使用 Scipy/Numpy-NDSolve 等效求解 Python 中的矩阵微分方程? [英] Solving Matrix Differential Equation in Python using Scipy/Numpy- NDSolve equivalent?

问题描述

我有两个 numpy 数组:9x9 和 9x1.我想在离散时间点求解微分方程，但无法让 ODEInt 工作.我不确定我是否在做正确的事情.

使用 Mathematica，等式是:

Solution = {A[t]}/.NDSolve[{A'[t] == Ab.A[t] &&A[0] == A0}, {A[t]}, {t, 0, .5}, MaxSteps ->\[无限]];时间 = 0.25;增量 = 0.05;MA = Table[Solution, {t, 0, time, increment}];

其中 Ab 是 9x9 矩阵，A0 是 9x1 矩阵(初始).在这里，我解决了时间和生活的美好.

在 Python 实现中，我有以下代码给出了错误的答案:

from scipy.integrate import odeint从 numpy 导入数组，点，pidef deriv(A, t, Ab):返回点(Ab，A)def MatrixBM3(k12,k21,k13,k31,k23,k32,delta1,delta2,delta3,w1、R1、R2):K = 数组([[-k21 -k23, k12, k32, 0., 0., 0., 0., 0., 0.],[k21, -k12 - k13, k31, 0., 0., 0., 0., 0., 0.],[k23, k13, -k31 - k32, 0., 0., 0., 0., 0., 0.],[0., 0., 0., -k21 - k23, k12, k32, 0., 0., 0.],[0., 0., 0., k21, -k12 - k13, k31, 0., 0., 0.],[0., 0., 0., k23, k13, -k31 - k32, 0., 0., 0.],[0., 0., 0., 0., 0., 0., -k21 - k23, k12, k32],[0., 0., 0., 0., 0., 0., k21, -k12 - k13, k31],[0., 0., 0., 0., 0., 0., k23, k13, -k31 - k32]])Der = array([[0., 0., 0., -delta2, 0., 0., 0., 0., 0.],[0., 0., 0., 0., -delta1, 0., 0., 0., 0.],[0., 0., 0., 0., 0., -delta3, 0., 0., 0.],[delta2, 0., 0., 0., 0., 0., 0., 0., 0.],[0., delta1, 0., 0., 0., 0., 0., 0., 0.],[0., 0., delta3, 0., 0., 0., 0., 0., 0.],[0., 0., 0., 0., 0., 0., 0., 0., 0.],[0., 0., 0., 0., 0., 0., 0., 0., 0.],[0., 0., 0., 0., 0., 0., 0., 0., 0.]])W = 数组([[0., 0., 0., 0., 0., 0., 0., 0., 0.],[0., 0., 0., 0., 0., 0., 0., 0., 0.],[0., 0., 0., 0., 0., 0., 0., 0., 0.],[0., 0., 0., 0., 0., 0., w1, 0., 0.],[0., 0., 0., 0., 0., 0., 0., w1, 0.],[0., 0., 0., 0., 0., 0., 0., 0., w1],[0., 0., 0., w1, 0., 0., 0., 0., 0.],[0., 0., 0., 0., w1, 0., 0., 0., 0.],[0., 0., 0., 0., 0., w1, 0., 0., 0.]])*2*piR = 数组([[-R2, 0., 0., 0., 0., 0., 0., 0., 0.],[0., -R2, 0., 0., 0., 0., 0., 0., 0.],[0., 0., -R2, 0., 0., 0., 0., 0., 0.],[0., 0., 0., -R2, 0., 0., 0., 0., 0.],[0., 0., 0., 0., -R2, 0., 0., 0., 0.],[0., 0., 0., 0., 0., -R2, 0., 0., 0.],[0., 0., 0., 0., 0., 0., -R1, 0., 0.],[0., 0., 0., 0., 0., 0., 0., -R1, 0.],[0., 0., 0., 0., 0., 0., 0., 0., -R1]])回报(K + Der + W + R)AB = MatrixBM3(21.218791062154633，17653.497151475527，40.50203461096454，93956.36617049483，0.0，0.0，-646.4238856161137，6727.748368359598，20919.132768439955，200.0，2.36787，5.39681)A0 =阵列([ -  0.001071585381162955，-0.89153191708755708，-0.00038431516707591748，0.0，0.0，0.0，0.00054009700135979673，0.4493470361764082，0.00019370128872934646])时间 = 数组([0.0,0.05,0.1,0.15,0.2,0.25])MA = odeint(deriv, A0, time, args=(Ab,), maxsteps=2000)

