"""Symbolic differential equation model."""
from __future__ import annotations
import contextlib
import copy
import itertools
import re
from itertools import chain
from typing import TYPE_CHECKING
from collections.abc import Callable
from collections.abc import Sequence
import numpy as np
import sympy as sp
from sympy import ImmutableDenseMatrix, MutableDenseMatrix
from ._codegen.cxx_functions import (
sparse_functions,
sensi_functions,
nobody_functions,
var_in_function_signature,
)
from .cxxcodeprinter import csc_matrix
from .de_model_components import (
DifferentialState,
AlgebraicState,
AlgebraicEquation,
Observable,
EventObservable,
SigmaY,
SigmaZ,
Parameter,
Constant,
LogLikelihoodY,
LogLikelihoodZ,
LogLikelihoodRZ,
Expression,
ConservationLaw,
Event,
State,
ModelQuantity,
)
from .import_utils import (
_default_simplify,
SBMLException,
toposort_symbols,
smart_subs_dict,
ObservableTransformation,
amici_time_symbol,
strip_pysb,
unique_preserve_order,
)
from .sympy_utils import (
smart_jacobian,
smart_is_zero_matrix,
smart_multiply,
_parallel_applyfunc,
)
from .logging import get_logger, log_execution_time, set_log_level
import logging
if TYPE_CHECKING:
from .splines import AbstractSpline
logger = get_logger(__name__, logging.ERROR)
DERIVATIVE_PATTERN = re.compile(r"^d(x_rdata|xdot|\w+?)d(\w+?)(?:_explicit)?$")
[docs]
class DEModel:
"""
Defines a Differential Equation as set of ModelQuantities.
This class provides general purpose interfaces to compute arbitrary
symbolic derivatives that are necessary for model simulation or
sensitivity computation.
:ivar _differential_states:
list of differential state variables
:ivar _algebraic_states:
list of algebraic state variables
:ivar _observables:
list of observables
:ivar _event_observables:
list of event observables
:ivar _sigma_ys:
list of sigmas for observables
:ivar _sigma_zs:
list of sigmas for event observables
:ivar _parameters:
list of parameters
:ivar _log_likelihood_ys:
list of loglikelihoods for observables
:ivar _log_likelihood_zs:
list of loglikelihoods for event observables
:ivar _log_likelihood_rzs:
list of loglikelihoods for event observable regularizations
:ivar _expressions:
list of expressions instances
:ivar _conservation_laws:
list of conservation laws
:ivar _symboldim_funs:
define functions that compute model dimensions, these
are functions as the underlying symbolic expressions have not been
populated at compile time
:ivar _eqs:
carries symbolic formulas of the symbolic variables of the model
:ivar _sparseeqs:
carries linear list of all symbolic formulas for sparsified
variables
:ivar _vals:
carries numeric values of symbolic identifiers of the symbolic
variables of the model
:ivar _names:
carries the names of symbolic identifiers of the symbolic variables
of the model
:ivar _syms:
carries symbolic identifiers of the symbolic variables of the
model
:ivar _sparsesyms:
carries linear list of all symbolic identifiers for sparsified
variables
:ivar _colptrs:
carries column pointers for sparsified variables. See
SUNMatrixContent_Sparse definition in ``sunmatrix/sunmatrix_sparse.h``
:ivar _rowvals:
carries row values for sparsified variables. See
SUNMatrixContent_Sparse definition in ``sunmatrix/sunmatrix_sparse.h``
:ivar _equation_prototype:
defines the attribute from which an equation should be generated via
list comprehension (see :meth:`OEModel._generate_equation`)
:ivar _variable_prototype:
defines the attribute from which a variable should be generated via
list comprehension (see :meth:`DEModel._generate_symbol`)
:ivar _value_prototype:
defines the attribute from which a value should be generated via
list comprehension (see :meth:`DEModel._generate_value`)
:ivar _total_derivative_prototypes:
defines how a total derivative equation is computed for an equation,
key defines the name and values should be arguments for
:meth:`DEModel.totalDerivative`
:ivar _lock_total_derivative:
add chainvariables to this set when computing total derivative from
a partial derivative call to enforce a partial derivative in the
next recursion. prevents infinite recursion
:ivar _simplify:
If not None, this function will be used to simplify symbolic
derivative expressions. Receives sympy expressions as only argument.
To apply multiple simplifications, wrap them in a lambda expression.
:ivar _x0_fixedParameters_idx:
Index list of subset of states for which x0_fixedParameters was
computed
:ivar _w_recursion_depth:
recursion depth in w, quantified as nilpotency of dwdw
:ivar _has_quadratic_nllh:
whether all observables have a gaussian noise model, i.e. whether
res and FIM make sense.
:ivar _static_indices:
dict of lists list of indices of static variables for different
model entities.
:ivar _z2event:
list of event indices for each event observable
"""
[docs]
def __init__(
self,
verbose: bool | int | None = False,
simplify: Callable | None = _default_simplify,
cache_simplify: bool = False,
):
"""
Create a new DEModel instance.
:param verbose:
verbosity level for logging, True/False default to
``logging.DEBUG``/``logging.ERROR``
:param simplify:
see :meth:`DEModel._simplify`
:param cache_simplify:
Whether to cache calls to the simplify method. Can e.g. decrease
import times for models with events.
"""
self._differential_states: list[DifferentialState] = []
self._algebraic_states: list[AlgebraicState] = []
self._algebraic_equations: list[AlgebraicEquation] = []
self._observables: list[Observable] = []
self._event_observables: list[EventObservable] = []
self._sigma_ys: list[SigmaY] = []
self._sigma_zs: list[SigmaZ] = []
self._parameters: list[Parameter] = []
self._constants: list[Constant] = []
self._log_likelihood_ys: list[LogLikelihoodY] = []
self._log_likelihood_zs: list[LogLikelihoodZ] = []
self._log_likelihood_rzs: list[LogLikelihoodRZ] = []
self._expressions: list[Expression] = []
self._conservation_laws: list[ConservationLaw] = []
self._events: list[Event] = []
self._splines: list[AbstractSpline] = []
self._symboldim_funs: dict[str, Callable[[], int]] = {
"sx": self.num_states_solver,
"v": self.num_states_solver,
"vB": self.num_states_solver,
"xB": self.num_states_solver,
"sigmay": self.num_obs,
"sigmaz": self.num_eventobs,
}
self._eqs: dict[
str,
(sp.Matrix | sp.SparseMatrix | list[sp.Matrix | sp.SparseMatrix]),
] = dict()
self._sparseeqs: dict[str, sp.Matrix | list[sp.Matrix]] = dict()
self._vals: dict[str, list[sp.Expr]] = dict()
self._names: dict[str, list[str]] = dict()
self._syms: dict[str, sp.Matrix | list[sp.Matrix]] = dict()
self._sparsesyms: dict[str, list[str] | list[list[str]]] = dict()
self._colptrs: dict[str, list[int] | list[list[int]]] = dict()
self._rowvals: dict[str, list[int] | list[list[int]]] = dict()
self._equation_prototype: dict[str, Callable] = {
"total_cl": self.conservation_laws,
"x0": self.states,
"y": self.observables,
"Jy": self.log_likelihood_ys,
"Jz": self.log_likelihood_zs,
"Jrz": self.log_likelihood_rzs,
"w": self.expressions,
"root": self.events,
"sigmay": self.sigma_ys,
"sigmaz": self.sigma_zs,
}
self._variable_prototype: dict[str, Callable] = {
"tcl": self.conservation_laws,
"x_rdata": self.states,
"y": self.observables,
"z": self.event_observables,
"p": self.parameters,
"k": self.constants,
"w": self.expressions,
"sigmay": self.sigma_ys,
"sigmaz": self.sigma_zs,
"h": self.events,
}
self._value_prototype: dict[str, Callable] = {
"p": self.parameters,
"k": self.constants,
}
self._total_derivative_prototypes: dict[
str, dict[str, str | list[str]]
] = {
"sroot": {
"eq": "root",
"chainvars": ["x"],
"var": "p",
"dxdz_name": "sx",
},
}
self._lock_total_derivative: list[str] = list()
self._simplify: Callable = simplify
if cache_simplify and simplify is not None:
def cached_simplify(
expr: sp.Expr,
_simplified: dict[str, sp.Expr] = {}, # noqa B006
_simplify: Callable = simplify,
) -> sp.Expr:
"""Speed up expression simplification with caching.
