2021 年高教社杯全国大学生数学建模竞赛 B题 乙醇偶合制备 C4 烯烃 求解思路2

由附件1“性能数据表”中A1~A14（用装料方式I）整理得下表“sheet2”：

使用全部影响乙醇转化率和C4 烯烃选择性的参数做拟合：

xx1：Co/SiO2

xx2：Co/SiO2 和 HAP 的质量比

xx3：催化剂总量

xx4：乙醇浓度

xx5：温度

乙醇转化率：   y1=f(xx1,xx2,xx3,xx4,xx5)（使用万能的泰勒公式-幂级数，拟合公式见代码）

C4 烯烃选择性：y2=f(xx1,xx2,xx3,xx4,xx5)（使用万能的泰勒公式-幂级数，拟合公式见代码）

三级泰勒公式展开
!!!using["luopt","math"]; //使用命名空间
目标函数(xx1,xx2,xx3,xx4,xx5,  //共61个拟合参数
    a,a1,a2,a3,a4,a5,
    a11,a12,a13,a14,a15,
        a22,a23,a24,a25,
            a33,a34,a35,
                a44,a45,
                    a55,
    a111,a112,a113,a114,a115,
         a122,a123,a124,a125,
              a133,a134,a135,
                   a144,a145,
                        a155,
         a222,a223,a224,a225,
              a233,a234,a235,
                   a244,a245,
                        a255,
              a333,a334,a335,
                   a344,a345,
                        a355,
                   a444,a445,
                        a455,
                        a555
    : i,s,x1,x2,x3,x4,x5,y1,y2,py,pc : tyArray,max)=
{
    i=-1, s=0, (++i<max).while{
        x1=tyArray(i,0)-xx1, x2=tyArray(i,1)-xx2, x3=tyArray(i,2)-xx3, x4=tyArray(i,3)-xx4, x5=tyArray(i,4)-xx5, py=tyArray(i,5), pc=tyArray(i,6),
        y1= a+a1*x1+a2*x2+a3*x3+a4*x4+a5*x5+
            a11*x1*x1+a12*x1*x2+a13*x1*x3+a14*x1*x4+a15*x1*x5+
                      a22*x2*x2+a23*x2*x3+a24*x2*x4+a25*x2*x5+
                                a33*x3*x3+a34*x3*x4+a35*x3*x5+
                                          a44*x4*x4+a45*x4*x5+
                                                    a55*x5*x5+ 
         a111*x1*x1*x1+a112*x1*x1*x2+a113*x1*x1*x3+a114*x1*x1*x4+a115*x1*x1*x5+
                       a122*x1*x2*x2+a123*x1*x2*x3+a124*x1*x2*x4+a125*x1*x2*x5+
                                     a133*x1*x3*x3+a134*x1*x3*x4+a135*x1*x3*x5+
                                                   a144*x1*x4*x4+a145*x1*x4*x5+
                                                                 a155*x1*x5*x5+
                       a222*x2*x2*x2+a223*x2*x2*x3+a224*x2*x2*x4+a225*x2*x2*x5+
                                     a233*x2*x3*x3+a234*x2*x3*x4+a235*x2*x3*x5+
                                                   a244*x2*x4*x4+a245*x2*x4*x5+
                                                                 a255*x2*x5*x5+
                                     a333*x3*x3*x3+a334*x3*x3*x4+a335*x3*x3*x5+
                                                   a344*x3*x4*x4+a345*x3*x4*x5+
                                                                 a355*x3*x5*x5+
                                                   a444*x4*x4*x4+a445*x4*x4*x5+
                                                                 a455*x4*x5*x5+
                                                                 a555*x5*x5*x5,
        s=s+[y1-py]^2    //拟合乙醇转化率
        //s=s+[y1-pc]^2  //拟合C4烯烃选择性
    },
    sqrt[s/max]          //均方差RMSE为目标函数值
};
main(::tyArray,max)=
{
    tyArray=matrix{ //存放实验数据：使用装料方式I全部数据
        "
0.5	1	400	1.68	250	4.0 	9.6 
0.5	1	400	1.68	275	12.1 	8.6 
0.5	1	400	1.68	300	29.5 	10.7 
0.5	1	400	1.68	325	43.3 	18.9 
0.5	1	400	1.68	350	60.5 	27.3 
0.5	1	400	1.68	400	88.4 	41.0 
1	0.4925	100	1.68	250	2.5 	1.9 
1	0.4925	100	1.68	275	5.3 	2.6 
... ...（略）
        "
    },
    len[tyArray,0,&max],
    Opt1[@目标函数, optmax,1000, optmode,20, optdeep,20, optwaysimdeep, optwayconfra, optwaylme] //Opt1函数全局优化
};

