Source code for pymc.ode.ode

#   Copyright 2020 The PyMC Developers
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#       http://www.apache.org/licenses/LICENSE-2.0
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import logging

import numpy as np
import pytensor
import pytensor.tensor as at
import scipy

from pytensor.graph.basic import Apply
from pytensor.graph.op import Op, get_test_value
from pytensor.tensor.type import TensorType

from pymc.exceptions import DtypeError, ShapeError
from pymc.ode import utils

_log = logging.getLogger("pymc")
floatX = pytensor.config.floatX


[docs]class DifferentialEquation(Op): r""" Specify an ordinary differential equation Due to the nature of the model (as well as included solvers), the process of ODE solution may perform slowly. A faster alternative library based on PyMC--sunode--has implemented Adams' method and BDF (backward differentation formula). More information about sunode is available at: https://github.com/aseyboldt/sunode. .. math:: \dfrac{dy}{dt} = f(y,t,p) \quad y(t_0) = y_0 Parameters ---------- func : callable Function specifying the differential equation. Must take arguments y (n_states,), t (scalar), p (n_theta,) times : array Array of times at which to evaluate the solution of the differential equation. n_states : int Dimension of the differential equation. For scalar differential equations, n_states=1. For vector valued differential equations, n_states = number of differential equations in the system. n_theta : int Number of parameters in the differential equation. t0 : float Time corresponding to the initial condition Examples -------- .. code-block:: python def odefunc(y, t, p): #Logistic differential equation return p[0] * y[0] * (1 - y[0]) times = np.arange(0.5, 5, 0.5) ode_model = DifferentialEquation(func=odefunc, times=times, n_states=1, n_theta=1, t0=0) """ _itypes = [ TensorType(floatX, (False,)), # y0 as 1D floatX vector TensorType(floatX, (False,)), # theta as 1D floatX vector ] _otypes = [ TensorType(floatX, (False, False)), # model states as floatX of shape (T, S) TensorType( floatX, (False, False, False) ), # sensitivities as floatX of shape (T, S, len(y0) + len(theta)) ] __props__ = ("func", "times", "n_states", "n_theta", "t0")
[docs] def __init__(self, func, times, *, n_states, n_theta, t0=0): if not callable(func): raise ValueError("Argument func must be callable.") if n_states < 1: raise ValueError("Argument n_states must be at least 1.") if n_theta <= 0: raise ValueError("Argument n_theta must be positive.") # Public self.func = func self.t0 = t0 self.times = tuple(times) self.n_times = len(times) self.n_states = n_states self.n_theta = n_theta self.n_p = n_states + n_theta # Private self._augmented_times = np.insert(times, 0, t0).astype(floatX) self._augmented_func = utils.augment_system(func, self.n_states, self.n_theta) self._sens_ic = utils.make_sens_ic(self.n_states, self.n_theta, floatX) # Cache symbolic sensitivities by the hash of inputs self._apply_nodes = {} self._output_sensitivities = {}
def _system(self, Y, t, p): r"""This is the function that will be passed to odeint. Solves both ODE and sensitivities. Parameters ---------- Y : array augmented state vector (n_states + n_states + n_theta) t : float current time p : array parameter vector (y0, theta) """ dydt, ddt_dydp = self._augmented_func(Y[: self.n_states], t, p, Y[self.n_states :]) derivatives = np.concatenate([dydt, ddt_dydp]) return derivatives def _simulate(self, y0, theta): # Initial condition comprised of state initial conditions and raveled sensitivity matrix s0 = np.concatenate([y0, self._sens_ic]) # perform the integration sol = scipy.integrate.odeint( func=self._system, y0=s0, t=self._augmented_times, args=(np.concatenate([y0, theta]),) ).astype(floatX) # The solution y = sol[1:, : self.n_states] # The sensitivities, reshaped to be a sequence of matrices sens = sol[1:, self.n_states :].reshape(self.n_times, self.n_states, self.n_p) return y, sens
[docs] def make_node(self, y0, theta): inputs = (y0, theta) _log.debug(f"make_node for inputs {hash(inputs)}") states = self._otypes[0]() sens = self._otypes[1]() # store symbolic output in dictionary such that it can be accessed in the grad method self._output_sensitivities[hash(inputs)] = sens return Apply(self, inputs, (states, sens))
def __call__(self, y0, theta, return_sens=False, **kwargs): if isinstance(y0, (list, tuple)) and not len(y0) == self.n_states: raise ShapeError("Length of y0 is wrong.", actual=(len(y0),), expected=(self.n_states,)) if isinstance(theta, (list, tuple)) and not len(theta) == self.n_theta: raise ShapeError( "Length of theta is wrong.", actual=(len(theta),), expected=(self.n_theta,) ) # convert inputs to tensors (and check their types) y0 = at.cast(at.as_tensor_variable(y0), floatX) theta = at.cast(at.as_tensor_variable(theta), floatX) inputs = [y0, theta] for i, (input_val, itype) in enumerate(zip(inputs, self._itypes)): if not itype.is_super(input_val.type): raise ValueError( f"Input {i} of type {input_val.type} does not have the expected type of {itype}" ) # use default implementation to prepare symbolic outputs (via make_node) states, sens = super().__call__(y0, theta, **kwargs) if pytensor.config.compute_test_value != "off": # compute test values from input test values test_states, test_sens = self._simulate( y0=get_test_value(y0), theta=get_test_value(theta) ) # check types of simulation result if not test_states.dtype == self._otypes[0].dtype: raise DtypeError( "Simulated states have the wrong type.", actual=test_states.dtype, expected=self._otypes[0].dtype, ) if not test_sens.dtype == self._otypes[1].dtype: raise DtypeError( "Simulated sensitivities have the wrong type.", actual=test_sens.dtype, expected=self._otypes[1].dtype, ) # check shapes of simulation result expected_states_shape = (self.n_times, self.n_states) expected_sens_shape = (self.n_times, self.n_states, self.n_p) if not test_states.shape == expected_states_shape: raise ShapeError( "Simulated states have the wrong shape.", test_states.shape, expected_states_shape, ) if not test_sens.shape == expected_sens_shape: raise ShapeError( "Simulated sensitivities have the wrong shape.", test_sens.shape, expected_sens_shape, ) # attach results as test values to the outputs states.tag.test_value = test_states sens.tag.test_value = test_sens if return_sens: return states, sens return states
[docs] def perform(self, node, inputs_storage, output_storage): y0, theta = inputs_storage[0], inputs_storage[1] # simulate states and sensitivities in one forward pass output_storage[0][0], output_storage[1][0] = self._simulate(y0, theta)
[docs] def infer_shape(self, fgraph, node, input_shapes): s_y0, s_theta = input_shapes output_shapes = [(self.n_times, self.n_states), (self.n_times, self.n_states, self.n_p)] return output_shapes
[docs] def grad(self, inputs, output_grads): _log.debug(f"grad w.r.t. inputs {hash(tuple(inputs))}") # fetch symbolic sensitivity output node from cache ihash = hash(tuple(inputs)) if ihash in self._output_sensitivities: sens = self._output_sensitivities[ihash] else: _log.debug("No cached sensitivities found!") _, sens = self.__call__(*inputs, return_sens=True) ograds = output_grads[0] # for each parameter, multiply sensitivities with the output gradient and sum the result # sens is (n_times, n_states, n_p) # ograds is (n_times, n_states) grads = [at.sum(sens[:, :, p] * ograds) for p in range(self.n_p)] # return separate gradient tensors for y0 and theta inputs result = at.stack(grads[: self.n_states]), at.stack(grads[self.n_states :]) return result