NB: This can decrease model import times for models that have
many repeated expressions during C++ file generation.
For example, this can be useful for models with events.
However, for other models, this may increase model import
times.
:param expr:
The SymPy expression.
:param _simplified:
The cache.
:param _simplify:
The simplification method.
:return:
The simplified expression.
"""
expr_str = repr(expr)
if expr_str not in _simplified:
_simplified[expr_str] = _simplify(expr)
return _simplified[expr_str]
self._simplify = cached_simplify
self._x0_fixedParameters_idx: None | Sequence[int]
self._w_recursion_depth: int = 0
self._has_quadratic_nllh: bool = True
set_log_level(logger, verbose)
self._static_indices: dict[str, list[int]] = {}
[docs]
def differential_states(self) -> list[DifferentialState]:
"""Get all differential states."""
return self._differential_states
[docs]
def algebraic_states(self) -> list[AlgebraicState]:
"""Get all algebraic states."""
return self._algebraic_states
[docs]
def observables(self) -> list[Observable]:
"""Get all observables."""
return self._observables
[docs]
def parameters(self) -> list[Parameter]:
"""Get all parameters."""
return self._parameters
[docs]
def constants(self) -> list[Constant]:
"""Get all constants."""
return self._constants
[docs]
def expressions(self) -> list[Expression]:
"""Get all expressions."""
return self._expressions
[docs]
def events(self) -> list[Event]:
"""Get all events."""
return self._events
[docs]
def event_observables(self) -> list[EventObservable]:
"""Get all event observables."""
return self._event_observables
[docs]
def sigma_ys(self) -> list[SigmaY]:
"""Get all observable sigmas."""
return self._sigma_ys
[docs]
def sigma_zs(self) -> list[SigmaZ]:
"""Get all event observable sigmas."""
return self._sigma_zs
[docs]
def conservation_laws(self) -> list[ConservationLaw]:
"""Get all conservation laws."""
return self._conservation_laws
[docs]
def log_likelihood_ys(self) -> list[LogLikelihoodY]:
"""Get all observable log likelihoodss."""
return self._log_likelihood_ys
[docs]
def log_likelihood_zs(self) -> list[LogLikelihoodZ]:
"""Get all event observable log likelihoods."""
return self._log_likelihood_zs
[docs]
def log_likelihood_rzs(self) -> list[LogLikelihoodRZ]:
"""Get all event observable regularization log likelihoods."""
return self._log_likelihood_rzs
[docs]
def is_ode(self) -> bool:
"""Check if model is ODE model."""
return len(self._algebraic_equations) == 0
[docs]
def states(self) -> list[State]:
"""Get all states."""
return self._differential_states + self._algebraic_states
def _process_sbml_rate_of(self) -> None:
"""Substitute any SBML-rateOf constructs in the model equations"""
rate_of_func = sp.core.function.UndefinedFunction("rateOf")
species_sym_to_xdot = dict(
zip(self.sym("x"), self.sym("xdot"), strict=True)
)
species_sym_to_idx = {x: i for i, x in enumerate(self.sym("x"))}
def get_rate(symbol: sp.Symbol):
"""Get rate of change of the given symbol"""
if symbol.find(rate_of_func):
raise SBMLException("Nesting rateOf() is not allowed.")
# Replace all rateOf(some_species) by their respective xdot equation
with contextlib.suppress(KeyError):
return self._eqs["xdot"][species_sym_to_idx[symbol]]
# For anything other than a state, rateOf(.) is 0 or invalid
return 0
# replace rateOf-instances in xdot by xdot symbols
made_substitutions = False
for i_state in range(len(self.eq("xdot"))):
if rate_ofs := self._eqs["xdot"][i_state].find(rate_of_func):
self._eqs["xdot"][i_state] = self._eqs["xdot"][i_state].subs(
{
# either the rateOf argument is a state, or it's 0
rate_of: species_sym_to_xdot.get(rate_of.args[0], 0)
for rate_of in rate_ofs
}
)
made_substitutions = True
if made_substitutions:
# substitute in topological order
subs = toposort_symbols(
dict(zip(self.sym("xdot"), self.eq("xdot"), strict=True))
)
self._eqs["xdot"] = smart_subs_dict(self.eq("xdot"), subs)
# replace rateOf-instances in x0 by xdot equation
for i_state in range(len(self.eq("x0"))):
if rate_ofs := self._eqs["x0"][i_state].find(rate_of_func):
self._eqs["x0"][i_state] = self._eqs["x0"][i_state].subs(
{
rate_of: get_rate(rate_of.args[0])
for rate_of in rate_ofs
}
)
# replace rateOf-instances in w by xdot equation
# here we may need toposort, as xdot may depend on w
made_substitutions = False
for i_expr in range(len(self.eq("w"))):
if rate_ofs := self._eqs["w"][i_expr].find(rate_of_func):
self._eqs["w"][i_expr] = self._eqs["w"][i_expr].subs(
{
rate_of: get_rate(rate_of.args[0])
for rate_of in rate_ofs
}
)
made_substitutions = True
if made_substitutions:
# Sort expressions in self._expressions, w symbols, and w equations
# in topological order. Ideally, this would already happen before
# adding the expressions to the model, but at that point, we don't
# have access to xdot yet.
# NOTE: elsewhere, conservations law expressions are expected to
# occur before any other w expressions, so we must maintain their
# position
# toposort everything but conservation law expressions,
# then prepend conservation laws
w_sorted = toposort_symbols(
dict(
zip(
self.sym("w")[self.num_cons_law() :, :],
self.eq("w")[self.num_cons_law() :, :],
strict=True,
)
)
)
w_sorted = (
dict(
zip(
self.sym("w")[: self.num_cons_law(), :],
self.eq("w")[: self.num_cons_law(), :],
strict=True,
)
)
| w_sorted
)
old_syms = tuple(self._syms["w"])
topo_expr_syms = tuple(w_sorted.keys())
new_order = [old_syms.index(s) for s in topo_expr_syms]
self._expressions = [self._expressions[i] for i in new_order]
self._syms["w"] = sp.Matrix(topo_expr_syms)
self._eqs["w"] = sp.Matrix(list(w_sorted.values()))
for component in chain(
self.observables(),
self.events(),
self._algebraic_equations,
):
if rate_ofs := component.get_val().find(rate_of_func):
if isinstance(component, Event):
# TODO froot(...) can currently not depend on `w`, so this substitution fails for non-zero rates
# see, e.g., sbml test case 01293
raise SBMLException(
"AMICI does currently not support rateOf(.) inside event trigger functions."
)
if isinstance(component, AlgebraicEquation):
# TODO IDACalcIC fails with
# "The linesearch algorithm failed: step too small or too many backtracks."
# see, e.g., sbml test case 01482
raise SBMLException(
"AMICI does currently not support rateOf(.) inside AlgebraicRules."
)
component.set_val(
component.get_val().subs(
{
rate_of: get_rate(rate_of.args[0])
for rate_of in rate_ofs
}
)
)
for event in self.events():
if event._state_update is None:
continue
for i_state in range(len(event._state_update)):
if rate_ofs := event._state_update[i_state].find(rate_of_func):
raise SBMLException(
"AMICI does currently not support rateOf(.) inside event state updates."
