# Based on https://github.com/tensorflow/tensorflow/tree/master/tensorflow/contrib/integrate
import torch
from .misc import (
_scaled_dot_product, _convert_to_tensor, _is_finite, _select_initial_step, _handle_unused_kwargs, _is_iterable,
_optimal_step_size, _compute_error_ratio
)
from .solvers import AdaptiveStepsizeODESolver
from .interp import _interp_fit, _interp_evaluate
from .rk_common import _RungeKuttaState, _ButcherTableau, _runge_kutta_step
_DORMAND_PRINCE_SHAMPINE_TABLEAU = _ButcherTableau(
alpha=[1 / 5, 3 / 10, 4 / 5, 8 / 9, 1., 1.],
beta=[
[1 / 5],
[3 / 40, 9 / 40],
[44 / 45, -56 / 15, 32 / 9],
[19372 / 6561, -25360 / 2187, 64448 / 6561, -212 / 729],
[9017 / 3168, -355 / 33, 46732 / 5247, 49 / 176, -5103 / 18656],
[35 / 384, 0, 500 / 1113, 125 / 192, -2187 / 6784, 11 / 84],
],
c_sol=[35 / 384, 0, 500 / 1113, 125 / 192, -2187 / 6784, 11 / 84, 0],
c_error=[
35 / 384 - 1951 / 21600,
0,
500 / 1113 - 22642 / 50085,
125 / 192 - 451 / 720,
-2187 / 6784 - -12231 / 42400,
11 / 84 - 649 / 6300,
-1. / 60.,
],
)
DPS_C_MID = [
6025192743 / 30085553152 / 2, 0, 51252292925 / 65400821598 / 2, -2691868925 / 45128329728 / 2,
187940372067 / 1594534317056 / 2, -1776094331 / 19743644256 / 2, 11237099 / 235043384 / 2
]
def _interp_fit_dopri5(y0, y1, k, dt, tableau=_DORMAND_PRINCE_SHAMPINE_TABLEAU):
"""Fit an interpolating polynomial to the results of a Runge-Kutta step."""
dt = dt.type_as(y0[0])
y_mid = tuple(y0_ + _scaled_dot_product(dt, DPS_C_MID, k_) for y0_, k_ in zip(y0, k))
f0 = tuple(k_[0] for k_ in k)
f1 = tuple(k_[-1] for k_ in k)
return _interp_fit(y0, y1, y_mid, f0, f1, dt)
def _abs_square(x):
return torch.mul(x, x)
def _ta_append(list_of_tensors, value):
"""Append a value to the end of a list of PyTorch tensors."""
list_of_tensors.append(value)
return list_of_tensors
class Dopri5Solver(AdaptiveStepsizeODESolver):
def __init__(
self, func, y0, rtol, atol, first_step=None, safety=0.9, ifactor=10.0, dfactor=0.2, max_num_steps=2**31 - 1,
**unused_kwargs
):
_handle_unused_kwargs(self, unused_kwargs)
del unused_kwargs
self.func = func
self.y0 = y0
self.rtol = rtol if _is_iterable(rtol) else [rtol] * len(y0)
self.atol = atol if _is_iterable(atol) else [atol] * len(y0)
self.first_step = first_step
self.safety = _convert_to_tensor(safety, dtype=torch.float64, device=y0[0].device)
self.ifactor = _convert_to_tensor(ifactor, dtype=torch.float64, device=y0[0].device)
self.dfactor = _convert_to_tensor(dfactor, dtype=torch.float64, device=y0[0].device)
self.max_num_steps = _convert_to_tensor(max_num_steps, dtype=torch.int32, device=y0[0].device)
def before_integrate(self, t):
f0 = self.func(t[0].type_as(self.y0[0]), self.y0)
if self.first_step is None:
first_step = _select_initial_step(self.func, t[0], self.y0, 4, self.rtol[0], self.atol[0], f0=f0).to(t)
else:
first_step = _convert_to_tensor(self.first_step, dtype=t.dtype, device=t.device)
self.rk_state = _RungeKuttaState(self.y0, f0, t[0], t[0], first_step, interp_coeff=[self.y0] * 5)
def advance(self, next_t):
"""Interpolate through the next time point, integrating as necessary."""
n_steps = 0
while next_t > self.rk_state.t1:
assert n_steps < self.max_num_steps, 'max_num_steps exceeded ({}>={})'.format(n_steps, self.max_num_steps)
self.rk_state = self._adaptive_dopri5_step(self.rk_state)
n_steps += 1
return _interp_evaluate(self.rk_state.interp_coeff, self.rk_state.t0, self.rk_state.t1, next_t)
def _adaptive_dopri5_step(self, rk_state):
"""Take an adaptive Runge-Kutta step to integrate the ODE."""
y0, f0, _, t0, dt, interp_coeff = rk_state
########################################################
# Assertions #
########################################################
assert t0 + dt > t0, 'underflow in dt {}'.format(dt.item())
for y0_ in y0:
assert _is_finite(torch.abs(y0_)), 'non-finite values in state `y`: {}'.format(y0_)
y1, f1, y1_error, k = _runge_kutta_step(self.func, y0, f0, t0, dt, tableau=_DORMAND_PRINCE_SHAMPINE_TABLEAU)
########################################################
# Error Ratio #
########################################################
mean_sq_error_ratio = _compute_error_ratio(y1_error, atol=self.atol, rtol=self.rtol, y0=y0, y1=y1)
accept_step = (torch.tensor(mean_sq_error_ratio) <= 1).all()
########################################################
# Update RK State #
########################################################
y_next = y1 if accept_step else y0
f_next = f1 if accept_step else f0
t_next = t0 + dt if accept_step else t0
interp_coeff = _interp_fit_dopri5(y0, y1, k, dt) if accept_step else interp_coeff
dt_next = _optimal_step_size(
dt, mean_sq_error_ratio, safety=self.safety, ifactor=self.ifactor, dfactor=self.dfactor, order=5
)
rk_state = _RungeKuttaState(y_next, f_next, t0, t_next, dt_next, interp_coeff)
return rk_state
