:
s = (tf.math.conj(self._scale) if self.adjoint
and dtype_util.is_complex(self._scale.dtype) else self._scale)
y = y * s
if self.shift is not None:
return y + self.shift
return y
with tf.control_dependencies(
self._maybe_check_scale() if self.validate_args else []):
y = self.scale.matvec(y, adjoint=self.adjoint)
if self.shift is not None:
y = y + self.shift
return y
def _sample_n(self, n, seed):
components_seed, mix_seed = samplers.split_seed(
seed, salt='MixtureSameFamily')
mixture_distribution, components_distribution = (
self._get_distributions_with_broadcast_batch_shape())
x = components_distribution.sample( # [n, B, k, E]
n, seed=components_seed)
event_ndims = ps.rank_from_shape(self.event_shape_tensor,
self.event_shape)
# We could also check if num_components can be computed statically from
# self.mixture_distribution's logits or probs.
num_components = ps.dimension_size(x, idx=-1 - event_ndims)
# TODO(jvdillon): Consider using tf.gather (by way of index unrolling).
npdt = dtype_util.as_numpy_dtype(x.dtype)
mix_sample = mixture_distribution.sample(n, seed=mix_seed) # [n, B]
mask = tf.one_hot(
indices=mix_sample, # [n, B]
depth=num_components,
on_value=npdt(1),
off_value=npdt(0)) # [n, B, k]
# Pad `mask` to [n, B, k, [1]*e].
batch_ndims = ps.rank(x) - event_ndims - 1
mask_batch_ndims = ps.rank(mask) - 1
pad_ndims = batch_ndims - mask_batch_ndims
mask_shape = ps.shape(mask)
target_shape = ps.concat([
mask_shape[:-1],
ps.ones([pad_ndims], dtype=tf.int32),
mask_shape[-1:],
ps.ones([event_ndims], dtype=tf.int32),
],
axis=0)
mask = tf.reshape(mask, shape=target_shape)
if dtype_util.is_floating(x.dtype) or dtype_util.is_complex(x.dtype):
masked = tf.math.multiply_no_nan(x, mask)
else:
masked = x * mask
ret = tf.reduce_sum(masked, axis=-1 - event_ndims) # [n, B, E]
if self._reparameterize:
ret = self._reparameterize_sample(
ret, event_shape=components_distribution.event_shape_tensor())
return ret
def auto_correlation(x,
axis=-1,
max_lags=None,
center=True,
normalize=True,
name='auto_correlation'):
"""Auto correlation along one axis.
Given a `1-D` wide sense stationary (WSS) sequence `X`, the auto correlation
`RXX` may be defined as (with `E` expectation and `Conj` complex conjugate)
```
RXX[m] := E{ W[m] Conj(W[0]) } = E{ W[0] Conj(W[-m]) },
W[n] := (X[n] - MU) / S,
MU := E{ X[0] },
S**2 := E{ (X[0] - MU) Conj(X[0] - MU) }.
```
This function takes the viewpoint that `x` is (along one axis) a finite
sub-sequence of a realization of (WSS) `X`, and then uses `x` to produce an
estimate of `RXX[m]` as follows:
After extending `x` from length `L` to `inf` by zero padding, the auto
correlation estimate `rxx[m]` is computed for `m = 0, 1, ..., max_lags` as
```
rxx[m] := (L - m)**-1 sum_n w[n + m] Conj(w[n]),
w[n] := (x[n] - mu) / s,
mu := L**-1 sum_n x[n],
s**2 := L**-1 sum_n (x[n] - mu) Conj(x[n] - mu)
```
The error in this estimate is proportional to `1 / sqrt(len(x) - m)`, so users
often set `max_lags` small enough so that the entire output is meaningful.
Note that since `mu` is an imperfect estimate of `E{ X[0] }`, and we divide by
`len(x) - m` rather than `len(x) - m - 1`, our estimate of auto correlation
contains a slight bias, which goes to zero as `len(x) - m --> infinity`.