)
# TODO here we need xdot sym, not eqs
# event._state_update[i_state] = event._state_update[i_state].subs(
# {rate_of: get_rate(rate_of.args[0]) for rate_of in rate_ofs}
# )
[docs]
def add_component(
self, component: ModelQuantity, insert_first: bool | None = False
) -> None:
"""
Adds a new ModelQuantity to the model.
:param component:
model quantity to be added
:param insert_first:
whether to add quantity first or last, relevant when components
may refer to other components of the same type.
"""
if type(component) not in {
Observable,
Expression,
Parameter,
Constant,
DifferentialState,
AlgebraicState,
AlgebraicEquation,
LogLikelihoodY,
LogLikelihoodZ,
LogLikelihoodRZ,
SigmaY,
SigmaZ,
ConservationLaw,
Event,
EventObservable,
}:
raise ValueError(f"Invalid component type {type(component)}")
component_list = getattr(
self,
"_"
+ "_".join(
s.lower()
for s in re.split(r"([A-Z][^A-Z]+)", type(component).__name__)
if s
)
+ "s",
)
if insert_first:
component_list.insert(0, component)
else:
component_list.append(component)
[docs]
def add_conservation_law(
self,
state: sp.Symbol,
total_abundance: sp.Symbol,
coefficients: dict[sp.Symbol, sp.Expr],
) -> None:
r"""
Adds a new conservation law to the model. A conservation law is defined
by the conserved quantity :math:`T = \sum_i(a_i * x_i)`, where
:math:`a_i` are coefficients and :math:`x_i` are different state
variables.
:param state:
symbolic identifier of the state that should be replaced by
the conservation law (:math:`x_j`)
:param total_abundance:
symbolic identifier of the total abundance (:math:`T/a_j`)
:param coefficients:
Dictionary of coefficients {x_i: a_i}
"""
try:
ix = next(
filter(
lambda is_s: is_s[1].get_id() == state,
enumerate(self._differential_states),
)
)[0]
except StopIteration:
raise ValueError(
f"Specified state {state} was not found in the "
f"model states."
)
state_id = self._differential_states[ix].get_id()
# \sum_{i≠j}(a_i * x_i)/a_j
target_expression = (
sp.Add(
*(
c_i * x_i
for x_i, c_i in coefficients.items()
if x_i != state
)
)
/ coefficients[state]
)
# x_j = T/a_j - \sum_{i≠j}(a_i * x_i)/a_j
state_expr = total_abundance - target_expression
# T/a_j = \sum_{i≠j}(a_i * x_i)/a_j + x_j
abundance_expr = target_expression + state_id
self.add_component(
Expression(state_id, str(state_id), state_expr), insert_first=True
)
cl = ConservationLaw(
total_abundance,
f"total_{state_id}",
abundance_expr,
coefficients,
state_id,
)
self.add_component(cl)
self._differential_states[ix].set_conservation_law(cl)
[docs]
def add_spline(self, spline: AbstractSpline, spline_expr: sp.Expr) -> None:
"""Add a spline to the model.
:param spline:
Spline instance to be added
:param spline_expr:
Sympy function representation of `spline` from
``spline.ode_model_symbol()``.
"""
self._splines.append(spline)
self.add_component(
Expression(
identifier=spline.sbml_id,
name=str(spline.sbml_id),
value=spline_expr,
)
)
[docs]
def num_states_rdata(self) -> int:
"""
Number of states.
:return:
number of state variable symbols
"""
return len(self.sym("x_rdata"))
[docs]
def num_states_solver(self) -> int:
"""
Number of states after applying conservation laws.
:return:
number of state variable symbols
"""
return len(self.sym("x"))
[docs]
def num_cons_law(self) -> int:
"""
Number of conservation laws.
:return:
number of conservation laws
"""
return self.num_states_rdata() - self.num_states_solver()
[docs]
def num_state_reinits(self) -> int:
"""
Number of solver states which would be reinitialized after
preequilibration
:return:
number of state variable symbols with reinitialization
"""
reinit_states = self.eq("x0_fixedParameters")
solver_states = self.eq("x_solver")
return sum(ix in solver_states for ix in reinit_states)
[docs]
def num_obs(self) -> int:
"""
Number of Observables.
:return:
number of observable symbols
"""
return len(self.sym("y"))
[docs]
def num_eventobs(self) -> int:
"""
Number of Event Observables.
:return:
number of event observable symbols
"""
return len(self.sym("z"))
[docs]
def num_const(self) -> int:
"""
Number of Constants.
:return:
number of constant symbols
"""
return len(self.sym("k"))
[docs]
def num_par(self) -> int:
"""
Number of Parameters.
:return:
number of parameter symbols
"""
return len(self.sym("p"))
[docs]
def num_expr(self) -> int:
"""
Number of Expressions.
:return:
number of expression symbols
"""
return len(self.sym("w"))
[docs]
def num_events(self) -> int:
"""
Total number of Events (those for which root-functions are added and those without).
:return:
number of events
"""
return len(self.sym("h"))
[docs]
def num_events_solver(self) -> int:
"""
Number of Events.
:return:
number of event symbols (length of the root vector in AMICI)
"""
return sum(
not event.triggers_at_fixed_timepoint() for event in self.events()
)
[docs]
def sym(self, name: str) -> sp.Matrix:
"""
Returns (and constructs if necessary) the identifiers for a symbolic
entity.
:param name:
name of the symbolic variable
:return:
matrix of symbolic identifiers
"""
if name not in self._syms:
self._generate_symbol(name)
return self._syms[name]
[docs]
def sparsesym(self, name: str, force_generate: bool = True) -> list[str]:
"""
Returns (and constructs if necessary) the sparsified identifiers for
a sparsified symbolic variable.
:param name:
name of the symbolic variable
:param force_generate:
whether the symbols should be generated if not available
:return:
linearized Matrix containing the symbolic identifiers
"""
if name not in sparse_functions:
raise ValueError(f"{name} is not marked as sparse")
if name not in self._sparsesyms and force_generate:
self._generate_sparse_symbol(name)
return self._sparsesyms.get(name, [])
[docs]
def eq(self, name: str) -> sp.Matrix:
"""
Returns (and constructs if necessary) the formulas for a symbolic
entity.
:param name:
name of the symbolic variable
:return:
matrix of symbolic formulas
"""
if name not in self._eqs:
dec = log_execution_time(f"computing {name}", logger)
dec(self._compute_equation)(name)
return self._eqs[name]
[docs]
def sparseeq(self, name) -> sp.Matrix:
"""
Returns (and constructs if necessary) the sparsified formulas for a
sparsified symbolic variable.
:param name:
name of the symbolic variable
:return:
linearized matrix containing the symbolic formulas
"""
if name not in sparse_functions:
raise ValueError(f"{name} is not marked as sparse")
if name not in self._sparseeqs:
self._generate_sparse_symbol(name)
return self._sparseeqs[name]
[docs]
def colptrs(self, name: str) -> list[sp.Number] | list[list[sp.Number]]:
"""
Returns (and constructs if necessary) the column pointers for
a sparsified symbolic variable.
:param name:
name of the symbolic variable
:return:
list containing the column pointers
"""
if name not in sparse_functions:
raise ValueError(f"{name} is not marked as sparse")
if name not in self._sparseeqs:
self._generate_sparse_symbol(name)
return self._colptrs[name]
[docs]
def rowvals(self, name: str) -> list[sp.Number] | list[list[sp.Number]]:
"""
Returns (and constructs if necessary) the row values for a
sparsified symbolic variable.
