def test_first_step_size_is_large_when_ode_fn_is_constant(self, dtype):
initial_state_vec = tf.constant([1.], dtype=dtype)
real_dtype = abs(initial_state_vec).dtype
atol = tf.constant(1e-12, dtype=real_dtype)
first_order_bdf_coefficient = -0.1850
first_order_error_coefficient = first_order_bdf_coefficient + 0.5
initial_time = tf.constant(0., dtype=real_dtype)
ode_fn_vec = lambda time, state: 1.
rtol = tf.constant(1e-8, dtype=real_dtype)
safety_factor = tf.constant(0.9, dtype=real_dtype)
max_step_size = 1.
step_size = bdf_util.first_step_size(atol,
first_order_error_coefficient,
initial_state_vec,
initial_time,
ode_fn_vec,
rtol,
safety_factor,
max_step_size=max_step_size)
# Step size should be maximal.
self.assertAllClose(self.evaluate(step_size), max_step_size)
def _initialize_solver_internal_state(
self,
ode_fn,
initial_time,
initial_state,
):
p = self._prepare_common_params(
ode_fn=ode_fn,
initial_state=initial_state,
initial_time=initial_time,
)
first_step_size = self._first_step_size
if first_step_size is None:
_, error_coefficients = self._prepare_coefficients(
p.common_state_dtype)
first_step_size = bdf_util.first_step_size(
p.atol, error_coefficients[1], p.initial_state_vec,
p.initial_time, p.ode_fn_vec, p.rtol, p.safety_factor)
first_step_size = tf.convert_to_tensor(first_step_size,
dtype=p.real_dtype)
if self._validate_args:
first_step_size = tf.ensure_shape(first_step_size, [])
first_order_backward_difference = p.ode_fn_vec(
p.initial_time, p.initial_state_vec) * tf.cast(
first_step_size, p.common_state_dtype)
backward_differences = tf.concat(
[
p.initial_state_vec[tf.newaxis, :],
first_order_backward_difference[tf.newaxis, :],
tf.zeros(ps.stack([bdf_util.MAX_ORDER + 1, p.num_odes]),
dtype=p.common_state_dtype),
],
axis=0,
)
return _BDFSolverInternalState(
backward_differences=backward_differences,
order=tf.ones([], tf.int32),
step_size=first_step_size)
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):
"""See `tfp.math.ode.Solver.solve`."""
# The `solve` 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 assert_ops():
"""Creates a list of assert operations."""
if not self._validate_args:
return []
assert_ops = []
if ((not initial_state_missing)
and (previous_solver_internal_state is not None)):
assert_initial_state_matches_previous_solver_internal_state = (
tf.assert_near(
tf.norm(
original_initial_state -
previous_solver_internal_state.
backward_differences[0], np.inf),
0.,
message=
'`previous_solver_internal_state` does not match '
'`initial_state`.'))
assert_ops.append(
assert_initial_state_matches_previous_solver_internal_state
)
if solution_times_chosen_by_solver:
assert_ops.append(
util.assert_positive(final_time - initial_time,
'final_time - initial_time'))
else:
assert_ops += [
util.assert_increasing(solution_times, 'solution_times'),
util.assert_nonnegative(
solution_times[0] - initial_time,
'solution_times[0] - initial_time'),
]
if max_num_steps is not None:
assert_ops.append(
util.assert_positive(max_num_steps, 'max_num_steps'))
if max_num_newton_iters is not None:
assert_ops.append(
util.assert_positive(max_num_newton_iters,
'max_num_newton_iters'))
assert_ops += [
util.assert_positive(rtol, 'rtol'),
util.assert_positive(atol, 'atol'),
util.assert_positive(first_step_size, 'first_step_size'),
util.assert_positive(safety_factor, 'safety_factor'),
util.assert_positive(min_step_size_factor,
'min_step_size_factor'),
util.assert_positive(max_step_size_factor,
'max_step_size_factor'),
tf.Assert((max_order >= 1) & (max_order <= bdf_util.MAX_ORDER),
[
'`max_order` must be between 1 and {}.'.format(
bdf_util.MAX_ORDER)
]),
util.assert_positive(newton_tol_factor, 'newton_tol_factor'),
util.assert_positive(newton_step_size_factor,
'newton_step_size_factor'),
]
return assert_ops
def advance_to_solution_time(n, diagnostics, iterand,
solver_internal_state, states_array,
times_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))
solution_times_n = solution_times_array.read(n)
[
_, diagnostics, iterand, solver_internal_state, states_array,
times_array
] = tf.while_loop(step_cond, step, [
solution_times_n, diagnostics, iterand, solver_internal_state,
states_array, times_array
])
states_array = states_array.write(
n, solver_internal_state.backward_differences[0])
times_array = times_array.write(n, solution_times_n)
return (n + 1, diagnostics, iterand, solver_internal_state,
states_array, times_array)