Args:
x: `float32` or `complex64` `Tensor`.
axis: Python `int`. The axis number along which to compute correlation.
Other dimensions index different batch members.
max_lags: Positive `int` tensor. The maximum value of `m` to consider (in
equation above). If `max_lags >= x.shape[axis]`, we effectively re-set
`max_lags` to `x.shape[axis] - 1`.
center: Python `bool`. If `False`, do not subtract the mean estimate `mu`
from `x[n]` when forming `w[n]`.
normalize: Python `bool`. If `False`, do not divide by the variance
estimate `s**2` when forming `w[n]`.
name: `String` name to prepend to created ops.
Returns:
`rxx`: `Tensor` of same `dtype` as `x`. `rxx.shape[i] = x.shape[i]` for
`i != axis`, and `rxx.shape[axis] = max_lags + 1`.
Raises:
TypeError: If `x` is not a supported type.
"""
# Implementation details:
# Extend length N / 2 1-D array x to length N by zero padding onto the end.
# Then, set
# F[x]_k := sum_n x_n exp{-i 2 pi k n / N }.
# It is not hard to see that
# F[x]_k Conj(F[x]_k) = F[R]_k, where
# R_m := sum_n x_n Conj(x_{(n - m) mod N}).
# One can also check that R_m / (N / 2 - m) is an unbiased estimate of RXX[m].
# Since F[x] is the DFT of x, this leads us to a zero-padding and FFT/IFFT
# based version of estimating RXX.
# Note that this is a special case of the Wiener-Khinchin Theorem.
with tf.name_scope(name):
x = tf.convert_to_tensor(x, name='x')
# Rotate dimensions of x in order to put axis at the rightmost dim.
# FFT op requires this.
rank = ps.rank(x)
if axis < 0:
axis = rank + axis
shift = rank - 1 - axis
# Suppose x.shape[axis] = T, so there are T 'time' steps.
# ==> x_rotated.shape = B + [T],
# where B is x_rotated's batch shape.
x_rotated = distribution_util.rotate_transpose(x, shift)
if center:
x_rotated = x_rotated - tf.reduce_mean(
x_rotated, axis=-1, keepdims=True)
# x_len = N / 2 from above explanation. The length of x along axis.
# Get a value for x_len that works in all cases.
x_len = ps.shape(x_rotated)[-1]
# TODO(langmore) Investigate whether this zero padding helps or hurts. At
# the moment is necessary so that all FFT implementations work.
# Zero pad to the next power of 2 greater than 2 * x_len, which equals
# 2**(ceil(Log_2(2 * x_len))). Note: Log_2(X) = Log_e(X) / Log_e(2).
x_len_float64 = ps.cast(x_len, np.float64)
target_length = ps.pow(np.float64(2.),
ps.ceil(ps.log(x_len_float64 * 2) / np.log(2.)))
pad_length = ps.cast(target_length - x_len_float64, np.int32)
# We should have:
# x_rotated_pad.shape = x_rotated.shape[:-1] + [T + pad_length]
# = B + [T + pad_length]
x_rotated_pad = distribution_util.pad(x_rotated,
axis=-1,
back=True,
count=pad_length)
dtype = x.dtype
if not dtype_util.is_complex(dtype):
if not dtype_util.is_floating(dtype):
raise TypeError(
'Argument x must have either float or complex dtype'
' found: {}'.format(dtype))
x_rotated_pad = tf.complex(
x_rotated_pad,
dtype_util.as_numpy_dtype(dtype_util.real_dtype(dtype))(0.))
