Source code for brian2.stateupdaters.exact

Exact integration for linear equations.

import itertools

import sympy as sp
from sympy import I, Symbol, Wild, im, re

from brian2.equations.codestrings import is_constant_over_dt
from brian2.parsing.sympytools import str_to_sympy, sympy_to_str
from brian2.stateupdaters.base import (
from brian2.utils.caching import cached
from brian2.utils.logger import get_logger

__all__ = ["linear", "exact", "independent"]

logger = get_logger(__name__)

[docs] def get_linear_system(eqs, variables): """ Convert equations into a linear system using sympy. Parameters ---------- eqs : `Equations` The model equations. Returns ------- (diff_eq_names, coefficients, constants) : (list of str, `sympy.Matrix`, `sympy.Matrix`) A tuple containing the variable names (`diff_eq_names`) corresponding to the rows of the matrix `coefficients` and the vector `constants`, representing the system of equations in the form M * X + B Raises ------ ValueError If the equations cannot be converted into an M * X + B form. """ diff_eqs = eqs.get_substituted_expressions(variables) diff_eq_names = [name for name, _ in diff_eqs] symbols = [Symbol(name, real=True) for name in diff_eq_names] coefficients = sp.zeros(len(diff_eq_names)) constants = sp.zeros(len(diff_eq_names), 1) for row_idx, (name, expr) in enumerate(diff_eqs): s_expr = str_to_sympy(expr.code, variables).expand() current_s_expr = s_expr for col_idx, symbol in enumerate(symbols): current_s_expr = current_s_expr.collect(symbol) constant_wildcard = Wild("c", exclude=[symbol]) factor_wildcard = Wild(f"c_{name}", exclude=symbols) one_pattern = factor_wildcard * symbol + constant_wildcard matches = current_s_expr.match(one_pattern) if matches is None: raise UnsupportedEquationsException( f"The expression '{expr}', " "defining the variable " f"'{name}', could not be " "separated into linear " "components." ) coefficients[row_idx, col_idx] = matches[factor_wildcard] current_s_expr = matches[constant_wildcard] # The remaining constant should be a true constant constants[row_idx] = current_s_expr return (diff_eq_names, coefficients, constants)
[docs] class IndependentStateUpdater(StateUpdateMethod): """ A state update for equations that do not depend on other state variables, i.e. 1-dimensional differential equations. The individual equations are solved by sympy. .. deprecated:: 2.1 This method might be removed from future versions of Brian. """
[docs] def __call__(self, equations, variables=None, method_options=None): logger.warn( "The 'independent' state updater is deprecated and might be " "removed in future versions of Brian.", "deprecated_independent", once=True, ) extract_method_options(method_options, {}) if equations.is_stochastic: raise UnsupportedEquationsException( "Cannot solve stochastic equations with this state updater" ) if variables is None: variables = {} diff_eqs = equations.get_substituted_expressions(variables) t = Symbol("t", real=True, positive=True) dt = Symbol("dt", real=True, positive=True) t0 = Symbol("t0", real=True, positive=True) code = [] for name, expression in diff_eqs: rhs = str_to_sympy(expression.code, variables) # We have to be careful and use the real=True assumption as well, # otherwise sympy doesn't consider the symbol a match to the content # of the equation var = Symbol(name, real=True) f = sp.Function(name) rhs = rhs.subs(var, f(t)) derivative = sp.Derivative(f(t), t) diff_eq = sp.Eq(derivative, rhs) # TODO: simplify=True sometimes fails with 0.7.4, see: # try: general_solution = sp.dsolve(diff_eq, f(t), simplify=True) except RuntimeError: general_solution = sp.dsolve(diff_eq, f(t), simplify=False) # Check whether this is an explicit solution if not getattr(general_solution, "lhs", None) == f(t): raise UnsupportedEquationsException( f"Cannot explicitly solve: {str(diff_eq)}" ) # Solve for C1 (assuming "var" as the initial value and "t0" as time) if general_solution.has(Symbol("C1")): if general_solution.has(Symbol("C2")): raise