def step(next_time, diagnostics, iterand, solver_internal_state,
states_array, times_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=tf.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=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:
states_array = states_array.write(
states_array.size(),
solver_internal_state.backward_differences[0])
times_array = times_array.write(times_array.size(),
iterand.time)
return (next_time, diagnostics, iterand, solver_internal_state,
states_array, times_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, 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, state_shape, step_size = solver_internal_state
if max_num_steps is not None:
status = tf.where(tf.equal(num_steps, max_num_steps), -1, 0)
backward_differences = tf.where(
should_update_step_size,
bdf_util.interpolate_backward_differences(
backward_differences, order, new_step_size / step_size),
backward_differences)
step_size = tf.where(should_update_step_size, new_step_size,
step_size)
should_update_factorization = should_update_step_size
num_steps_same_size = tf.where(should_update_step_size, 0,
num_steps_same_size)
def update_factorization():
return bdf_util.newton_qr(
jacobian, newton_coefficients_array.read(order), step_size)
if self._evaluate_jacobian_lazily:
def update_jacobian_and_factorization():
new_jacobian = jacobian_fn_mat(time,
backward_differences[0])
new_unitary, new_upper = update_factorization()
return [
new_jacobian, 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, jacobian_is_up_to_date,
num_jacobian_evaluations, new_unitary, new_upper
]
[
jacobian, 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 = atol + rtol * tf.abs(backward_differences[0])
newton_tol = newton_tol_factor * tf.norm(tol)
[
newton_converged, next_backward_difference, next_state,
newton_num_iters
] = bdf_util.newton(backward_differences, max_num_newton_iters,
newton_coefficients_array.read(order),
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 * tf.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 = tf.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 = tf.where(
converged_and_rejected,
util.next_step_size(step_size, order, error_ratio,
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 = tf.where(accepted, time + step_size, time)
backward_differences = tf.where(
accepted,
bdf_util.update_backward_differences(backward_differences,
next_backward_difference,
next_state, order),
backward_differences)
jacobian_is_up_to_date = jacobian_is_up_to_date & tf.logical_not(
accepted)
num_steps_same_size = tf.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.get_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 = tf.where(
should_update_order_and_step_size
& proposed_error_ratio_is_lower, proposed_order, new_order)
new_error_ratio = tf.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 = tf.where(
should_update_order_and_step_size,
util.next_step_size(step_size, order, error_ratio,
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, 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, state_shape, step_size)
return accepted, diagnostics, iterand, solver_internal_state
# (1) Make static assertions.
# TODO(parsiad): 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))
initial_state_missing = initial_state is None
if initial_state_missing and previous_solver_internal_state is None:
raise ValueError(
'At least one of `initial_state` or `previous_solver_internal_state` '
'must be specified')
with tf.name_scope(self._name):
# (2) Initialize variables.
original_initial_state = initial_state
if previous_solver_internal_state is None:
initial_state = tf.convert_to_tensor(initial_state)
original_state_shape = tf.shape(initial_state)
else:
initial_state = previous_solver_internal_state.backward_differences[
0]
original_state_shape = previous_solver_internal_state.state_shape
state_dtype = initial_state.dtype
util.error_if_not_real_or_complex(initial_state, 'initial_state')
# TODO(parsiad): Support complex automatic Jacobians.
if jacobian_fn is None and state_dtype.is_complex:
raise NotImplementedError(
'The BDF solver does not support automatic '
'Jacobian computations for complex dtypes.')
num_odes = tf.size(initial_state)
original_state_tensor_shape = initial_state.get_shape()
initial_state = tf.reshape(initial_state, [-1])
ode_fn_vec = util.get_ode_fn_vec(ode_fn, original_state_shape)
