def step_func(self, func, t, dt, y):
self._update_history(t, func(t, y))
order = min(len(self.prev_f), self.max_order - 1)
if order < _MIN_ORDER - 1:
# Compute using RK4.
dy = rk_common.rk4_alt_step_func(func, t, dt, y, k1=self.prev_f[0])
return dy
else:
# Adams-Bashforth predictor.
bashforth_coeffs = _BASHFORTH_COEFFICIENTS[order]
ab_div = _DIVISOR[order]
dy = tuple(dt * _scaled_dot_product(1 / ab_div, bashforth_coeffs, f_) for f_ in zip(*self.prev_f))
# Adams-Moulton corrector.
if self.implicit:
moulton_coeffs = _MOULTON_COEFFICIENTS[order + 1]
am_div = _DIVISOR[order + 1]
delta = tuple(dt * _scaled_dot_product(1 / am_div, moulton_coeffs[1:], f_) for f_ in zip(*self.prev_f))
converged = False
for _ in range(self.max_iters):
dy_old = dy
f = func(t + dt, tuple(y_ + dy_ for y_, dy_ in zip(y, dy)))
dy = tuple(dt * (moulton_coeffs[0] / am_div) * f_ + delta_ for f_, delta_ in zip(f, delta))
converged = _has_converged(dy_old, dy, self.rtol, self.atol)
if converged:
break
if not converged:
print('Warning: Functional iteration did not converge. Solution may be incorrect.', file=sys.stderr)
self.prev_f.pop()
self._update_history(t, f)
return dy
def _interp_fit_adaptive_heun(y0, y1, k, dt, tableau=_ADAPTIVE_HEUN_TABLEAU):
"""Fit an interpolating polynomial to the results of a Runge-Kutta step."""
dt = tf.cast(dt, y0[0].dtype)
y_mid = tuple(y0_ + _scaled_dot_product(dt, AH_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 _interp_fit_bosh3(y0, y1, k, dt):
"""Fit an interpolating polynomial to the results of a Runge-Kutta step."""
dt = tf.cast(dt, y0[0].dtype)
y_mid = tuple(y0_ + _scaled_dot_product(dt, BS_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 _interp_fit_dopri8(y0, y1, k, dt, tableau=_DOPRI8_TABLEAU):
"""Fit an interpolating polynomial to the results of a Runge-Kutta step."""
y_mid = tuple(y0_ + _scaled_dot_product(dt, 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 _interp_eval_tsit5(t0, t1, k, eval_t):
dt = t1 - t0
y0 = tuple(k_[0] for k_ in k)
interp_coeff = _interp_coeff_tsit5(t0, dt, eval_t)
y_t = tuple(y0_ + _scaled_dot_product(dt, interp_coeff, k_)
for y0_, k_ in zip(y0, k))
return y_t
def _runge_kutta_step(func, y0, f0, t0, dt, tableau):
"""Take an arbitrary Runge-Kutta step and estimate error.
Args:
func: Function to evaluate like `func(t, y)` to compute the time derivative
of `y`.
y0: Tensor initial value for the state.
f0: Tensor initial value for the derivative, computed from `func(t0, y0)`.
t0: float64 scalar Tensor giving the initial time.
dt: float64 scalar Tensor giving the size of the desired time step.
tableau: optional _ButcherTableau describing how to take the Runge-Kutta
step.
name: optional name for the operation.
Returns:
Tuple `(y1, f1, y1_error, k)` giving the estimated function value after
the Runge-Kutta step at `t1 = t0 + dt`, the derivative of the state at `t1`,
estimated error at `t1`, and a list of Runge-Kutta coefficients `k` used for
calculating these terms.
"""
y0 = cast_double(y0)
f0 = cast_double(f0)
dtype = y0[0].dtype
device = y0[0].device
t0 = _convert_to_tensor(t0, dtype=dtype, device=device)
dt = _convert_to_tensor(dt, dtype=dtype, device=device)
k = tuple(map(lambda x: [x], f0))
for alpha_i, beta_i in zip(tableau.alpha, tableau.beta):
ti = t0 + alpha_i * dt
yi = tuple(y0_ + _scaled_dot_product(dt, cast_double(beta_i), k_)
for y0_, k_ in zip(y0, k))
tuple(k_.append(cast_double(f_)) for k_, f_ in zip(k, func(ti, yi)))
if not (tableau.c_sol[-1] == 0 and tableau.c_sol[:-1] == tableau.beta[-1]):