:param name:
name of the symbolic variable
:return:
list containing the row values
"""
if name not in sparse_functions:
raise ValueError(f"{name} is not marked as sparse")
if name not in self._sparseeqs:
self._generate_sparse_symbol(name)
return self._rowvals[name]
[docs]
def val(self, name: str) -> list[sp.Number]:
"""
Returns (and constructs if necessary) the numeric values of a
symbolic entity
:param name:
name of the symbolic variable
:return:
list containing the numeric values
"""
if name not in self._vals:
self._generate_value(name)
return self._vals[name]
[docs]
def name(self, name: str) -> list[str]:
"""
Returns (and constructs if necessary) the names of a symbolic
variable
:param name:
name of the symbolic variable
:return:
list of names
"""
if name not in self._names:
self._generate_name(name)
return self._names[name]
[docs]
def free_symbols(self) -> set[sp.Basic]:
"""
Returns list of free symbols that appear in RHS and initial
conditions.
"""
return set(
chain.from_iterable(
state.get_free_symbols() for state in self.states()
)
)
[docs]
def static_indices(self, name: str) -> list[int]:
"""
Returns the indices of static expressions in the given model entity.
Static expressions are those that do not depend on time,
neither directly nor indirectly.
:param name: Name of the model entity.
:return: List of indices of static expressions.
"""
# already computed?
if (res := self._static_indices.get(name)) is not None:
return res
if name == "w":
dwdx = self.sym("dwdx")
dwdw = self.sym("dwdw")
w = self.eq("w")
# Check for direct (via `t`) or indirect (via `x`, `h`, or splines)
# time dependency.
# To avoid lengthy symbolic computations, we only check if we have
# any non-zeros in hierarchy. We currently neglect the case where
# different hierarchy levels may cancel out. Treating a static
# expression as dynamic in such rare cases shouldn't be a problem.
dynamic_dependency = np.asarray(
dwdx.applyfunc(lambda x: int(not x.is_zero))
).astype(np.int64)
# to check for other time-dependence, we add a column to the dwdx
# matrix
dynamic_syms = [
# FIXME: see spline comment below
# *self.sym("spl"),
*self.sym("h"),
amici_time_symbol,
]
dynamic_dependency = np.hstack(
(
dynamic_dependency,
np.array(
[
expr.has(*dynamic_syms)
# FIXME: the current spline implementation is a giant pita
# currently, the splines occur in the form of sympy functions, e.g.:
# AmiciSpline(y0, time, y0_3, y0_1)
# AmiciSplineSensitivity(y0, time, y0_1, y0_3, y0_1)
# until it uses the proper self.sym("spl") / self.sym("sspl")
# symbols, which will require quite some refactoring,
# we just do dumb string checks :|
or (
bool(self._splines)
and "AmiciSpline" in str(expr)
)
for expr in w
]
)[:, np.newaxis],
)
)
nonzero_dwdw = np.asarray(
dwdw.applyfunc(lambda x: int(not x.is_zero))
).astype(np.int64)
# `w` is made up an expression hierarchy. Any given entry is only
# static if all its dependencies are static. Here, we unravel
# the hierarchical structure of `w`.
# If for an entry in `w`, the row sum of the intermediate products
# is 0 across all levels, the expression is static.
tmp = dynamic_dependency
res = np.sum(tmp, axis=1)
while np.any(tmp != 0):
tmp = nonzero_dwdw.dot(tmp)
res += np.sum(tmp, axis=1)
self._static_indices[name] = (
np.argwhere(res == 0).flatten().tolist()
)
return self._static_indices[name]
if name in ("dwdw", "dwdx", "dwdp"):
static_indices_w = set(self.static_indices("w"))
dynamic_syms = [
*(
sym
for i, sym in enumerate(self.sym("w"))
if i not in static_indices_w
),
amici_time_symbol,
*self.sym("x"),
*self.sym("h"),
# FIXME see spline comment above
# *(self.sym("spl") if name in ("dwdw", "dwdx") else ()),
# *(self.sym("sspl") if name == "dwdp" else ()),
]
dynamic_syms = sp.Matrix(dynamic_syms)
rowvals = self.rowvals(name)
sparseeq = self.sparseeq(name)
# collect the indices of static expressions of dwd* from the list
# of non-zeros entries of the sparse matrix
self._static_indices[name] = [
i
for i, (expr, row_idx) in enumerate(
zip(sparseeq, rowvals, strict=True)
)
# derivative of a static expression is static
if row_idx in static_indices_w
# constant expressions
or expr.is_Number
# check for dependencies on non-static entities
or (
# FIXME see spline comment above
# (check str before diff, as diff will fail on spline functions)
(
# splines: non-static
not self._splines or "AmiciSpline" not in str(expr)
)
and (
# If the expression contains dynamic symbols, it might
# still be static. However, checking the derivative
# is currently too expensive, and we rather accept
# false negatives.
not expr.has(*dynamic_syms)
# or all(
# expr.diff(dyn_sym).is_zero
# for dyn_sym in dynamic_syms
# )
)
)
]
return self._static_indices[name]
raise NotImplementedError(name)
[docs]
def dynamic_indices(self, name: str) -> list[int]:
"""
Return the indices of dynamic expressions in the given model entity.
:param name: Name of the model entity.
:return: List of indices of dynamic expressions.
"""
static_idxs = set(self.static_indices(name))
length = len(
self.sparsesym(name)
if name in sparse_functions
else self.sym(name)
)
return [i for i in range(length) if i not in static_idxs]
def _generate_symbol(self, name: str) -> None:
"""
Generates the symbolic identifiers for a symbolic variable
:param name:
name of the symbolic variable
"""
if name in self._variable_prototype:
components = self._variable_prototype[name]()
self._syms[name] = sp.Matrix(
[comp.get_id() for comp in components]
)
if name == "y":
self._syms["my"] = sp.Matrix(
[comp.get_measurement_symbol() for comp in components]
)
if name == "z":
self._syms["mz"] = sp.Matrix(
[comp.get_measurement_symbol() for comp in components]
)
self._syms["rz"] = sp.Matrix(
[comp.get_regularization_symbol() for comp in components]
)
return
elif name == "x":
self._syms[name] = sp.Matrix(
[
state.get_id()
for state in self.states()
if not state.has_conservation_law()
]
)
return
elif name == "xdot":
self._syms[name] = sp.Matrix(
[
f"d{x.get_id()}dt" if self.is_ode() else f"de_{ix}"
for ix, x in enumerate(self._differential_states)
if not x.has_conservation_law()
]
+ [f"ae_{ix}" for ix in range(len(self._algebraic_equations))]
)
return
elif name == "dx":
self._syms[name] = sp.Matrix(
[
f"d{state.get_id()}dt"
for state in self.states()
if not state.has_conservation_law()
]
)
return
elif name == "sx0":
self._syms[name] = sp.Matrix(
[
f"s{state.get_id()}_0"
for state in self.states()
if not state.has_conservation_law()
]
)
return
elif name == "sx_rdata":
self._syms[name] = sp.Matrix(
[f"sx_rdata_{i}" for i in range(len(self.states()))]
)
return
elif name == "dtcldp":
# check, whether the CL consists of only one state. Then,
# sensitivities drop out, otherwise generate symbols
self._syms[name] = sp.Matrix(
[
[
sp.Symbol(
f"s{strip_pysb(tcl.get_id())}__"
f"{strip_pysb(par.get_id())}",
real=True,
)
for par in self._parameters
]
if self.conservation_law_has_multispecies(tcl)
else [0] * self.num_par()
for tcl in self._conservation_laws
]
)
return
elif name == "xdot_old":
length = len(self.eq("xdot"))
elif name in sparse_functions:
self._generate_sparse_symbol(name)
return
elif name in self._symboldim_funs:
length = self._symboldim_funs[name]()
elif name == "stau":
length = self.eq(name)[0].shape[1]
elif name in sensi_functions:
length = self.eq(name).shape[0]
elif name == "spl":