# Autocorrelation is IFFT of power-spectral density (up to some scaling).
fft_x_rotated_pad = tf.signal.fft(x_rotated_pad)
spectral_density = fft_x_rotated_pad * tf.math.conj(fft_x_rotated_pad)
# shifted_product is R[m] from above detailed explanation.
# It is the inner product sum_n X[n] * Conj(X[n - m]).
shifted_product = tf.signal.ifft(spectral_density)
# Cast back to real-valued if x was real to begin with.
shifted_product = tf.cast(shifted_product, dtype)
# Figure out if we can deduce the final static shape, and set max_lags.
# Use x_rotated as a reference, because it has the time dimension in the far
# right, and was created before we performed all sorts of crazy shape
# manipulations.
know_static_shape = True
if not tensorshape_util.is_fully_defined(x_rotated.shape):
know_static_shape = False
if max_lags is None:
max_lags = x_len - 1
else:
max_lags = tf.convert_to_tensor(max_lags, name='max_lags')
max_lags_ = tf.get_static_value(max_lags)
if max_lags_ is None or not know_static_shape:
know_static_shape = False
max_lags = tf.minimum(x_len - 1, max_lags)
else:
max_lags = min(x_len - 1, max_lags_)
# Chop off the padding.
# We allow users to provide a huge max_lags, but cut it off here.
# shifted_product_chopped.shape = x_rotated.shape[:-1] + [max_lags]
shifted_product_chopped = shifted_product[..., :max_lags + 1]
# If possible, set shape.
if know_static_shape:
chopped_shape = tensorshape_util.as_list(x_rotated.shape)
chopped_shape[-1] = min(x_len, max_lags + 1)
tensorshape_util.set_shape(shifted_product_chopped, chopped_shape)
# Recall R[m] is a sum of N / 2 - m nonzero terms x[n] Conj(x[n - m]). The
# other terms were zeros arising only due to zero padding.
# `denominator = (N / 2 - m)` (defined below) is the proper term to
# divide by to make this an unbiased estimate of the expectation
# E[X[n] Conj(X[n - m])].
x_len = ps.cast(x_len, dtype_util.real_dtype(dtype))
max_lags = ps.cast(max_lags, dtype_util.real_dtype(dtype))
denominator = x_len - ps.range(0., max_lags + 1.)
denominator = ps.cast(denominator, dtype)
shifted_product_rotated = shifted_product_chopped / denominator
if normalize:
shifted_product_rotated /= shifted_product_rotated[..., :1]
# Transpose dimensions back to those of x.
return distribution_util.rotate_transpose(shifted_product_rotated,
-shift)
def _sample_n(self, n, seed):
components_seed, mix_seed = samplers.split_seed(
seed, salt='MixtureSameFamily')
try:
seed_stream = SeedStream(seed, salt='MixtureSameFamily')
except TypeError as e: # Can happen for Tensor seeds.
seed_stream = None
seed_stream_err = e
try:
x = self.components_distribution.sample( # [n, B, k, E]
n, seed=components_seed)
if seed_stream is not None:
seed_stream() # Advance even if unused.
except TypeError as e:
if ('Expected int for argument' not in str(e)
and TENSOR_SEED_MSG_PREFIX not in str(e)):
raise
if seed_stream is None:
raise seed_stream_err
msg = (
'Falling back to stateful sampling for `components_distribution` '
'{} of type `{}`. Please update to use `tf.random.stateless_*` '
'RNGs. This fallback may be removed after 20-Aug-2020. {}')
warnings.warn(
msg.format(self.components_distribution.name,
type(self.components_distribution), str(e)))
x = self.components_distribution.sample( # [n, B, k, E]
n, seed=seed_stream())
event_shape = None
event_ndims = tensorshape_util.rank(self.event_shape)
if event_ndims is None:
event_shape = self.components_distribution.event_shape_tensor()
event_ndims = ps.rank_from_shape(event_shape)
event_ndims_static = tf.get_static_value(event_ndims)
num_components = None
if event_ndims_static is not None:
num_components = tf.compat.dimension_value(
x.shape[-1 - event_ndims_static])
# We could also check if num_components can be computed statically from
# self.mixture_distribution's logits or probs.
if num_components is None:
num_components = tf.shape(x)[-1 - event_ndims]