结果及讨论：

对应乙醇转化率Y1的最优参数及均方差RMSE：

2.417402744750639         -0.2039147139106143       275.8011306772221         0.4156384853693174        260.1013882930106         14.83901857201672         0.317318572539172         -19.50790611027291        -3.588076207774742e-002   -6.855005447168541        -4.086740657820882e-002   -1.460941918741034        -0.4367844784685392       3.425230738907303e-002    11.49855862581616         4.904074627155117e-002    12.37311562403754         9.190390656233782e-002    2.893081399660135         -8.277525297568886e-002   -5.116674454929938e-004   -3.922081102157155e-003   3.238624290254206e-005    -2.034325083811818        5.339523080291108e-003    3.387818581763618e-003    -8.649776365931871        1.907021895486433         3.717192575220874e-003    0.2081378685138888        -2.614230849096533e-002   0.8020982822085792        1.067869051018692e-002    13.65608817413937         8.195945752882006e-003    -1.559228490839049e-004   5.447393901609546e-003    -6.670850845389642e-005   3.286334739062546         -3.95352575062304e-002    7.496760379706923e-005    -1.666984727364169        -2.45598856334926e-004    -2.942456600663122        4.646162907471563e-002    5.272516733906026e-005    2.86629773902457e-002     5.44992912475403e-004     26.46808705338533         -1.218629532938329e-003   1.496886943412408e-005    2.675602420290519e-007    2.753847472266319e-004    1.455617526774928e-005    -4.091460730469629e-002   9.190733722458817e-004    -1.477766443432227e-006   -8.220395243069316        -8.389116545320428e-003   3.364704951845783e-004    -1.132804688837243e-005   3.685019008260015

对应C4烯烃选择性Y2的最优参数及均方差RMSE：

1.986158004301724         2.450433071976323         67.49819932455158         0.208662313445064         256.4968573838999         1.799578875493056         -5.974066210514014        -11.56553740063614        -0.1545396393604948       5.172523119752178         0.1540143576135561        -17.45297800630631        -0.1090062772880443       -1.310984675110006e-002   -14.03392383507257        -4.289169392186376e-002   22.25748000512774         -2.245456583454652e-002   8.212272604609786e-002    6.271792925982265e-003    -6.669459749834531e-005   -1.903100059080562e-002   7.217058706506922e-004    -9.15304420446407         -3.741345572514077e-002   2.559970423512096e-003    11.47373282665268         9.341921817829952         3.264886165514432e-002    -6.472974026168921e-003   -2.121310819655539e-003   5.51933035952529          -5.171538944184857e-003   4.989155322559992         0.1533949251601508        -1.698477968040691e-004   -2.204068286556028e-002   2.761963091700759e-004    4.0798863013763           9.376124672502875e-002    2.641556051018874e-004    4.100002442739125         -4.734441692273163e-003   -1.961348488352317        -0.1706020449016992       2.700080312269564e-004    -9.322610320728019e-004   1.622536988804513e-003    11.72316720890722         -0.1463931085480807       -5.872822055720774e-004   1.205698763703183e-006    1.286795200473773e-004    6.839371475233182e-006    3.721430144278198e-002    -4.335815325906642e-004   -1.157491548269355e-007   9.860323047148855         -1.120116777723885e-002   -1.625346889836123e-004   -1.228545092598462e-005   2.394257044586305

因参数多，耗时较长，等Opt::ShowMin输出稳定（半天多时间），取以上结果。

因耗时较长，未多次运行；若多台电脑同时运行，或许有更优结果。

因目标函数值稍大（均方差RMSE为3.69（对应Y1）和2.39（对应Y2）），说明三级泰勒公式展开精度欠佳，四级泰勒公式展开将有更好结果，但拟合参数将增加到131个，耗时将更长。