# placeholders for the numeric spline values.
# Need to create symbols
self._syms[name] = sp.Matrix(
[[f"spl_{isp}" for isp in range(len(self._splines))]]
)
return
elif name == "sspl":
# placeholders for spline sensitivities. Need to create symbols
self._syms[name] = sp.Matrix(
[
[f"sspl_{isp}_{ip}" for ip in range(len(self._syms["p"]))]
for isp in range(len(self._splines))
]
)
return
else:
length = len(self.eq(name))
self._syms[name] = sp.Matrix(
[
sp.Symbol(f'{name}{0 if name == "stau" else i}', real=True)
for i in range(length)
]
)
[docs]
def generate_basic_variables(self) -> None:
"""
Generates the symbolic identifiers for all variables in
``DEModel._variable_prototype``
"""
# We need to process events and Heaviside functions in the ``DEModel`,
# before adding it to DEExporter
self.parse_events()
for var in self._variable_prototype:
if var not in self._syms:
self._generate_symbol(var)
# symbols for spline values need to be created in addition
for var in ["spl", "sspl"]:
self._generate_symbol(var)
self._generate_symbol("x")
[docs]
def parse_events(self) -> None:
"""
This function checks the right-hand side for roots of Heaviside
functions or events, collects the roots, removes redundant roots,
and replaces the formulae of the found roots by identifiers of AMICI's
Heaviside function implementation in the right-hand side
"""
# Track all roots functions in the right-hand side
roots = copy.deepcopy(self._events)
for state in self._differential_states:
state.set_dt(self._process_heavisides(state.get_dt(), roots))
for expr in self._expressions:
expr.set_val(self._process_heavisides(expr.get_val(), roots))
# remove all possible Heavisides from roots, which may arise from
# the substitution of `'w'` in `_collect_heaviside_roots`
for root in roots:
root.set_val(self._process_heavisides(root.get_val(), roots))
# Now add the found roots to the model components
for root in roots:
# skip roots of SBML events, as these have already been added
if root in self._events:
continue
# add roots of heaviside functions
self.add_component(root)
# re-order events - first those that require root tracking, then the others
self._events = list(
chain(
itertools.filterfalse(
Event.triggers_at_fixed_timepoint, self._events
),
filter(Event.triggers_at_fixed_timepoint, self._events),
)
)
[docs]
def get_appearance_counts(self, idxs: list[int]) -> list[int]:
"""
Counts how often a state appears in the time derivative of
another state and expressions for a subset of states
:param idxs:
list of state indices for which counts are to be computed
:return:
list of counts for the states ordered according to the provided
indices
"""
free_symbols_dt = list(
itertools.chain.from_iterable(
[str(symbol) for symbol in state.get_dt().free_symbols]
for state in self.states()
)
)
free_symbols_expr = list(
itertools.chain.from_iterable(
[str(symbol) for symbol in expr.get_val().free_symbols]
for expr in self._expressions
)
)
return [
free_symbols_dt.count(str(self._differential_states[idx].get_id()))
+ free_symbols_expr.count(
str(self._differential_states[idx].get_id())
)
for idx in idxs
]
def _generate_sparse_symbol(self, name: str) -> None:
"""
Generates the sparse symbolic identifiers, symbolic identifiers,
sparse equations, column pointers and row values for a symbolic
variable
:param name:
name of the symbolic variable
"""
matrix = self.eq(name)
if match_deriv := DERIVATIVE_PATTERN.match(name):
eq = match_deriv[1]
var = match_deriv[2]
rownames = self.sym(eq)
colnames = self.sym(var)
if name == "dJydy":
# One entry per y-slice
self._colptrs[name] = []
self._rowvals[name] = []
self._sparseeqs[name] = []
self._sparsesyms[name] = []
self._syms[name] = []
for iy in range(self.num_obs()):
(
symbol_col_ptrs,
symbol_row_vals,
sparse_list,
symbol_list,
sparse_matrix,
) = csc_matrix(
matrix[iy, :],
rownames=rownames,
colnames=colnames,
identifier=iy,
)
self._colptrs[name].append(symbol_col_ptrs)
self._rowvals[name].append(symbol_row_vals)
self._sparseeqs[name].append(sparse_list)
self._sparsesyms[name].append(symbol_list)
self._syms[name].append(sparse_matrix)
else:
(
symbol_col_ptrs,
symbol_row_vals,
sparse_list,
symbol_list,
sparse_matrix,
) = csc_matrix(
matrix,
rownames=rownames,
colnames=colnames,
pattern_only=name in nobody_functions,
)
self._colptrs[name] = symbol_col_ptrs
self._rowvals[name] = symbol_row_vals
self._sparseeqs[name] = sparse_list
self._sparsesyms[name] = symbol_list
self._syms[name] = sparse_matrix
def _compute_equation(self, name: str) -> None:
"""
Computes the symbolic formula for a symbolic variable
:param name:
name of the symbolic variable
"""
# replacement ensures that we don't have to adapt name in abstract
# model and keep backwards compatibility with matlab
match_deriv = DERIVATIVE_PATTERN.match(
re.sub(r"dJ(y|z|rz)dsigma", r"dJ\1dsigma\1", name)
.replace("sigmarz", "sigmaz")
.replace("dJrzdz", "dJrzdrz")
)
time_symbol = sp.Matrix([amici_time_symbol])
if name in self._equation_prototype:
self._equation_from_components(
name, self._equation_prototype[name]()
)
elif name in self._total_derivative_prototypes:
args = self._total_derivative_prototypes[name]
args["name"] = name
self._lock_total_derivative += args["chainvars"]
self._total_derivative(**args)
for cv in args["chainvars"]:
self._lock_total_derivative.remove(cv)
elif name == "xdot":
if self.is_ode():
self._eqs[name] = sp.Matrix(
[
state.get_dt()
for state in self._differential_states
if not state.has_conservation_law()
]
)
else:
self._eqs[name] = sp.Matrix(
[
x.get_dt() - dx
for x, dx in zip(
(
s
for s in self._differential_states
if not s.has_conservation_law()
),
self.sym("dx"),
# dx contains extra elements for algebraic states
strict=False,
)
]
+ [eq.get_val() for eq in self._algebraic_equations]
)
elif name == "x_rdata":
self._eqs[name] = sp.Matrix(
[state.get_x_rdata() for state in self.states()]
)
elif name == "x_solver":
self._eqs[name] = sp.Matrix(
[
state.get_id()
for state in self.states()
if not state.has_conservation_law()
]
)
elif name == "sx_solver":
self._eqs[name] = sp.Matrix(
[
self.sym("sx_rdata")[ix]
for ix, state in enumerate(self.states())
if not state.has_conservation_law()
]
)
elif name == "sx0":
self._derivative(name[1:], "p", name=name)
elif name == "sx0_fixedParameters":
# deltax = -x+x0_fixedParameters if x0_fixedParameters>0 else 0
# deltasx = -sx+dx0_fixed_parametersdx*sx+dx0_fixedParametersdp
# if x0_fixedParameters>0 else 0
# sx0_fixedParameters = sx+deltasx =
# dx0_fixed_parametersdx*sx+dx0_fixedParametersdp
self._eqs[name] = smart_jacobian(
self.eq("x0_fixedParameters"), self.sym("p")
)
dx0_fixed_parametersdx = smart_jacobian(
self.eq("x0_fixedParameters"), self.sym("x")
)