# TODO(jvdillon): Consider using tf.gather (by way of index unrolling).
npdt = dtype_util.as_numpy_dtype(x.dtype)
try:
mix_sample = self.mixture_distribution.sample(
n, seed=mix_seed) # [n, B] or [n]
except TypeError as e:
if ('Expected int for argument' not in str(e)
and TENSOR_SEED_MSG_PREFIX not in str(e)):
raise
if seed_stream is None:
raise seed_stream_err
msg = (
'Falling back to stateful sampling for `mixture_distribution` '
'{} of type `{}`. Please update to use `tf.random.stateless_*` '
'RNGs. This fallback may be removed after 20-Aug-2020. ({})')
warnings.warn(
msg.format(self.mixture_distribution.name,
type(self.mixture_distribution), str(e)))
mix_sample = self.mixture_distribution.sample(
n, seed=seed_stream()) # [n, B] or [n]
mask = tf.one_hot(
indices=mix_sample, # [n, B] or [n]
depth=num_components,
on_value=npdt(1),
off_value=npdt(0)) # [n, B, k] or [n, k]
# Pad `mask` to [n, B, k, [1]*e] or [n, [1]*b, k, [1]*e] .
batch_ndims = ps.rank(x) - event_ndims - 1
mask_batch_ndims = ps.rank(mask) - 1
pad_ndims = batch_ndims - mask_batch_ndims
mask_shape = ps.shape(mask)
target_shape = ps.concat([
mask_shape[:-1],
ps.ones([pad_ndims], dtype=tf.int32),
mask_shape[-1:],
ps.ones([event_ndims], dtype=tf.int32),
],
axis=0)
mask = tf.reshape(mask, shape=target_shape)
if dtype_util.is_floating(x.dtype) or dtype_util.is_complex(x.dtype):
masked = tf.math.multiply_no_nan(x, mask)
else:
masked = x * mask
ret = tf.reduce_sum(masked, axis=-1 - event_ndims) # [n, B, E]
if self._reparameterize:
if event_shape is None:
event_shape = self.components_distribution.event_shape_tensor()
ret = self._reparameterize_sample(ret, event_shape=event_shape)
return ret
def error_if_complex(dtype):
if dtype_util.is_complex(dtype):
raise NotImplementedError(
'The adjoint sensitivity method does '
'not support complex dtypes.')
def _solve(
self,
ode_fn,
initial_time,
initial_state,
solution_times,
jacobian_fn=None,
jacobian_sparsity=None,
batch_ndims=None,
previous_solver_internal_state=None,
):
# This function is comprised of the following sequential stages:
# (1) Make static assertions.
# (2) Initialize variables.
# (3) Make non-static assertions.
# (4) Solve up to final time.
# (5) Return `Results` object.
#
# The stages can be found in the code by searching for (n) where n=1..5.
#
# By static vs. non-static assertions (see stages 1 and 3), we mean
# assertions that can be made before the graph is run vs. those that can
# only be made at run time. The latter are constructed as a list of
# tf.Assert operations by the function `assert_ops` (see below).
#
# If `solution_times` is specified as a `Tensor`, stage 4 consists of three
# nested loops, which can be conceptually understood as follows:
# ```
# current_time, current_state = initial_time, initial_state
# order, step_size = 1, first_step_size
# for solution_time in solution_times:
# while current_time < solution_time:
# while True:
# next_time = current_time + step_size
# next_state, error = (
# solve_nonlinear_equation_to_get_approximate_state_at_next_time(
# current_time, current_state, next_time, order))
# if error < tolerance:
# current_time, current_state = next_time, next_state
# order, step_size = (
# maybe_update_order_and_step_size(order, step_size))
# break
# else:
# step_size = decrease_step_size(step_size)
# ```
# The outermost loop advances the solver to the next `solution_time` (see
# `advance_to_solution_time`). The middle loop advances the solver by a
# small timestep (see `step`). The innermost loop determines the size of
# that timestep (see `maybe_step`).
#
# If `solution_times` is specified as
# `tfp.math.ode.ChosenBySolver(final_time)`, the outermost loop is skipped
# and `solution_time` in the middle loop is replaced by `final_time`.
def advance_to_solution_time(n, diagnostics, iterand,
solver_internal_state, state_vec_array,
time_array):
"""Takes multiple steps to advance time to `solution_times[n]`."""