输出为:

[[ -1.07158538e-003 -8.91531917e-001 -3.84315167e-004 0.00000000e+0000.00000000e+000 0.00000000e+000 5.40097001e-004 4.49347036e-0011.93701289e-004][ 3.09311322e+019 9.45061860e+022 2.35327270e+019 2.11901406e+0201.63784238e+023 7.60569684e+019 2.29098804e+020 1.89766602e+0238.18752241e+019][ 9.84409730e+042 3.00774018e+046 7.48949158e+042 6.74394343e+0435.21257342e+046 2.42057805e+043 7.29126532e+043 6.03948436e+0462.60574901e+043][ 3.13296814e+066 9.57239028e+069 2.38359473e+066 2.14631766e+0671.65894606e+070 7.70369662e+066 2.32050753e+067 1.92211754e+0708.29301904e+066][ 9.97093898e+089 3.04649506e+093 7.58599405e+089 6.83083947e+0905.27973769e+093 2.45176732e+090 7.38521364e+090 6.11730342e+0932.63932422e+090][ 3.17333659e+113 9.69573101e+116 2.41430747e+113 2.17397307e+1141.68032166e+117 7.80295913e+113 2.35040739e+114 1.94688412e+1178.39987500e+113]]

但正确答案应该是:

<预> <代码> { - 0.0010733126291998989，-0.8929689437405254，-0.0003849346301906338，0.1，0.1，0.1，0.0005366563145999495，0.4464844718702628，0.00019246731509531696}{-0.000591095648651598，-0.570032546156741，-0.00023381082725213798，-0.00024790706920038567，0.00010389803046880286，-0.00005361569187144767，0.0003273277204077012，0.2870035216110215，0.00012300339326137006}{-0.0003770535829276868，-0.364106358478121，-0.0001492324135668267，-0.0001596072774600538，-0.0011479989178276948，-0.000034744485507007025，0.00020965172928479557，0.18378613639965447，0.00007876820247280559}{-0.00024100792803807562，-0.23298939195213314，-0.00009543704274825206，-0.00010271831380730501，-0.0013205519868311284，-0.000022472380871477824，0.00013326471695185768，0.11685506361394844，0.00005008078740423007}{-0.00015437993249587976，-0.1491438843823813，-0.00006111736454518403，-0.00006545797627466387，-0.0005705018939767294，-0.000014272382451480663，0.00008455890984798549，0.0741820536557778，0.00003179071165818503}{-0.00009882799610556456，-0.09529950309336405，-0.00003909275555213336，-0.00004138741286392128，0.00006303116741431477，-8.944610716890746 * ^ - 6，0.00005406263888971806，0.04743157303933772，0.00002032674776723143}

谁能指出我可能做错了什么?

解决方案

在对 odeint 的调用中，尝试将 tuple(array[Ab]) 更改为 (array(Ab),)，甚至只是 (Ab,).即使用

MA = odeint(deriv, A0, time, (Ab,))

在没有看到您如何定义 A0 和 Ab 的情况下，我不能确定这会解决问题，但是您的代码的以下变体将起作用.我使用了 3x3 数组而不是 9x9.

将 numpy 导入为 np从 scipy.integrate 导入 o​​deintdef deriv(A, t, Ab):返回 np.dot(Ab, A)Ab = np.array([[-0.25, 0, 0],[ 0.25, -0.2, 0],[ 0, 0.2, -0.1]])时间 = np.linspace(0, 25, 101)A0 = np.array([10, 20, 30])MA = odeint(deriv, A0, 时间, args=(Ab,))

I have two numpy arrays: 9x9 and 9x1. I'd like to solve the differential equation at discrete time points, but am having trouble getting ODEInt to work. I do am unsure if I'm even doing the right thing.