if not smart_is_zero_matrix(dx0_fixed_parametersdx):
if isinstance(self._eqs[name], ImmutableDenseMatrix):
self._eqs[name] = MutableDenseMatrix(self._eqs[name])
tmp = smart_multiply(dx0_fixed_parametersdx, self.sym("sx0"))
for ip in range(self._eqs[name].shape[1]):
self._eqs[name][:, ip] += tmp
elif name == "x0_fixedParameters":
k = self.sym("k")
self._x0_fixedParameters_idx = [
ix
for ix, eq in enumerate(self.eq("x0"))
if any(sym in eq.free_symbols for sym in k)
]
eq = self.eq("x0")
self._eqs[name] = sp.Matrix(
[eq[ix] for ix in self._x0_fixedParameters_idx]
)
elif name == "dtotal_cldx_rdata":
x_rdata = self.sym("x_rdata")
self._eqs[name] = sp.Matrix(
[
[cl.get_ncoeff(xr) for xr in x_rdata]
for cl in self._conservation_laws
]
)
elif name == "dtcldx":
# this is always zero
self._eqs[name] = sp.zeros(
self.num_cons_law(), self.num_states_solver()
)
elif name == "dtcldp":
# force symbols
self._eqs[name] = self.sym(name)
elif name == "dx_rdatadx_solver":
if self.num_cons_law():
x_solver = self.sym("x")
self._eqs[name] = sp.Matrix(
[
[state.get_dx_rdata_dx_solver(xs) for xs in x_solver]
for state in self.states()
]
)
else:
# so far, dx_rdatadx_solver is only required for sx_rdata
# in case of no conservation laws, C++ code will directly use
# sx, we don't need this
self._eqs[name] = sp.zeros(
self.num_states_rdata(), self.num_states_solver()
)
elif name == "dx_rdatadp":
if self.num_cons_law():
self._eqs[name] = smart_jacobian(
self.eq("x_rdata"), self.sym("p")
)
else:
# so far, dx_rdatadp is only required for sx_rdata
# in case of no conservation laws, C++ code will directly use
# sx, we don't need this
self._eqs[name] = sp.zeros(
self.num_states_rdata(), self.num_par()
)
elif name == "dx_rdatadtcl":
self._eqs[name] = smart_jacobian(
self.eq("x_rdata"), self.sym("tcl")
)
elif name == "dxdotdx_explicit":
# force symbols
self._derivative("xdot", "x", name=name)
elif name == "dxdotdp_explicit":
# force symbols
self._derivative("xdot", "p", name=name)
elif name == "spl":
self._eqs[name] = self.sym(name)
elif name == "sspl":
# force symbols
self._eqs[name] = self.sym(name)
elif name == "spline_values":
# force symbols
self._eqs[name] = sp.Matrix(
[y for spline in self._splines for y in spline.values_at_nodes]
)
elif name == "spline_slopes":
# force symbols
self._eqs[name] = sp.Matrix(
[
d
for spline in self._splines
for d in (
sp.zeros(len(spline.derivatives_at_nodes), 1)
if spline.derivatives_by_fd
else spline.derivatives_at_nodes
)
]
)
elif name == "drootdt":
self._eqs[name] = smart_jacobian(self.eq("root"), time_symbol)
elif name == "drootdt_total":
# backsubstitution of optimized right-hand side terms into RHS
# calling subs() is costly. Due to looping over events though, the
# following lines are only evaluated if a model has events
w_sorted = toposort_symbols(
dict(zip(self.sym("w"), self.eq("w"), strict=True))
)
tmp_xdot = smart_subs_dict(self.eq("xdot"), w_sorted)
self._eqs[name] = self.eq("drootdt")
if self.num_states_solver():
self._eqs[name] += smart_multiply(self.eq("drootdx"), tmp_xdot)
elif name == "deltax":
# fill boluses for Heaviside functions, as empty state updates
# would cause problems when writing the function file later
event_eqs = []
for event in self._events:
if event._state_update is None:
event_eqs.append(sp.zeros(self.num_states_solver(), 1))
else:
event_eqs.append(event._state_update)
self._eqs[name] = event_eqs
elif name == "z":
event_observables = [
sp.zeros(self.num_eventobs(), 1) for _ in self._events
]
event_ids = [e.get_id() for e in self._events]
# TODO: get rid of this stupid 1-based indexing as soon as we can
# the matlab interface
z2event = [
event_ids.index(event_obs.get_event()) + 1
for event_obs in self._event_observables
]
for (iz, ie), event_obs in zip(
enumerate(z2event), self._event_observables, strict=True
):
event_observables[ie - 1][iz] = event_obs.get_val()
self._eqs[name] = event_observables
self._z2event = z2event
elif name in ["ddeltaxdx", "ddeltaxdp", "ddeltaxdt", "dzdp", "dzdx"]:
if match_deriv[2] == "t":
var = time_symbol
else:
var = self.sym(match_deriv[2])
self._eqs[name] = [
smart_jacobian(self.eq(match_deriv[1])[ie], var)
for ie in range(self.num_events())
]
if name == "dzdx":
for ie in range(self.num_events()):
dtaudx = (
-self.eq("drootdx")[ie, :]
/ self.eq("drootdt_total")[ie]
)
for iz in range(self.num_eventobs()):
if ie != self._z2event[iz] - 1:
continue
dzdt = sp.diff(self.eq("z")[ie][iz], time_symbol)
self._eqs[name][ie][iz, :] += dzdt * dtaudx
elif name in ["rz", "drzdx", "drzdp"]:
eq_events = []
for ie in range(self.num_events()):
val = sp.zeros(
self.num_eventobs(),
1 if name == "rz" else len(self.sym(match_deriv[2])),
)
# match event observables to root function
for iz in range(self.num_eventobs()):
if ie == self._z2event[iz] - 1:
val[iz, :] = self.eq(name.replace("rz", "root"))[ie, :]
eq_events.append(val)
self._eqs[name] = eq_events
elif name == "stau":
self._eqs[name] = [
-self.eq("sroot")[ie, :] / self.eq("drootdt_total")[ie]
if not self.eq("drootdt_total")[ie].is_zero
else sp.zeros(*self.eq("sroot")[ie, :].shape)
for ie in range(self.num_events())
]
elif name == "deltasx":
if self.num_states_solver() * self.num_par() == 0:
self._eqs[name] = []
return
event_eqs = []
for ie, event in enumerate(self._events):
tmp_eq = sp.zeros(self.num_states_solver(), self.num_par())
# need to check if equations are zero since we are using
# symbols
if not smart_is_zero_matrix(
self.eq("stau")[ie]
) and not smart_is_zero_matrix(self.eq("xdot")):
tmp_eq += smart_multiply(
self.sym("xdot_old") - self.sym("xdot"),
self.sym("stau").T,
)
# only add deltax part if there is state update
if event._state_update is not None:
# partial derivative for the parameters
tmp_eq += self.eq("ddeltaxdp")[ie]
# initial part of chain rule state variables
tmp_dxdp = self.sym("sx") * sp.ones(1, self.num_par())
# need to check if equations are zero since we are using
# symbols
if not smart_is_zero_matrix(self.eq("stau")[ie]):
# chain rule for the time point
tmp_eq += smart_multiply(
self.eq("ddeltaxdt")[ie], self.sym("stau").T
)
# additional part of chain rule state variables
tmp_dxdp += smart_multiply(
self.sym("xdot_old"), self.sym("stau").T
)
# finish chain rule for the state variables
tmp_eq += smart_multiply(
self.eq("ddeltaxdx")[ie], tmp_dxdp
)
event_eqs.append(tmp_eq)
self._eqs[name] = event_eqs
elif name == "xdot_old":
# force symbols
self._eqs[name] = self.sym(name)
elif name == "dwdx":
if (
expected := list(
map(
ConservationLaw.get_x_rdata,
reversed(self.conservation_laws()),
)
)
) != (actual := self.eq("w")[: self.num_cons_law()]):
raise AssertionError(
"Conservation laws are not at the beginning of 'w'. "
f"Got {actual}, expected {expected}."