def step_cond(next_time, diagnostics, iterand, *_):
return (iterand.time < next_time) & (tf.equal(
diagnostics.status, 0))
nth_solution_time = solution_time_array.read(n)
[
_, diagnostics, iterand, solver_internal_state,
state_vec_array, time_array
] = tf.while_loop(step_cond, step, [
nth_solution_time, diagnostics, iterand, solver_internal_state,
state_vec_array, time_array
])
state_vec_array = state_vec_array.write(
n, solver_internal_state.backward_differences[0])
time_array = time_array.write(n, nth_solution_time)
return (n + 1, diagnostics, iterand, solver_internal_state,
state_vec_array, time_array)
def step(next_time, diagnostics, iterand, solver_internal_state,
state_vec_array, time_array):
"""Takes a single step."""
distance_to_next_time = next_time - iterand.time
overstepped = iterand.new_step_size > distance_to_next_time
iterand = iterand._replace(new_step_size=tf1.where(
overstepped, distance_to_next_time, iterand.new_step_size),
should_update_step_size=overstepped
| iterand.should_update_step_size)
if not self._evaluate_jacobian_lazily:
diagnostics = diagnostics._replace(
num_jacobian_evaluations=diagnostics.
num_jacobian_evaluations + 1)
iterand = iterand._replace(jacobian_mat=jacobian_fn_mat(
iterand.time,
solver_internal_state.backward_differences[0]),
jacobian_is_up_to_date=True)
def maybe_step_cond(accepted, diagnostics, *_):
return tf.logical_not(accepted) & tf.equal(
diagnostics.status, 0)
_, diagnostics, iterand, solver_internal_state = tf.while_loop(
maybe_step_cond, maybe_step,
[False, diagnostics, iterand, solver_internal_state])
if solution_times_chosen_by_solver:
state_vec_array = state_vec_array.write(
state_vec_array.size(),
solver_internal_state.backward_differences[0])
time_array = time_array.write(time_array.size(), iterand.time)
return (next_time, diagnostics, iterand, solver_internal_state,
state_vec_array, time_array)
def maybe_step(accepted, diagnostics, iterand, solver_internal_state):
"""Takes a single step only if the outcome has a low enough error."""
[
num_jacobian_evaluations, num_matrix_factorizations,
num_ode_fn_evaluations, status
] = diagnostics
[
jacobian_mat, jacobian_is_up_to_date, new_step_size, num_steps,
num_steps_same_size, should_update_jacobian,
should_update_step_size, time, unitary, upper
] = iterand
[backward_differences, order, step_size] = solver_internal_state
if max_num_steps is not None:
status = tf1.where(tf.equal(num_steps, max_num_steps), -1, 0)
backward_differences = tf1.where(
should_update_step_size,
bdf_util.interpolate_backward_differences(
backward_differences, order, new_step_size / step_size),
backward_differences)
step_size = tf1.where(should_update_step_size, new_step_size,
step_size)
should_update_factorization = should_update_step_size
num_steps_same_size = tf1.where(should_update_step_size, 0,
num_steps_same_size)
def update_factorization():
return bdf_util.newton_qr(
jacobian_mat, newton_coefficients_array.read(order),
step_size)
if self._evaluate_jacobian_lazily:
def update_jacobian_and_factorization():
new_jacobian_mat = jacobian_fn_mat(time,
backward_differences[0])
new_unitary, new_upper = update_factorization()
return [
new_jacobian_mat, True, num_jacobian_evaluations + 1,
new_unitary, new_upper
]
def maybe_update_factorization():
new_unitary, new_upper = tf.cond(
should_update_factorization, update_factorization,
lambda: [unitary, upper])
return [
jacobian_mat, jacobian_is_up_to_date,
num_jacobian_evaluations, new_unitary, new_upper
]
[
jacobian_mat, jacobian_is_up_to_date,
num_jacobian_evaluations, unitary, upper
] = tf.cond(should_update_jacobian,
update_jacobian_and_factorization,
maybe_update_factorization)
else:
unitary, upper = update_factorization()
num_matrix_factorizations += 1
tol = p.atol + p.rtol * tf.abs(backward_differences[0])
newton_tol = newton_tol_factor * tf.norm(tol)
[
newton_converged, next_backward_difference, next_state_vec,
newton_num_iters
] = bdf_util.newton(backward_differences, max_num_newton_iters,
newton_coefficients_array.read(order),
p.ode_fn_vec, order, step_size, time,
newton_tol, unitary, upper)