# `real_dtype` is the floating point `dtype` associated with
# `initial_state.dtype` (recall that the latter can be complex).
real_dtype = tf.abs(initial_state).dtype
initial_time = tf.ensure_shape(
tf.convert_to_tensor(initial_time, dtype=real_dtype), [])
num_solution_times = 0
if solution_times_chosen_by_solver:
final_time = solution_times.final_time
final_time = tf.ensure_shape(
tf.convert_to_tensor(final_time, dtype=real_dtype), [])
else:
solution_times = tf.convert_to_tensor(solution_times,
dtype=real_dtype)
num_solution_times = tf.size(solution_times)
solution_times_array = tf.TensorArray(
solution_times.dtype,
size=num_solution_times,
element_shape=[]).unstack(solution_times)
util.error_if_not_vector(solution_times, 'solution_times')
jacobian_fn_mat = util.get_jacobian_fn_mat(
jacobian_fn,
ode_fn_vec,
original_state_shape,
use_pfor=self._use_pfor_to_compute_jacobian)
rtol = tf.convert_to_tensor(self._rtol, dtype=real_dtype)
atol = tf.convert_to_tensor(self._atol, dtype=real_dtype)
safety_factor = tf.ensure_shape(
tf.convert_to_tensor(self._safety_factor, dtype=real_dtype),
[])
min_step_size_factor = tf.ensure_shape(
tf.convert_to_tensor(self._min_step_size_factor,
dtype=real_dtype), [])
max_step_size_factor = tf.ensure_shape(
tf.convert_to_tensor(self._max_step_size_factor,
dtype=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.ensure_shape(
tf.convert_to_tensor(self._newton_tol_factor,
dtype=real_dtype), [])
newton_step_size_factor = tf.ensure_shape(
tf.convert_to_tensor(self._newton_step_size_factor,
dtype=real_dtype), [])
bdf_coefficients = tf.cast(
tf.concat([[0.],
tf.convert_to_tensor(self._bdf_coefficients,
dtype=real_dtype)], 0),
state_dtype)
util.error_if_not_vector(bdf_coefficients, 'bdf_coefficients')
newton_coefficients = 1. / (
(1. - bdf_coefficients) * bdf_util.RECIPROCAL_SUMS)
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 = bdf_coefficients * bdf_util.RECIPROCAL_SUMS + 1. / (
bdf_util.ORDERS + 1)
error_coefficients_array = tf.TensorArray(
error_coefficients.dtype,
size=bdf_util.MAX_ORDER + 1,
clear_after_read=False,
element_shape=[]).unstack(error_coefficients)
first_step_size = self._first_step_size
if first_step_size is None:
first_step_size = bdf_util.first_step_size(
atol, error_coefficients_array.read(1), initial_state,
initial_time, ode_fn_vec, rtol, safety_factor)
elif previous_solver_internal_state is not None:
tf.logging.warn(
'`first_step_size` is ignored since'
'`previous_solver_internal_state` was specified.')
first_step_size = tf.convert_to_tensor(first_step_size,
dtype=real_dtype)
if self._validate_args:
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, [])
bdf_coefficients = tf.ensure_shape(bdf_coefficients, [6])
first_step_size = tf.ensure_shape(first_step_size, [])
solver_internal_state = previous_solver_internal_state
if solver_internal_state is None:
first_order_backward_difference = ode_fn_vec(
initial_time, initial_state) * tf.cast(
first_step_size, state_dtype)
backward_differences = tf.concat([
tf.reshape(initial_state, [1, -1]),
first_order_backward_difference[tf.newaxis, :],
tf.zeros(tf.stack([bdf_util.MAX_ORDER + 1, num_odes]),
dtype=state_dtype),
], 0)
solver_internal_state = _BDFSolverInternalState(
backward_differences=backward_differences,
order=1,
state_shape=original_state_shape,
step_size=first_step_size)
states_array = tf.TensorArray(
state_dtype,
size=num_solution_times,
dynamic_size=solution_times_chosen_by_solver,
element_shape=initial_state.get_shape())
times_array = tf.TensorArray(
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=tf.zeros([num_odes, num_odes], dtype=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=initial_time,
unitary=tf.zeros([num_odes, num_odes], dtype=state_dtype),
upper=tf.zeros([num_odes, num_odes], dtype=state_dtype))
# (3) Make non-static assertions.
with tf.control_dependencies(assert_ops()):
# (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,
states_array, times_array
] = tf.while_loop(step_cond, step, [
final_time, diagnostics, iterand,
solver_internal_state, states_array, times_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,
states_array, times_array
] = tf.while_loop(
advance_to_solution_time_cond,
advance_to_solution_time, [
0, diagnostics, iterand, solver_internal_state,
states_array, times_array
])
# (6) Return `Results` object.
states = tf.reshape(states_array.stack(),
tf.concat([[-1], original_state_shape], 0))
times = times_array.stack()
if not solution_times_chosen_by_solver:
times.set_shape(solution_times.get_shape())
states.set_shape(solution_times.get_shape().concatenate(
original_state_tensor_shape))
return base.Results(
times=times,
states=states,
diagnostics=diagnostics,
solver_internal_state=solver_internal_state)