)
x = self.sym("x")
self._eqs[name] = sp.Matrix(
[
[-cl.get_ncoeff(xs) for xs in x]
# the insert first in ode_model._add_conservation_law() means
# that we need to reverse the order here
for cl in reversed(self._conservation_laws)
]
).col_join(
smart_jacobian(self.eq("w")[self.num_cons_law() :, :], x)
)
elif match_deriv:
self._derivative(match_deriv[1], match_deriv[2], name)
else:
raise ValueError(f"Unknown equation {name}")
if name in ("sigmay", "sigmaz"):
# check for states in sigma{y,z}, which is currently not supported
syms_x = self.sym("x")
syms_yz = self.sym(name.removeprefix("sigma"))
xs_in_sigma = {}
for sym_yz, eq_yz in zip(syms_yz, self._eqs[name], strict=True):
yz_free_syms = eq_yz.free_symbols
if tmp := {x for x in syms_x if x in yz_free_syms}:
xs_in_sigma[sym_yz] = tmp
if xs_in_sigma:
msg = ", ".join(
[f"{yz} depends on {xs}" for yz, xs in xs_in_sigma.items()]
)
raise NotImplementedError(
f"State-dependent observables are not supported, but {msg}."
)
if name == "root":
# Events are processed after the model has been set up.
# Equations are there, but symbols for roots must be added
self.sym("h")
if name in {"Jy", "dydx"}:
# do not transpose if we compute the partial derivative as part of
# a total derivative
if not len(self._lock_total_derivative):
self._eqs[name] = self._eqs[name].transpose()
if name in {"dzdx", "drzdx"}:
self._eqs[name] = [e.T for e in self._eqs[name]]
if self._simplify:
dec = log_execution_time(f"simplifying {name}", logger)
if isinstance(self._eqs[name], list):
self._eqs[name] = [
dec(_parallel_applyfunc)(sub_eq, self._simplify)
for sub_eq in self._eqs[name]
]
else:
self._eqs[name] = dec(_parallel_applyfunc)(
self._eqs[name], self._simplify
)
[docs]
def sym_names(self) -> list[str]:
"""
Returns a list of names of generated symbolic variables
:return:
list of names
"""
return list(self._syms.keys())
def _derivative(self, eq: str, var: str, name: str = None) -> None:
"""
Creates a new symbolic variable according to a derivative
:param eq:
name of the symbolic variable that defines the formula
:param var:
name of the symbolic variable that defines the identifiers
with respect to which the derivatives are to be computed
:param name:
name of resulting symbolic variable, default is ``d{eq}d{var}``
"""
if not name:
name = f"d{eq}d{var}"
ignore_chainrule = {
("xdot", "p"): "w", # has generic implementation in c++ code
("xdot", "x"): "w", # has generic implementation in c++ code
("w", "w"): "tcl", # dtcldw = 0
("w", "x"): "tcl", # dtcldx = 0
}
# automatically detect chainrule
chainvars = [
cv
for cv in ["w", "tcl"]
if var_in_function_signature(eq, cv, self.is_ode())
and cv not in self._lock_total_derivative
and var != cv
and min(self.sym(cv).shape)
and (
(eq, var) not in ignore_chainrule
or ignore_chainrule[(eq, var)] != cv
)
]
if len(chainvars):
self._lock_total_derivative += chainvars
self._total_derivative(name, eq, chainvars, var)
for cv in chainvars:
self._lock_total_derivative.remove(cv)
return
# partial derivative
sym_eq = self.eq(eq).transpose() if eq == "Jy" else self.eq(eq)
sym_var = self.sym(var)
derivative = smart_jacobian(sym_eq, sym_var)
self._eqs[name] = derivative
# compute recursion depth based on nilpotency of jacobian. computing
# nilpotency can be done more efficiently on numerical sparsity pattern
if name == "dwdw":
nonzeros = np.asarray(
derivative.applyfunc(lambda x: int(not x.is_zero))
).astype(np.int64)
recursion = nonzeros.copy()
if max(recursion.shape):
while recursion.max():
recursion = recursion.dot(nonzeros)
self._w_recursion_depth += 1
if self._w_recursion_depth > len(sym_eq):
raise RuntimeError(
"dwdw is not nilpotent. Something, somewhere went "
"terribly wrong. Please file a bug report at "
"https://github.com/AMICI-dev/AMICI/issues and "
"attach this model."
)
if name == "dydw" and not smart_is_zero_matrix(derivative):
dwdw = self.eq("dwdw")
# h(k) = d{eq}dw*dwdw^k* (k=1)
h = smart_multiply(derivative, dwdw)
while not smart_is_zero_matrix(h):
self._eqs[name] += h
# h(k+1) = d{eq}dw*dwdw^(k+1) = h(k)*dwdw
h = smart_multiply(h, dwdw)
def _total_derivative(
self,
name: str,
eq: str,
chainvars: list[str],
var: str,
dydx_name: str = None,
dxdz_name: str = None,
) -> None:
"""
Creates a new symbolic variable according to a total derivative
using the chain rule
:param name:
name of resulting symbolic variable
:param eq:
name of the symbolic variable that defines the formula
:param chainvars:
names of the symbolic variable that define the
identifiers with respect to which the chain rules are applied
:param var:
name of the symbolic variable that defines the identifiers
with respect to which the derivatives are to be computed
:param dydx_name:
defines the name of the symbolic variable that
defines the derivative of the ``eq`` with respect to ``chainvar``,
default is ``d{eq}d{chainvar}``
:param dxdz_name:
defines the name of the symbolic variable that
defines the derivative of the ``chainvar`` with respect to ``var``,
default is d{chainvar}d{var}
"""
# compute total derivative according to chainrule
# Dydz = dydx*dxdz + dydz
# initialize with partial derivative dydz without chain rule
self._eqs[name] = self.sym_or_eq(name, f"d{eq}d{var}")
if not isinstance(self._eqs[name], sp.Symbol):
# if not a Symbol, create a copy using sympy API
# NB deepcopy does not work safely, see sympy issue #7672
self._eqs[name] = self._eqs[name].copy()
for chainvar in chainvars:
if dydx_name is None:
dydx_name = f"d{eq}d{chainvar}"
if dxdz_name is None:
dxdz_name = f"d{chainvar}d{var}"
dydx = self.sym_or_eq(name, dydx_name)
dxdz = self.sym_or_eq(name, dxdz_name)
# Save time for large models if one multiplicand is zero,
# which is not checked for by sympy
if not smart_is_zero_matrix(dydx) and not smart_is_zero_matrix(
dxdz
):
dydx_times_dxdz = smart_multiply(dydx, dxdz)
if (
dxdz.shape[1] == 1
and self._eqs[name].shape[1] != dxdz.shape[1]
):
for iz in range(self._eqs[name].shape[1]):
self._eqs[name][:, iz] += dydx_times_dxdz
else:
self._eqs[name] += dydx_times_dxdz
[docs]
def sym_or_eq(self, name: str, varname: str) -> sp.Matrix:
"""
Returns symbols or equations depending on whether a given
variable appears in the function signature or not.
:param name:
name of function for which the signature should be checked
:param varname:
name of the variable which should be contained in the
function signature
:return:
the variable symbols if the variable is part of the signature and
the variable equations otherwise.