num_steps += 1
num_ode_fn_evaluations += newton_num_iters
# If Newton's method failed and the Jacobian was up to date, decrease the
# step size.
newton_failed = tf.logical_not(newton_converged)
should_update_step_size = newton_failed & jacobian_is_up_to_date
new_step_size = step_size * tf1.where(should_update_step_size,
newton_step_size_factor, 1.)
# If Newton's method failed and the Jacobian was NOT up to date, update
# the Jacobian.
should_update_jacobian = newton_failed & tf.logical_not(
jacobian_is_up_to_date)
error_ratio = tf1.where(
newton_converged,
bdf_util.error_ratio(next_backward_difference,
error_coefficients_array.read(order),
tol), np.nan)
accepted = error_ratio < 1.
converged_and_rejected = newton_converged & tf.logical_not(
accepted)
# If Newton's method converged but the solution was NOT accepted, decrease
# the step size.
new_step_size = tf1.where(
converged_and_rejected,
util.next_step_size(step_size, order, error_ratio,
p.safety_factor, min_step_size_factor,
max_step_size_factor), new_step_size)
should_update_step_size = should_update_step_size | converged_and_rejected
# If Newton's method converged and the solution was accepted, update the
# matrix of backward differences.
time = tf1.where(accepted, time + step_size, time)
backward_differences = tf1.where(
accepted,
bdf_util.update_backward_differences(backward_differences,
next_backward_difference,
next_state_vec, order),
backward_differences)
jacobian_is_up_to_date = jacobian_is_up_to_date & tf.logical_not(
accepted)
num_steps_same_size = tf1.where(accepted, num_steps_same_size + 1,
num_steps_same_size)
# Order and step size are only updated if we have taken strictly more than
# order + 1 steps of the same size. This is to prevent the order from
# being throttled.
should_update_order_and_step_size = accepted & (num_steps_same_size
> order + 1)
backward_differences_array = tf.TensorArray(
backward_differences.dtype,
size=bdf_util.MAX_ORDER + 3,
clear_after_read=False,
element_shape=next_backward_difference.shape).unstack(
backward_differences)
new_order = order
new_error_ratio = error_ratio
for offset in [-1, +1]:
proposed_order = tf.clip_by_value(order + offset, 1, max_order)
proposed_error_ratio = bdf_util.error_ratio(
backward_differences_array.read(proposed_order + 1),
error_coefficients_array.read(proposed_order), tol)
proposed_error_ratio_is_lower = proposed_error_ratio < new_error_ratio
new_order = tf1.where(
should_update_order_and_step_size
& proposed_error_ratio_is_lower, proposed_order, new_order)
new_error_ratio = tf1.where(
should_update_order_and_step_size
& proposed_error_ratio_is_lower, proposed_error_ratio,
new_error_ratio)
order = new_order
error_ratio = new_error_ratio
new_step_size = tf1.where(
should_update_order_and_step_size,
util.next_step_size(step_size, order, error_ratio,
p.safety_factor, min_step_size_factor,
max_step_size_factor), new_step_size)
should_update_step_size = (should_update_step_size
| should_update_order_and_step_size)
diagnostics = _BDFDiagnostics(num_jacobian_evaluations,
num_matrix_factorizations,
num_ode_fn_evaluations, status)
iterand = _BDFIterand(jacobian_mat, jacobian_is_up_to_date,
new_step_size, num_steps,
num_steps_same_size, should_update_jacobian,
should_update_step_size, time, unitary,
upper)
solver_internal_state = _BDFSolverInternalState(
backward_differences, order, step_size)