由以上拟合参数求C4烯烃收率（Y1*Y2）最大值，可求最佳参数xx1,xx2,xx3,xx4,xx5。

三级泰勒公式展开，优化求最佳C4烯烃收率
!!!using["luopt","math"];; //使用命名空间
init(::       //拟合参数赋值
    aa1,aa2,aa3,aa4,aa5,
    a,a1,a2,a3,a4,a5,
    a11,a12,a13,a14,a15,
        a22,a23,a24,a25,
            a33,a34,a35,
                a44,a45,
                    a55,
    a111,a112,a113,a114,a115,
         a122,a123,a124,a125,
              a133,a134,a135,
                   a144,a145,
                        a155,
         a222,a223,a224,a225,
              a233,a234,a235,
                   a244,a245,
                        a255,
              a333,a334,a335,
                   a344,a345,
                        a355,
                   a444,a445,
                        a455,
                        a555,
    bb1,bb2,bb3,bb4,bb5,
    b,b1,b2,b3,b4,b5,
    b11,b12,b13,b14,b15,
        b22,b23,b24,b25,
            b33,b34,b35,
                b44,b45,
                    b55,
    b111,b112,b113,b114,b115,
         b122,b123,b124,b125,
              b133,b134,b135,
                   b144,b145,
                        b155,
         b222,b223,b224,b225,
              b233,b234,b235,
                   b244,b245,
                        b255,
              b333,b334,b335,
                   b344,b345,
                        b355,
                   b444,b445,
                        b455,
                        b555) =
  new[real_s,61].SetArray[
  "
  1.605961168999771         -0.4410695905557486       235.7757505227786  ... ...（这里放对应Y1的输出结果）
  "].in[0 :
    &aa1,&aa2,&aa3,&aa4,&aa5,
    &a,&a1,&a2,&a3,&a4,&a5,
    &a11,&a12,&a13,&a14,&a15,
        &a22,&a23,&a24,&a25,
            &a33,&a34,&a35,
                &a44,&a45,
                    &a55,
    &a111,&a112,&a113,&a114,&a115,
         &a122,&a123,&a124,&a125,
              &a133,&a134,&a135,
                   &a144,&a145,
                        &a155,
         &a222,&a223,&a224,&a225,
              &a233,&a234,&a235,
                   &a244,&a245,
                        &a255,
              &a333,&a334,&a335,
                   &a344,&a345,
                        &a355,
                   &a444,&a445,
                        &a455,
                        &a555],
  new[real_s,61].SetArray[
  "
  0.6291493182031664        0.2958183812514781        -0.4652893843549317  ... ...（这里放对应Y2的输出结果）