With Mathematica, the equation is:

Solution = {A[t]} /. NDSolve[{A'[t] == Ab.A[t] && A[0] == A0}, {A[t]}, {t, 0, .5}, MaxSteps -> \[Infinity]];

time = 0.25;
increment = 0.05;
MA = Table[Solution, {t, 0, time, increment}];

Where Ab is the 9x9 matrix, A0 is the 9x1 matrix (initial). Here, I solve for time and life is good.

In Python implementation I have the following code which gives me the wrong answer:

from scipy.integrate import odeint
from numpy import array, dot, pi

def deriv(A, t, Ab):
    return dot(Ab, A)

def MatrixBM3(k12,k21,k13,k31,k23,k32,delta1,delta2,delta3,
              w1, R1, R2):

  K = array([[-k21 -k23, k12, k32, 0., 0., 0., 0., 0., 0.],
                [k21, -k12 - k13, k31, 0., 0., 0., 0., 0., 0.],
                [k23, k13, -k31 - k32, 0., 0., 0., 0., 0., 0.],
                [0., 0., 0., -k21 - k23, k12, k32, 0., 0., 0.],
                [0., 0., 0., k21, -k12 - k13, k31, 0., 0., 0.],
                [0., 0., 0., k23, k13, -k31 - k32, 0., 0., 0.],
                [0., 0., 0., 0., 0., 0., -k21 - k23, k12, k32],
                [0., 0., 0., 0., 0., 0., k21, -k12 - k13, k31],
                [0., 0., 0., 0., 0., 0., k23, k13, -k31 - k32]])

  Der = array([[0., 0., 0., -delta2, 0., 0., 0., 0., 0.],
                  [0., 0., 0., 0., -delta1, 0., 0., 0., 0.],
                  [0., 0., 0., 0., 0., -delta3, 0., 0., 0.],
                  [delta2, 0., 0., 0., 0., 0., 0., 0., 0.],
                  [0., delta1, 0., 0., 0., 0., 0., 0., 0.],
                  [0., 0., delta3, 0., 0., 0., 0., 0., 0.],
                  [0., 0., 0., 0., 0., 0., 0., 0., 0.],
                  [0., 0., 0., 0., 0., 0., 0., 0., 0.],
                  [0., 0., 0., 0., 0., 0., 0., 0., 0.]])

  W = array([[0., 0., 0., 0., 0., 0., 0., 0., 0.],
                [0., 0., 0., 0., 0., 0., 0., 0., 0.],
                [0., 0., 0., 0., 0., 0., 0., 0., 0.], 
                [0., 0., 0., 0., 0., 0., w1, 0., 0.],
                [0., 0., 0., 0., 0., 0., 0., w1, 0.],
                [0., 0., 0., 0., 0., 0., 0., 0., w1],
                [0., 0., 0., w1, 0., 0., 0., 0., 0.], 
                [0., 0., 0., 0., w1, 0., 0., 0., 0.],
                [0., 0., 0., 0., 0., w1, 0., 0., 0.]])*2*pi

  R = array([[-R2, 0., 0., 0., 0., 0., 0., 0., 0.],
                [0., -R2, 0., 0., 0., 0., 0., 0., 0.],
                [0., 0., -R2, 0., 0., 0., 0., 0., 0.],
                [0., 0., 0., -R2, 0., 0., 0., 0., 0.],
                [0., 0., 0., 0., -R2, 0., 0., 0., 0.], 
                [0., 0., 0., 0., 0., -R2, 0., 0., 0.], 
                [0., 0., 0., 0., 0., 0., -R1, 0., 0.],
                [0., 0., 0., 0., 0., 0., 0., -R1, 0.],
                [0., 0., 0., 0., 0., 0., 0., 0., -R1]]) 
  return(K + Der + W + R)