return accepted, diagnostics, iterand, solver_internal_state
# (1) Make static assertions.
# TODO(b/138304296): Support specifying Jacobian sparsity patterns.
if jacobian_sparsity is not None:
raise NotImplementedError(
'The BDF solver does not support specifying '
'Jacobian sparsity patterns.')
if batch_ndims is not None and batch_ndims != 0:
raise NotImplementedError(
'The BDF solver does not support batching.')
solution_times_chosen_by_solver = (isinstance(solution_times,
base.ChosenBySolver))
with tf.name_scope(self._name):
# (2) Convert to tensors.
p = self._prepare_common_params(
ode_fn=ode_fn,
initial_state=initial_state,
initial_time=initial_time,
)
if jacobian_fn is None and dtype_util.is_complex(
p.common_state_dtype):
raise NotImplementedError(
'The BDF solver does not support automatic '
'Jacobian computations for complex dtypes.')
# Convert everything to operate on a single, concatenated vector form.
jacobian_fn_mat = util.get_jacobian_fn_mat(
jacobian_fn,
p.ode_fn_vec,
p.state_shape,
dtype=p.common_state_dtype,
)
num_solution_times = 0
if solution_times_chosen_by_solver:
final_time = tf.cast(solution_times.final_time, p.real_dtype)
else:
solution_times = tf.cast(solution_times, p.real_dtype)
final_time = tf.reduce_max(solution_times)
num_solution_times = tf.size(solution_times)
solution_time_array = tf.TensorArray(
solution_times.dtype,
size=num_solution_times,
element_shape=[]).unstack(solution_times)
util.error_if_not_vector(solution_times, 'solution_times')
min_step_size_factor = tf.convert_to_tensor(
self._min_step_size_factor, dtype=p.real_dtype)
max_step_size_factor = tf.convert_to_tensor(
self._max_step_size_factor, dtype=p.real_dtype)
max_num_steps = self._max_num_steps
if max_num_steps is not None:
max_num_steps = tf.convert_to_tensor(max_num_steps,
dtype=tf.int32)
max_order = tf.convert_to_tensor(self._max_order, dtype=tf.int32)
max_num_newton_iters = self._max_num_newton_iters
if max_num_newton_iters is not None:
max_num_newton_iters = tf.convert_to_tensor(
max_num_newton_iters, dtype=tf.int32)
newton_tol_factor = tf.convert_to_tensor(self._newton_tol_factor,
dtype=p.real_dtype)
newton_step_size_factor = tf.convert_to_tensor(
self._newton_step_size_factor, dtype=p.real_dtype)
newton_coefficients, error_coefficients = self._prepare_coefficients(
p.common_state_dtype)
if self._validate_args:
final_time = tf.ensure_shape(final_time, [])
min_step_size_factor = tf.ensure_shape(min_step_size_factor,
[])
max_step_size_factor = tf.ensure_shape(max_step_size_factor,
[])
if max_num_steps is not None:
max_num_steps = tf.ensure_shape(max_num_steps, [])
max_order = tf.ensure_shape(max_order, [])
if max_num_newton_iters is not None:
max_num_newton_iters = tf.ensure_shape(
max_num_newton_iters, [])
newton_tol_factor = tf.ensure_shape(newton_tol_factor, [])
newton_step_size_factor = tf.ensure_shape(
newton_step_size_factor, [])
newton_coefficients_array = tf.TensorArray(
newton_coefficients.dtype,
size=bdf_util.MAX_ORDER + 1,
clear_after_read=False,
element_shape=[]).unstack(newton_coefficients)
error_coefficients_array = tf.TensorArray(
error_coefficients.dtype,
size=bdf_util.MAX_ORDER + 1,
