def _prepare_coefficients(self, dtype):
bdf_coefficients = tf.concat(
[[0.], tf.cast(self._bdf_coefficients, dtype=dtype)], 0)
if self._validate_args:
bdf_coefficients = tf.ensure_shape(bdf_coefficients, [6])
util.error_if_not_vector(bdf_coefficients, 'bdf_coefficients')
newton_coefficients = 1. / (
(1. - bdf_coefficients) * bdf_util.RECIPROCAL_SUMS)
error_coefficients = bdf_coefficients * bdf_util.RECIPROCAL_SUMS + 1. / (
bdf_util.ORDERS + 1)
return newton_coefficients, error_coefficients
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,
):
# Static assertions
del jacobian_fn, jacobian_sparsity # not used by DormandPrince
if batch_ndims is not None and batch_ndims != 0:
raise NotImplementedError(
'For homogeneous batching use `batch_ndims=0`.')
solution_times_by_solver = isinstance(solution_times,
base.ChosenBySolver)
with tf.name_scope(self._name):
# (2) Convert to tensors, determine dtypes.
p = self._prepare_common_params(initial_state, initial_time)
max_num_steps = self._max_num_steps
max_ode_fn_evals = 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_ode_fn_evals = max_num_steps * self.ODE_FN_EVALS_PER_STEP
step_size = tf.convert_to_tensor(self._step_size,
dtype=p.real_dtype)
rtol = tf.convert_to_tensor(tf.cast(self._rtol, p.real_dtype))
atol = tf.convert_to_tensor(tf.cast(self._atol, p.real_dtype))
# Use i(d)factor notation for increasing and decreasing factors.
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=p.initial_state,
initial_time=p.initial_time,
)
num_solution_times = 0
if solution_times_by_solver:
final_time = tf.cast(solution_times.final_time, p.real_dtype)
times_array = tf.TensorArray(p.real_dtype,
size=num_solution_times,
dynamic_size=True,
element_shape=tf.TensorShape([]))
else:
solution_times = tf.cast(solution_times, p.real_dtype)
util.error_if_not_vector(solution_times, 'solution_times')
num_solution_times = tf.size(solution_times)
times_array = tf.TensorArray(
p.real_dtype,
size=num_solution_times,
dynamic_size=False,
element_shape=[]).unstack(solution_times)
solutions_arrays = [
tf.TensorArray(dtype=component_dtype,
size=num_solution_times,
dynamic_size=solution_times_by_solver)
for component_dtype in tf.nest.flatten(p.state_dtypes)
]
solutions_arrays = tf.nest.pack_sequence_as(
p.initial_state, solutions_arrays)
rk_step = functools.partial(self._step,
max_ode_fn_evals=max_ode_fn_evals,
ode_fn=ode_fn,
atol=atol,
rtol=rtol,
safety=safety,
ifactor=ifactor,
dfactor=dfactor)
advance_to_solution_time = functools.partial(
_advance_to_solution_time,
times_array=solution_times,
step_fn=rk_step,
validate_args=self._validate_args)
assert_ops = self._assert_ops(
ode_fn=ode_fn,
initial_time=p.initial_time,
initial_state=p.initial_state,
solution_times=solution_times,
previous_solver_state=previous_solver_internal_state,
rtol=rtol,
atol=atol,
first_step_size=step_size,
safety_factor=safety,
min_step_size_factor=ifactor,
max_step_size_factor=dfactor,
max_num_steps=max_num_steps,
solution_times_by_solver=solution_times_by_solver)
with tf.control_dependencies(assert_ops):
ode_evals_by_now = 1 if self._validate_args else 0
ode_evals_by_now += 1 if solver_internal_state is None else 0
diagnostics = _DopriDiagnostics(
num_ode_fn_evaluations=ode_evals_by_now,
num_jacobian_evaluations=0,
num_matrix_factorizations=0,
status=0)
if solution_times_by_solver:
r = _dense_solutions_to_final_time(
final_time=final_time,
solver_state=solver_internal_state,
diagnostics=diagnostics,
step_fn=rk_step,
ode_fn=ode_fn,
times_array=times_array,
solutions_arrays=solutions_arrays,
validate_args=self._validate_args)
solver_internal_state, diagnostics, times_array, solutions_arrays = r
else:
def iterate_cond(time_id, *_):
return time_id < num_solution_times
[
_, solver_internal_state, diagnostics, solutions_arrays
] = tf.while_loop(iterate_cond, advance_to_solution_time, [
0, solver_internal_state, diagnostics, solutions_arrays
])
times = times_array.stack()
stack_components = lambda x: x.stack()
states = tf.nest.map_structure(stack_components,
solutions_arrays)
return base.Results(
times=times,
states=states,
diagnostics=diagnostics,
solver_internal_state=solver_internal_state)
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,
):
# Static assertions
del jacobian_fn, jacobian_sparsity # not used by DormandPrince
if batch_ndims is not None and batch_ndims != 0:
raise NotImplementedError('For homogeneous batching use `batch_ndims=0`.')