Ab = MatrixBM3(21.218791062154633, 17653.497151475527, 40.50203461096454, 93956.36617049483, 0.0, 0.0, -646.4238856161137, 6727.748368359598, 20919.132768439955, 200.0, 2.36787, 5.39681)
A0 = array([-0.001071585381162955, -0.89153191708755708, -0.00038431516707591748, 0.0, 0.0, 0.0, 0.00054009700135979673, 0.4493470361764082, 0.00019370128872934646])
time = array([0.0,0.05,0.1,0.15,0.2,0.25])
MA = odeint(deriv, A0, time, args=(Ab,), maxsteps=2000)

Output is:

[[ -1.07158538e-003  -8.91531917e-001  -3.84315167e-004   0.00000000e+000
    0.00000000e+000   0.00000000e+000   5.40097001e-004   4.49347036e-001
    1.93701289e-004]
 [  3.09311322e+019   9.45061860e+022   2.35327270e+019   2.11901406e+020
    1.63784238e+023   7.60569684e+019   2.29098804e+020   1.89766602e+023
    8.18752241e+019]
 [  9.84409730e+042   3.00774018e+046   7.48949158e+042   6.74394343e+043
    5.21257342e+046   2.42057805e+043   7.29126532e+043   6.03948436e+046
    2.60574901e+043]
 [  3.13296814e+066   9.57239028e+069   2.38359473e+066   2.14631766e+067
    1.65894606e+070   7.70369662e+066   2.32050753e+067   1.92211754e+070
    8.29301904e+066]
 [  9.97093898e+089   3.04649506e+093   7.58599405e+089   6.83083947e+090
    5.27973769e+093   2.45176732e+090   7.38521364e+090   6.11730342e+093
    2.63932422e+090]
 [  3.17333659e+113   9.69573101e+116   2.41430747e+113   2.17397307e+114
    1.68032166e+117   7.80295913e+113   2.35040739e+114   1.94688412e+117
    8.39987500e+113]]

But the correct answer should be:

{-0.0010733126291998989, -0.8929689437405254, -0.0003849346301906338, 0., 0., 0., 0.0005366563145999495, 0.4464844718702628, 0.00019246731509531696}
{-0.000591095648651598, -0.570032546156741, -0.00023381082725213798, -0.00024790706920038567, 0.00010389803046880286, -0.00005361569187144767, 0.0003273277204077012, 0.2870035216110215, 0.00012300339326137006}
{-0.0003770535829276868, -0.364106358478121, -0.0001492324135668267, -0.0001596072774600538, -0.0011479989178276948, -0.000034744485507007025, 0.00020965172928479557, 0.18378613639965447, 0.00007876820247280559}
{-0.00024100792803807562, -0.23298939195213314, -0.00009543704274825206, -0.00010271831380730501, -0.0013205519868311284, -0.000022472380871477824, 0.00013326471695185768, 0.11685506361394844, 0.00005008078740423007}
{-0.00015437993249587976, -0.1491438843823813, -0.00006111736454518403, -0.00006545797627466387, -0.0005705018939767294, -0.000014272382451480663, 0.00008455890984798549, 0.0741820536557778, 0.00003179071165818503}
{-0.00009882799610556456, -0.09529950309336405, -0.00003909275555213336, -0.00004138741286392128, 0.00006303116741431477, -8.944610716890746*^-6, 0.00005406263888971806, 0.04743157303933772, 0.00002032674776723143}

Can anyone point me to what I may be doing wrong?

解决方案

In the call to odeint, try changing tuple(array[Ab]) to (array(Ab),), or even just (Ab,). That is, use

MA = odeint(deriv, A0, time, (Ab,))

Without seeing how you defined A0 and Ab, I can't be sure that this will fix the problem, but the following variation of your code will work. I used a 3x3 array instead of 9x9.

import numpy as np
from scipy.integrate import odeint


def deriv(A, t, Ab):
    return np.dot(Ab, A)


Ab = np.array([[-0.25,    0,    0],
               [ 0.25, -0.2,    0],
               [    0,  0.2, -0.1]])

time = np.linspace(0, 25, 101)
A0 = np.array([10, 20, 30])

MA = odeint(deriv, A0, time, args=(Ab,))

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