  "].in[0 :
    &bb1,&bb2,&bb3,&bb4,&bb5,
    &b,&b1,&b2,&b3,&b4,&b5,
    &b11,&b12,&b13,&b14,&b15,
        &b22,&b23,&b24,&b25,
            &b33,&b34,&b35,
                &b44,&b45,
                    &b55,
    &b111,&b112,&b113,&b114,&b115,
         &b122,&b123,&b124,&b125,
              &b133,&b134,&b135,
                   &b144,&b145,
                        &b155,
         &b222,&b223,&b224,&b225,
              &b233,&b234,&b235,
                   &b244,&b245,
                        &b255,
              &b333,&b334,&b335,
                   &b344,&b345,
                        &b355,
                   &b444,&b445,
                        &b455,
                        &b555];;
f(xx1,xx2,xx3,xx4,xx5  //目标函数
  : x1,x2,x3,x4,x5, y1, y2
  : aa1,aa2,aa3,aa4,aa5,
    a,a1,a2,a3,a4,a5,
    a11,a12,a13,a14,a15,
        a22,a23,a24,a25,
            a33,a34,a35,
                a44,a45,
                    a55,
    a111,a112,a113,a114,a115,
         a122,a123,a124,a125,
              a133,a134,a135,
                   a144,a145,
                        a155,
         a222,a223,a224,a225,
              a233,a234,a235,
                   a244,a245,
                        a255,
              a333,a334,a335,
                   a344,a345,
                        a355,
                   a444,a445,
                        a455,
                        a555,
    bb1,bb2,bb3,bb4,bb5,
    b,b1,b2,b3,b4,b5,
    b11,b12,b13,b14,b15,
        b22,b23,b24,b25,
            b33,b34,b35,
                b44,b45,
                    b55,
    b111,b112,b113,b114,b115,
         b122,b123,b124,b125,
              b133,b134,b135,
                   b144,b145,
                        b155,
         b222,b223,b224,b225,
              b233,b234,b235,
                   b244,b245,
                        b255,
              b333,b334,b335,
                   b344,b345,
                        b355,
                   b444,b445,
                        b455,
                        b555) =
{
    x1=xx1-aa1, x2=xx2-aa2, x3=xx3-aa3, x4=xx4-aa4, x5=xx5-aa5,
    y1= a+a1*x1+a2*x2+a3*x3+a4*x4+a5*x5+
        a11*x1*x1+a12*x1*x2+a13*x1*x3+a14*x1*x4+a15*x1*x5+
                  a22*x2*x2+a23*x2*x3+a24*x2*x4+a25*x2*x5+
                            a33*x3*x3+a34*x3*x4+a35*x3*x5+
                                      a44*x4*x4+a45*x4*x5+
                                                a55*x5*x5+ 
     a111*x1*x1*x1+a112*x1*x1*x2+a113*x1*x1*x3+a114*x1*x1*x4+a115*x1*x1*x5+
                   a122*x1*x2*x2+a123*x1*x2*x3+a124*x1*x2*x4+a125*x1*x2*x5+
                                 a133*x1*x3*x3+a134*x1*x3*x4+a135*x1*x3*x5+
                                               a144*x1*x4*x4+a145*x1*x4*x5+
                                                             a155*x1*x5*x5+
                   a222*x2*x2*x2+a223*x2*x2*x3+a224*x2*x2*x4+a225*x2*x2*x5+
                                 a233*x2*x3*x3+a234*x2*x3*x4+a235*x2*x3*x5+
                                               a244*x2*x4*x4+a245*x2*x4*x5+
                                                             a255*x2*x5*x5+
                                 a333*x3*x3*x3+a334*x3*x3*x4+a335*x3*x3*x5+
                                               a344*x3*x4*x4+a345*x3*x4*x5+
                                                             a355*x3*x5*x5+
                                               a444*x4*x4*x4+a445*x4*x4*x5+
                                                             a455*x4*x5*x5+
                                                             a555*x5*x5*x5,
    x1=xx1-bb1, x2=xx2-bb2, x3=xx3-bb3, x4=xx4-bb4, x5=xx5-bb5,
    y2= b+b1*x1+b2*x2+b3*x3+b4*x4+b5*x5+
        b11*x1*x1+b12*x1*x2+b13*x1*x3+b14*x1*x4+b15*x1*x5+
                  b22*x2*x2+b23*x2*x3+b24*x2*x4+b25*x2*x5+
                            b33*x3*x3+b34*x3*x4+b35*x3*x5+
                                      b44*x4*x4+b45*x4*x5+
                                                b55*x5*x5+ 
     b111*x1*x1*x1+b112*x1*x1*x2+b113*x1*x1*x3+b114*x1*x1*x4+b115*x1*x1*x5+
                   b122*x1*x2*x2+b123*x1*x2*x3+b124*x1*x2*x4+b125*x1*x2*x5+
                                 b133*x1*x3*x3+b134*x1*x3*x4+b135*x1*x3*x5+
                                               b144*x1*x4*x4+b145*x1*x4*x5+
                                                             b155*x1*x5*x5+
                   b222*x2*x2*x2+b223*x2*x2*x3+b224*x2*x2*x4+b225*x2*x2*x5+
                                 b233*x2*x3*x3+b234*x2*x3*x4+b235*x2*x3*x5+
                                               b244*x2*x4*x4+b245*x2*x4*x5+
                                                             b255*x2*x5*x5+
                                 b333*x3*x3*x3+b334*x3*x3*x4+b335*x3*x3*x5+
                                               b344*x3*x4*x4+b345*x3*x4*x5+
                                                             b355*x3*x5*x5+
                                               b444*x4*x4*x4+b445*x4*x4*x5+
                                                             b455*x4*x5*x5+
                                                             b555*x5*x5*x5,
    -y1*y2/100  //C4烯烃收率的负值为目标函数值
};
Opt1[@f, optrange: 0.4,5.1; 0.4,2.1; 90.0,410.0; 0.2,2.2; 230.0, 470.0]; //指定范围内求解，超出范围恐误差加大

结果（xx1,xx2,xx3,xx4,xx5及目标函数值）及讨论：

5.09999900000004          2.099999992453416         409.999999                2.199999982171348         469.9999990026478         -1951.17328362952