solution_times_by_solver = isinstance(solution_times, base.ChosenBySolver)
with tf.name_scope(self._name):
# (2) Convert to tensors, determined dtypes.
initial_state = tf.convert_to_tensor(initial_state)
util.error_if_not_real_or_complex(initial_state, 'initial_state')
state_dtype = initial_state.dtype
real_dtype = tf.abs(initial_state).dtype
initial_time = tf.cast(initial_time, real_dtype)
max_num_steps = self._max_num_steps
max_ode_fn_evals = 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_ode_fn_evals = max_num_steps * self.ODE_FN_EVALS_PER_STEP
step_size = tf.convert_to_tensor(self._first_step_size, dtype=real_dtype)
rtol = tf.convert_to_tensor(tf.cast(self._rtol, real_dtype))
atol = tf.convert_to_tensor(tf.cast(self._atol, real_dtype))
safety = tf.convert_to_tensor(self._safety_factor, dtype=real_dtype)
# Use i(d)factor notation for increasing and decreasing factors.
ifactor, dfactor = self._max_step_size_factor, self._min_step_size_factor
ifactor = tf.convert_to_tensor(ifactor, dtype=real_dtype)
dfactor = tf.convert_to_tensor(dfactor, dtype=real_dtype)
solver_internal_state = previous_solver_internal_state
if solver_internal_state is None:
initial_derivative = ode_fn(initial_time, initial_state)
initial_derivative = tf.convert_to_tensor(initial_derivative)
solver_internal_state = _RungeKuttaSolverInternalState(
current_state=initial_state,
current_derivative=initial_derivative,
last_step_start=initial_time,
current_time=initial_time,
step_size=step_size,
interpolating_coefficients=[initial_state] * self.ORDER
)
num_solution_times = 0
if solution_times_by_solver:
final_time = tf.cast(solution_times.final_time, real_dtype)
times_array = tf.TensorArray(
real_dtype,
size=num_solution_times,
dynamic_size=True,
element_shape=tf.TensorShape([]))
else:
solution_times = tf.cast(solution_times, real_dtype)
util.error_if_not_vector(solution_times, 'solution_times')
num_solution_times = tf.size(solution_times)
times_array = tf.TensorArray(
real_dtype,
size=num_solution_times,
dynamic_size=False,
element_shape=[]).unstack(solution_times)
solutions_array = tf.TensorArray(
state_dtype,
size=num_solution_times,
dynamic_size=solution_times_by_solver)
rk_step = functools.partial(
self._step,
max_ode_fn_evals=max_ode_fn_evals,
ode_fn=ode_fn,
atol=atol,
rtol=rtol,
safety=safety,
ifactor=ifactor,
dfactor=dfactor
)
advance_to_solution_time = functools.partial(
_advance_to_solution_time,
times_array=solution_times,
step_fn=rk_step
)
assert_ops = self._assert_ops(
ode_fn=ode_fn,
initial_time=initial_time,
initial_state=initial_state,
solution_times=solution_times,
previous_solver_state=previous_solver_internal_state,
rtol=rtol,
atol=atol,
first_step_size=step_size,
safety_factor=safety,
min_step_size_factor=ifactor,
max_step_size_factor=dfactor,
max_num_steps=max_num_steps,
solution_times_by_solver=solution_times_by_solver
)
with tf.control_dependencies(assert_ops):
ode_evals_by_now = 1 if self._validate_args else 0
ode_evals_by_now += 1 if solver_internal_state is None else 0
diagnostics = _DopriDiagnostics(
num_ode_fn_evaluations=ode_evals_by_now, num_jacobian_evaluations=0,
num_matrix_factorizations=0, status=0)
if solution_times_by_solver:
res = _dense_solutions_to_final_time(
final_time=final_time,
solver_state=solver_internal_state,
diagnostics=diagnostics,
step_fn=rk_step,
ode_fn=ode_fn,
times_array=times_array,
solutions_array=solutions_array
)
solver_internal_state, diagnostics, times_array, solutions_array = res
else:
def iterate_cond(time_id, *_):
return time_id < num_solution_times
[
_, solver_internal_state, diagnostics, solutions_array
] = tf.while_loop(iterate_cond, advance_to_solution_time, [
0, solver_internal_state, diagnostics, solutions_array
])
times = times_array.stack()
states = solutions_array.stack()
return base.Results(
times=times,
states=states,
diagnostics=diagnostics,
solver_internal_state=solver_internal_state)
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)
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)