UnsupportedEquationsException( f"Too many constants in solution: {str(general_solution)}" ) constant_solution = sp.solve(general_solution, Symbol("C1")) if len(constant_solution) != 1: raise UnsupportedEquationsException( "Couldn't solve for the constant C1 in : %s " % str(general_solution) ) constant = constant_solution[0].subs(t, t0).subs(f(t0), var) solution = general_solution.rhs.subs("C1", constant) else: solution = general_solution.rhs.subs(t, t0).subs(f(t0), var) # Evaluate the expression for one timestep solution = solution.subs(t, t + dt).subs(t0, t) # only try symplifying it -- it sometimes raises an error try: solution = solution.simplify() except ValueError: pass code.append(f"{name} = {sympy_to_str(solution)}") return "\n".join(code)
[docs] class LinearStateUpdater(StateUpdateMethod): """ A state updater for linear equations. Derives a state updater step from the analytical solution given by sympy. Uses the matrix exponential (which is only implemented for diagonalizable matrices in sympy). """
[docs] @cached def __call__(self, equations, variables=None, method_options=None): method_options = extract_method_options(method_options, {"simplify": True}) if equations.is_stochastic: raise UnsupportedEquationsException( "Cannot solve stochastic equations with this state updater." ) if variables is None: variables = {} # Get a representation of the ODE system in the form of # dX/dt = M*X + B varnames, matrix, constants = get_linear_system(equations, variables) # No differential equations, nothing to do (this occurs sometimes in the # test suite where the whole model is nothing more than something like # 'v : 1') if matrix.shape == (0, 0): return "" # Make sure that the matrix M is constant, i.e. it only contains # external variables or constant variables # Check for time dependence dt_value = variables["dt"].get_value()[0] if "dt" in variables else None # This will raise an error if we meet the symbol "t" anywhere # except as an argument of a locally constant function for entry in itertools.chain(matrix, constants): if not is_constant_over_dt(entry, variables, dt_value): raise UnsupportedEquationsException( f"Expression '{sympy_to_str(entry)}' is not guaranteed to be " "constant over a time step." ) symbols = [Symbol(variable, real=True) for variable in varnames] solution = sp.solve_linear_system(matrix.row_join(constants), *symbols) if solution is None or set(symbols) != set(solution.keys()): raise UnsupportedEquationsException( "Cannot solve the given equations with this stateupdater." ) b = sp.ImmutableMatrix([solution[symbol] for symbol in symbols]) # Solve the system dt = Symbol("dt", real=True, positive=True) try: A = (matrix * dt).exp() except NotImplementedError: raise UnsupportedEquationsException( "Cannot solve the given equations with this stateupdater." ) if method_options["simplify"]: A = A.applyfunc(lambda x: sp.factor_terms(sp.cancel(sp.signsimp(x)))) C = sp.ImmutableMatrix(A * b) - b _S = sp.MatrixSymbol("_S", len(varnames), 1) updates = A * _S + C updates = updates.as_explicit() # The solution contains _S[0, 0], _S[1, 0] etc. for the state variables, # replace them with the state variable names abstract_code = [] for variable, update in zip(varnames, updates): rhs = update if rhs.has(I, re, im): raise UnsupportedEquationsException( "The solution to the linear system " "contains complex values " "which is currently not implemented." ) for row_idx, varname in enumerate(varnames): rhs = rhs.subs(_S[row_idx, 0], varname) # Do not overwrite the real state variables yet, the update step # of other state variables might still need the original values abstract_code.append(f"_{variable} = {sympy_to_str(rhs)}") # Update the state variables for variable in varnames: abstract_code.append(f"{variable} = _{variable}") return "\n".join(abstract_code)
def __repr__(self): return f"{self.__class__.__name__}()"
independent = IndependentStateUpdater() linear = LinearStateUpdater() exact = linear