从拟合结果可以看出，目标函数值C4烯烃收率超过了100%，明显不符合实际情况，但仍对实验设计有指导意义。

xx1：Co/SiO2（实验数据提供了0.5、1、2、5共4个参数）最佳值在设定参数上限，应增加该参数进行实验。

xx2：Co/SiO2 和 HAP 的质量比（实验数据提供了0.4925、0.5556、1、2.0303共4个参数）最佳值在设定参数上限，应增加该参数进行实验。

xx3：催化剂总量（实验数据提供了100、140、400共3个参数）最佳值在设定参数上限，应增加该参数进行实验。

xx4：乙醇浓度（实验数据提供了0.3、0.9、1.68、2.1共4个参数）最佳值在设定参数上限，应增加该参数进行实验。

xx5：温度（实验数据提供较多）最佳值在设定参数上限，如果条件允许应增加该参数进行实验。

指定条件（Co/SiO2 和 HAP 的质量比为1，催化剂总量为100）求最优值：

Opt1[@f, optwaycom, optrange: 0.4,5.1; 0.999,1.001; 99.999,100.0; 0.3,2.2; 230.0, 470.0]; //指定范围内求解，超出范围恐误差加大

结果（xx1,xx2,xx3,xx4,xx5及目标函数值）：

2.059379393474595         1.000999997151315         99.9990005847001          2.199999994350263         452.9225113865244         -49.9568032788798

猜想：如果 xx1,xx2,xx3,xx4 数据更多些，拟合结果或许会更好。

下面使用Lu脚本优化库中新增的泰勒展开式计算函数FunTaylor，不仅代码量大幅减少，运行速度也大幅提高：

三级泰勒公式展开
!!!using["luopt","math"]; //使用命名空间
目标函数(a :: tyArray,Array,max)=
{
    FunTaylor[2,5,3,a,tyArray], //计算泰勒展开式的函数值
    sqrt{sum{[(tyArray[all:5]-Array).reshape()].^2.0}/max}  //均方差RMSE为目标函数值
};
main(:a:tyArray,Array,max)=
{
    //实验数据：使用装料方式I全部数据，最后一列为乙醇转化率（或者C4 烯烃选择性）
    tyArray=matrix{
        "
0.5	1	400	1.68	250	4.0 
0.5	1	400	1.68	275	12.1
0.5	1	400	1.68	300	29.5 
0.5	1	400	1.68	325	43.3 
0.5	1	400	1.68	350	60.5
0.5	1	400	1.68	400	88.4
1	0.4925	100	1.68	250	2.5 
1	0.4925	100	1.68	275	5.3 
        "
    },
    len[tyArray,0,&max], Array=tyArray[all:5],
    a=new[real_s,FunTaylor(0,5,3)],  //申请数组，存放泰勒展开式系数
    Opt1[a, @目标函数, optmax,1000, optmode,20, optdeep,20, optwaysimdeep, optwayconfra, optwaylme] //Opt1函数全局优化
};

运行结果与前述相同。

下面的代码可输出泰勒公式展开式：

输出泰勒公式展开式
!!!using["luopt","math"]; //使用命名空间
main(:a)=
{
    a=new[real_s,FunTaylor(0,5,3)].SetArray{  //申请数组
        "
2.417496983520035         -0.2038158183643675       275.7788602686833 ...  //存放泰勒展开式系数
        "
    },
    FunTaylor(1,5,3,a) //输出泰勒展开式
};

