将sympy表达式转换为numpy数组的函数 [英] Convert sympy expressions to function of numpy arrays

问题描述

我有一个用sympy编写的ODE系统:

I have a system of ODEs written in sympy:

from sympy.parsing.sympy_parser import parse_expr

xs = symbols('x1 x2')
ks = symbols('k1 k2')
strs = ['-k1 * x1**2 + k2 * x2', 'k1 * x1**2 - k2 * x2']
syms = [parse_expr(item) for item in strs]

我想将其转换为向量值函数，接受x值的1D numpy数组，k值的1D numpy数组，返回在这些点求值的方程的1D numpy数组.签名看起来像这样:

I would like to convert this into a vector valued function, accepting a 1D numpy array of the x value, a 1D numpy array of the k values, returning a 1D numpy array of the equations evaluated at those points. The signature would look something like this:

import numpy as np
x = np.array([3.5, 1.5])
k = np.array([4, 2])
xdot = my_odes(x, k)

我想要这样的原因是要将此功能提供给scipy.integrate.odeint，所以它需要快速.

The reason I want something like this is to give this function to scipy.integrate.odeint, so it needs to be fast.

尝试1:潜艇

当然，我可以在subs周围写一个包装器:

Of course, I can write a wrapper around subs:

def my_odes(x, k):
    all_dict = dict(zip(xs, x))
    all_dict.update(dict(zip(ks, k)))
    return np.array([sym.subs(all_dict) for sym in syms])

但这非常慢，特别是对于我的实际系统而言，该系统要大得多并且可以运行很多次.我需要将此操作编译为C代码.

But this is super slow, especially for my real system which is much bigger and is run many times. I need to compile this operation to C code.

尝试2:theano

我可以通过 sympy与theano的集成来:

I can get close with sympy's integration with theano:

from sympy.printing.theanocode import theano_function

f = theano_function(xs + ks, syms)

def my_odes(x, k):
    return np.array(f(*np.concatenate([x,k]))))

这将编译每个表达式，但是对输入和输出的所有这种打包和拆包都会使它变慢. theano_function返回的函数接受numpy数组作为参数，但是每个符号需要一个数组，而不是每个符号需要一个元素.对于functify和ufunctify，这也是相同的行为.我不需要广播行为；我需要它来将数组的每个元素解释为不同的符号.

This compiles each expression, but all this packing and unpacking of the inputs and outputs slows it back down. The function returned by theano_function accepts numpy arrays as arguments, but it needs one array for each symbol rather than one element for each symbol. This is the same behavior for functify and ufunctify as well. I don't need the broadcast behavior; I need it to interpret each element of the array as a different symbol.

尝试3:DeferredVector

如果使用DeferredVector，我可以创建一个接受numpy数组的函数，但是我不能将其编译为C代码或返回numpy数组而不自行打包.

If I use DeferredVector I can make a function that accepts numpy arrays, but I can't compile it to C code or return a numpy array without packaging it myself.

import numpy as np
import sympy as sp
from sympy import DeferredVector

x = sp.DeferredVector('x')
k =  sp.DeferredVector('k')
deferred_syms = [s.subs({'x1':x[0], 'x2':x[1], 'k1':k[0], 'k2':k[1]}) for s in syms]
f = [lambdify([x,k], s) for s in deferred_syms]

def my_odes(x, k):
    return np.array([f_i(x, k) for f_i in f])

使用DeferredVector我不需要解压缩输入，但仍然需要解压缩输出.另外，我可以使用lambdify，但是ufuncify和theano_function版本会消失，因此不会生成快速的C代码.

Using DeferredVector I do not need to unpack the inputs, but I still need to pack the outputs. Also, I can use lambdify, but the ufuncify and theano_function versions perish, so no fast C code is generated.

from sympy.utilities.autowrap import ufuncify
f = [ufuncify([x,k], s) for s in deferred_syms] # error

from sympy.printing.theanocode import theano_function
f = theano_function([x,k], deferred_syms) # error

推荐答案

您可以使用sympy函数 lambdify .例如，

You can use the sympy function lambdify. For example,

from sympy import symbols, lambdify
from sympy.parsing.sympy_parser import parse_expr
import numpy as np

xs = symbols('x1 x2')
ks = symbols('k1 k2')
strs = ['-k1 * x1**2 + k2 * x2', 'k1 * x1**2 - k2 * x2']
syms = [parse_expr(item) for item in strs]

# Convert each expression in syms to a function with signature f(x1, x2, k1, k2):
funcs = [lambdify(xs + ks, f) for f in syms]


# This is not exactly the same as the `my_odes` in the question.
# `t` is included so this can be used with `scipy.integrate.odeint`.
# The value returned by `sym.subs` is wrapped in a call to `float`
# to ensure that the function returns python floats and not sympy Floats.
def my_odes(x, t, k):
    all_dict = dict(zip(xs, x))
    all_dict.update(dict(zip(ks, k)))
    return np.array([float(sym.subs(all_dict)) for sym in syms])

def lambdified_odes(x, t, k):
    x1, x2 = x
    k1, k2 = k
    xdot = [f(x1, x2, k1, k2) for f in funcs]
    return xdot


if __name__ == "__main__":
    from scipy.integrate import odeint

    k1 = 0.5
    k2 = 1.0
    init = [1.0, 0.0]
    t = np.linspace(0, 1, 6)
    sola = odeint(lambdified_odes, init, t, args=((k1, k2),))
    solb = odeint(my_odes, init, t, args=((k1, k2),))
    print(np.allclose(sola, solb))

运行脚本时会打印

True.

True is printed when the script is run.

速度更快(请注意计时结果单位的变化):

It is much faster (note the change in units of the timing results):

In [79]: t = np.linspace(0, 10, 1001)

In [80]: %timeit sol = odeint(my_odes, init, t, args=((k1, k2),))
1 loops, best of 3: 239 ms per loop

In [81]: %timeit sol = odeint(lambdified_odes, init, t, args=((k1, k2),))
1000 loops, best of 3: 610 µs per loop

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