# This property (true for Dormand-Prince) lets us save a few FLOPs.
yi = tuple(y0_ + _scaled_dot_product(dt, tableau.c_sol, k_)
for y0_, k_ in zip(y0, k))
y1 = yi
f1 = tuple(k_[-1] for k_ in k)
y1_error = tuple(_scaled_dot_product(dt, tableau.c_error, k_) for k_ in k)
return (y1, f1, y1_error, k)
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 = tf.cast(dt, y0[0].dtype)
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 _adaptive_adams_step(self, vcabm_state, final_t):
y0, prev_f, prev_t, next_t, prev_phi, order = vcabm_state
if next_t > final_t:
next_t = final_t
dt = (next_t - prev_t[0])
dt_cast = move_to_device(dt, y0[0].device)
dt_cast = tf.cast(dt_cast, y0[0].dtype)
# Explicit predictor step.
g, phi = g_and_explicit_phi(prev_t, next_t, prev_phi, order)
# g = move_to_device(g, y0[0].device)
g = tf.cast(g, dt_cast.dtype)
# phi = [tf.cast(phi_, dt_cast.dtype) for phi_ in phi]
p_next = tuple(
y0_ +
_scaled_dot_product(dt_cast, g[:max(1, order -
1)], phi_[:max(1, order - 1)])
for y0_, phi_ in zip(y0, tuple(zip(*phi))))
# Update phi to implicit.
next_t = move_to_device(next_t, p_next[0].device)
next_f0 = self.func(tf.cast(next_t, self.y0[0].dtype), p_next)
implicit_phi_p = compute_implicit_phi(phi, next_f0, order + 1)
# Implicit corrector step.
y_next = tuple(
p_next_ + dt_cast * g[order - 1] * tf.cast(iphi_, dt_cast.dtype)
for p_next_, iphi_ in zip(p_next, implicit_phi_p[order - 1]))
# Error estimation.
tolerance = tuple(
atol_ +
rtol_ * tf.reduce_max([tf.abs(y0_), tf.abs(y1_)])
for atol_, rtol_, y0_, y1_ in zip(self.atol, self.rtol, y0,
y_next))
local_error = tuple(dt_cast * (g[order] - g[order - 1]) *
tf.cast(iphi_, dt_cast.dtype)
for iphi_ in implicit_phi_p[order])
error_k = _compute_error_ratio(local_error, tolerance)
accept_step = tf.reduce_all((tf.convert_to_tensor(error_k) <= 1))
if not accept_step:
# Retry with adjusted step size if step is rejected.
dt_next = _optimal_step_size(dt,
error_k,
self.safety,
self.ifactor,
self.dfactor,
order=order)
return _VCABMState(y0,
prev_f,
prev_t,
prev_t[0] + dt_next,
prev_phi,
order=order)
# We accept the step. Evaluate f and update phi.
next_t = move_to_device(next_t, p_next[0].device)
next_f0 = self.func(tf.cast(next_t, self.y0[0].dtype), y_next)
implicit_phi = compute_implicit_phi(phi, next_f0, order + 2)
next_order = order
if len(prev_t) <= 4 or order < 3:
next_order = min(order + 1, 3, self.max_order)
else:
error_km1 = _compute_error_ratio(
tuple(dt_cast * (g[order - 1] - g[order - 2]) * iphi_
for iphi_ in implicit_phi_p[order - 1]), tolerance)
error_km2 = _compute_error_ratio(
tuple(dt_cast * (g[order - 2] - g[order - 3]) * iphi_
for iphi_ in implicit_phi_p[order - 2]), tolerance)
if min(error_km1 + error_km2) < max(error_k):
next_order = order - 1
elif order < self.max_order:
error_kp1 = _compute_error_ratio(
tuple(dt_cast * gamma_star[order] * iphi_
for iphi_ in implicit_phi_p[order]), tolerance)
if max(error_kp1) < max(error_k):
next_order = order + 1
# Keep step size constant if increasing order. Else use adaptive step size.
dt_next = dt if next_order > order else _optimal_step_size(
dt,
error_k,
self.safety,
self.ifactor,
self.dfactor,
order=order + 1)
prev_f.appendleft(next_f0)
prev_t.appendleft(next_t)
return _VCABMState(p_next,
prev_f,
prev_t,
next_t + dt_next,
implicit_phi,
order=next_order)