"""
# dwdx and dwdp will be dynamically computed and their ordering
# within a column may differ from the initialization of symbols here,
# so those are not safe to use. Not removing them from signature as
# this would break backwards compatibility.
if var_in_function_signature(
name, varname, self.is_ode()
) and varname not in [
"dwdx",
"dwdp",
]:
return self.sym(varname)
else:
return self.eq(varname)
def _multiplication(
self,
name: str,
x: str,
y: str,
transpose_x: bool | None = False,
sign: int | None = 1,
):
"""
Creates a new symbolic variable according to a multiplication
:param name:
name of resulting symbolic variable, default is ``d{eq}d{var}``
:param x:
name of the symbolic variable that defines the first factor
:param y:
name of the symbolic variable that defines the second factor
:param transpose_x:
indicates whether the first factor should be
transposed before multiplication
:param sign:
defines the sign of the product, should be +1 or -1
"""
if sign not in [-1, 1]:
raise TypeError(f"sign must be +1 or -1, was {sign}")
variables = {
varname: self.sym(varname)
if var_in_function_signature(name, varname, self.is_ode())
else self.eq(varname)
for varname in [x, y]
}
xx = variables[x].transpose() if transpose_x else variables[x]
yy = variables[y]
self._eqs[name] = sign * smart_multiply(xx, yy)
def _equation_from_components(
self, name: str, components: list[ModelQuantity]
) -> None:
"""
Generates the formulas of a symbolic variable from the attributes
:param name:
name of resulting symbolic variable
:param component:
name of the attribute
"""
self._eqs[name] = sp.Matrix([comp.get_val() for comp in components])
[docs]
def get_conservation_laws(self) -> list[tuple[sp.Symbol, sp.Expr]]:
"""Returns a list of states with conservation law set
:return:
list of state identifiers
"""
return [
(state.get_id(), state.get_x_rdata())
for state in self.states()
if state.has_conservation_law()
]
def _generate_value(self, name: str) -> None:
"""
Generates the numeric values of a symbolic variable from value
prototypes
:param name:
name of resulting symbolic variable
"""
if name in self._value_prototype:
components = self._value_prototype[name]()
else:
raise ValueError(f"No values for {name}")
self._vals[name] = [comp.get_val() for comp in components]
def _generate_name(self, name: str) -> None:
"""
Generates the names of a symbolic variable from variable prototypes or
equation prototypes
:param name:
name of resulting symbolic variable
"""
if name in self._variable_prototype:
components = self._variable_prototype[name]()
elif name in self._equation_prototype:
components = self._equation_prototype[name]()
else:
raise ValueError(f"No names for {name}")
self._names[name] = [comp.get_name() for comp in components]
[docs]
def state_has_fixed_parameter_initial_condition(self, ix: int) -> bool:
"""
Checks whether the state at specified index has a fixed parameter
initial condition
:param ix:
state index
:return:
boolean indicating if any of the initial condition free
variables is contained in the model constants
"""
ic = self.states()[ix].get_val()
if not isinstance(ic, sp.Basic):
return False
return any(
fp in (c.get_id() for c in self._constants)
for fp in ic.free_symbols
)
[docs]
def state_has_conservation_law(self, ix: int) -> bool:
"""
Checks whether the state at specified index has a conservation
law set
:param ix:
state index
:return:
boolean indicating if conservation_law is not None
"""
return self.states()[ix].has_conservation_law()
[docs]
def get_solver_indices(self) -> dict[int, int]:
"""
Returns a mapping that maps rdata species indices to solver indices
:return:
dictionary mapping rdata species indices to solver indices
"""
solver_index = {}
ix_solver = 0
for ix in range(len(self.states())):
if self.state_has_conservation_law(ix):
continue
solver_index[ix] = ix_solver
ix_solver += 1
return solver_index
[docs]
def state_is_constant(self, ix: int) -> bool:
"""
Checks whether the temporal derivative of the state is zero
:param ix:
state index
:return:
boolean indicating if constant over time
"""
state = self.states()[ix]
if isinstance(state, AlgebraicState):
return False
return state.get_dt().is_zero
[docs]
def conservation_law_has_multispecies(self, tcl: ConservationLaw) -> bool:
"""
Checks whether a conservation law has multiple species or it just
defines one constant species
:param tcl:
conservation law
:return:
boolean indicating if conservation_law is not None
"""
state_set = set(self.sym("x_rdata"))
n_species = len(state_set.intersection(tcl.get_val().free_symbols))
return n_species > 1
def _expr_is_time_dependent(self, expr: sp.Expr) -> bool:
"""Determine whether an expression is time-dependent.
:param expr:
The expression.
:returns:
Whether the expression is time-dependent.
"""
# `expr.free_symbols` will be different to `self._states.keys()`, so
# it's easier to compare as `str`.
expr_syms = {str(sym) for sym in expr.free_symbols}
# Check if the time variable is in the expression.
if "t" in expr_syms:
return True
# Check if any time-dependent states are in the expression.
state_syms = [str(sym) for sym in self.states()]
return any(
not self.state_is_constant(state_syms.index(state))
for state in expr_syms.intersection(state_syms)
)
def _get_unique_root(
self,
root_found: sp.Expr,
roots: list[Event],
) -> sp.Symbol | None:
"""
Collects roots of Heaviside functions and events and stores them in
the roots list. It checks for redundancy to not store symbolically
equivalent root functions more than once.
:param root_found:
equation of the root function
:param roots:
list of already known root functions with identifier
:returns:
unique identifier for root, or ``None`` if the root is not
time-dependent
"""
if not self._expr_is_time_dependent(root_found):
return None
for root in roots:
if sp.simplify(root_found - root.get_val()).is_zero:
return root.get_id()
# create an event for a new root function
root_symstr = f"Heaviside_{len(roots)}"
roots.append(
Event(
identifier=sp.Symbol(root_symstr),
name=root_symstr,
value=root_found,
state_update=None,
)
)
return roots[-1].get_id()
def _collect_heaviside_roots(
self,
args: Sequence[sp.Expr],
) -> list[sp.Expr]:
"""
Recursively checks an expression for the occurrence of Heaviside
functions and return all roots found
:param args:
args attribute of the expanded expression
:returns:
root functions that were extracted from Heaviside function
arguments
"""
root_funs = []
for arg in args:
if arg.func == sp.Heaviside:
root_funs.append(arg.args[0])
elif arg.has(sp.Heaviside):
root_funs.extend(self._collect_heaviside_roots(arg.args))
if not root_funs:
return []
# substitute 'w' expressions into root expressions now, to avoid
# rewriting 'root.cpp' and 'stau.cpp' headers
# to include 'w.h'
w_sorted = toposort_symbols(
dict(
zip(
[expr.get_id() for expr in self._expressions],
[expr.get_val() for expr in self._expressions],
strict=True,
)
)
)
root_funs = [r.subs(w_sorted) for r in root_funs]
return root_funs
def _process_heavisides(
self,
dxdt: sp.Expr,
roots: list[Event],
) -> sp.Expr:
"""
Parses the RHS of a state variable, checks for Heaviside functions,
collects unique roots functions that can be tracked by SUNDIALS and
replaces Heaviside Functions by amici helper variables that will be
updated based on SUNDIALS root tracking.
:param dxdt:
right-hand side of state variable
:param roots:
list of known root functions with identifier
:returns:
dxdt with Heaviside functions replaced by amici helper variables
"""
# track all the old Heaviside expressions in tmp_roots_old
# replace them later by the new expressions
heavisides = []
# run through the expression tree and get the roots
tmp_roots_old = self._collect_heaviside_roots(dxdt.args)
for tmp_old in unique_preserve_order(tmp_roots_old):
# we want unique identifiers for the roots
tmp_new = self._get_unique_root(tmp_old, roots)
# `tmp_new` is None if the root is not time-dependent.
if tmp_new is None:
continue
# For Heavisides, we need to add the negative function as well
self._get_unique_root(sp.sympify(-tmp_old), roots)
heavisides.append((sp.Heaviside(tmp_old), tmp_new))
if heavisides:
# only apply subs if necessary
for heaviside_sympy, heaviside_amici in heavisides:
dxdt = dxdt.subs(heaviside_sympy, heaviside_amici)
return dxdt