clear_after_read=False,
element_shape=[]).unstack(error_coefficients)
solver_internal_state = previous_solver_internal_state
if solver_internal_state is None:
solver_internal_state = self._initialize_solver_internal_state(
ode_fn=ode_fn,
initial_state=initial_state,
initial_time=initial_time,
)
state_vec_array = tf.TensorArray(
p.common_state_dtype,
size=num_solution_times,
dynamic_size=solution_times_chosen_by_solver,
element_shape=p.initial_state_vec.shape)
time_array = tf.TensorArray(
p.real_dtype,
size=num_solution_times,
dynamic_size=solution_times_chosen_by_solver,
element_shape=tf.TensorShape([]))
diagnostics = _BDFDiagnostics(num_jacobian_evaluations=0,
num_matrix_factorizations=0,
num_ode_fn_evaluations=0,
status=0)
iterand = _BDFIterand(
jacobian_mat=tf.zeros([p.num_odes, p.num_odes],
dtype=p.common_state_dtype),
jacobian_is_up_to_date=False,
new_step_size=solver_internal_state.step_size,
num_steps=0,
num_steps_same_size=0,
should_update_jacobian=True,
should_update_step_size=False,
time=p.initial_time,
unitary=tf.zeros([p.num_odes, p.num_odes],
dtype=p.common_state_dtype),
upper=tf.zeros([p.num_odes, p.num_odes],
dtype=p.common_state_dtype),
)
# (3) Make non-static assertions.
with tf.control_dependencies(
self._assert_ops(
previous_solver_internal_state=
previous_solver_internal_state,
initial_state_vec=p.initial_state_vec,
final_time=final_time,
initial_time=p.initial_time,
solution_times=solution_times,
max_num_steps=max_num_steps,
max_num_newton_iters=max_num_newton_iters,
atol=p.atol,
rtol=p.rtol,
first_step_size=solver_internal_state.step_size,
safety_factor=p.safety_factor,
min_step_size_factor=min_step_size_factor,
max_step_size_factor=max_step_size_factor,
max_order=max_order,
newton_tol_factor=newton_tol_factor,
newton_step_size_factor=newton_step_size_factor,
solution_times_chosen_by_solver=
solution_times_chosen_by_solver,
)):
# (4) Solve up to final time.
if solution_times_chosen_by_solver:
def step_cond(next_time, diagnostics, iterand, *_):
return (iterand.time < next_time) & (tf.equal(
diagnostics.status, 0))
[
_, diagnostics, iterand, solver_internal_state,
state_vec_array, time_array
] = tf.while_loop(step_cond, step, [
final_time, diagnostics, iterand,
solver_internal_state, state_vec_array, time_array
])
else:
def advance_to_solution_time_cond(n, diagnostics, *_):
return (n < num_solution_times) & (tf.equal(
diagnostics.status, 0))
[
_, diagnostics, iterand, solver_internal_state,
state_vec_array, time_array
] = tf.while_loop(
advance_to_solution_time_cond,
advance_to_solution_time, [
0, diagnostics, iterand, solver_internal_state,
state_vec_array, time_array
])
# (6) Return `Results` object.
states = util.get_state_from_vec(state_vec_array.stack(),
p.state_shape)
times = time_array.stack()
if not solution_times_chosen_by_solver:
tensorshape_util.set_shape(times, solution_times.shape)
tf.nest.map_structure(
lambda s, ini_s: tensorshape_util.set_shape( # pylint: disable=g-long-lambda
s,
tensorshape_util.concatenate(
solution_times.shape, ini_s.shape)),
states,
p.initial_state)
return base.Results(
times=times,
states=states,
diagnostics=diagnostics,
solver_internal_state=solver_internal_state)