结果：y=f(y0,y1,y2,y3,y4) 乙醇转化率 的三级泰勒展开

x0=y0-(2.417496983520035), x1=y1-(-0.2038158183643675), x2=y2-(275.7788602686833), x3=y3-(0.4157150578775843), x4=y4-(260.1130205122635), 
y=(14.8339916081916)+(0.3166092689262277)*x0+(-19.5191890572145)*x1+(-3.579769353416268e-002)*x2+(-6.851231734885069)*x3+(-4.081252504690269e-002)*x4
  +(-1.462185975049673)*x0*x0+(-0.4399498529512698)*x0*x1+(3.424728880639916e-002)*x0*x2+(11.50149493634983)*x0*x3+(4.899775046562916e-002)*x0*x4+(12.37566086214654)*x1*x1+(9.179605162184748e-002)*x1*x2+(2.890613812179718)*x1*x3+(-8.280205689469142e-002)*x1*x4+(-5.112768904844806e-004)*x2*x2+(-3.945340831718832e-003)*x2*x3+(3.216395934710691e-005)*x2*x4+(-2.02763843455804)*x3*x3+(5.330066275526738e-003)*x3*x4+(3.38768061375748e-003)*x4*x4
  +(-8.64941211238461)*x0*x0*x0+(1.906979904073245)*x0*x0*x1+(3.715034007924377e-003)*x0*x0*x2+(0.2072410484718987)*x0*x0*x3+(-2.614438938622829e-002)*x0*x0*x4+(0.8077179983441646)*x0*x1*x1+(1.067299977263131e-002)*x0*x1*x2+(13.64855878020005)*x0*x1*x3+(8.218774646838642e-003)*x0*x1*x4+(-1.558999302693181e-004)*x0*x2*x2+(5.46432176380897e-003)*x0*x2*x3+(-6.669493109849741e-005)*x0*x2*x4+(3.289772620061781)*x0*x3*x3+(-3.952670192258448e-002)*x0*x3*x4+(7.497001157885084e-005)*x0*x4*x4+(-1.664062662934021)*x1*x1*x1+(-2.11898392118816e-004)*x1*x1*x2+(-2.94415280401644)*x1*x1*x3+(4.647403064153688e-002)*x1*x1*x4+(5.293074439992591e-005)*x1*x2*x2+(2.868693302201956e-002)*x1*x2*x3+(5.447745594835923e-004)*x1*x2*x4+(26.46740180486084)*x1*x3*x3+(-1.230102336289828e-003)*x1*x3*x4+(1.493155915585734e-005)*x1*x4*x4+(2.693155624482926e-007)*x2*x2*x2+(2.753552958995272e-004)*x2*x2*x3+(1.455718067712246e-005)*x2*x2*x4+(-4.09069242216141e-002)*x2*x3*x3+(9.190931133968552e-004)*x2*x3*x4+(-1.477757220874119e-006)*x2*x4*x4+(-8.219787163776919)*x3*x3*x3+(-8.378418878388916e-003)*x3*x3*x4+(3.364395602770745e-004)*x3*x4*x4+(-1.132849569198336e-005)*x4*x4*x4

结果：y=f(y0,y1,y2,y3,y4) C4 烯烃选择性 的三级泰勒展开

x0=y0-(1.986158004301724), x1=y1-(2.450433071976323), x2=y2-(67.49819932455158), x3=y3-(0.208662313445064), x4=y4-(256.4968573838999), 
y=(1.799578875493056)+(-5.974066210514014)*x0+(-11.56553740063614)*x1+(-0.1545396393604948)*x2+(5.172523119752178)*x3+(0.1540143576135561)*x4
  +(-17.45297800630631)*x0*x0+(-0.1090062772880443)*x0*x1+(-1.310984675110006e-002)*x0*x2+(-14.03392383507257)*x0*x3+(-4.289169392186376e-002)*x0*x4+(22.25748000512774)*x1*x1+(-2.245456583454652e-002)*x1*x2+(8.212272604609786e-002)*x1*x3+(6.271792925982265e-003)*x1*x4+(-6.669459749834531e-005)*x2*x2+(-1.903100059080562e-002)*x2*x3+(7.217058706506922e-004)*x2*x4+(-9.15304420446407)*x3*x3+(-3.741345572514077e-002)*x3*x4+(2.559970423512096e-003)*x4*x4
  +(11.47373282665268)*x0*x0*x0+(9.341921817829952)*x0*x0*x1+(3.264886165514432e-002)*x0*x0*x2+(-6.472974026168921e-003)*x0*x0*x3+(-2.121310819655539e-003)*x0*x0*x4+(5.51933035952529)*x0*x1*x1+(-5.171538944184857e-003)*x0*x1*x2+(4.989155322559992)*x0*x1*x3+(0.1533949251601508)*x0*x1*x4+(-1.698477968040691e-004)*x0*x2*x2+(-2.204068286556028e-002)*x0*x2*x3+(2.761963091700759e-004)*x0*x2*x4+(4.0798863013763)*x0*x3*x3+(9.376124672502875e-002)*x0*x3*x4+(2.641556051018874e-004)*x0*x4*x4+(4.100002442739125)*x1*x1*x1+(-4.734441692273163e-003)*x1*x1*x2+(-1.961348488352317)*x1*x1*x3+(-0.1706020449016992)*x1*x1*x4+(2.700080312269564e-004)*x1*x2*x2+(-9.322610320728019e-004)*x1*x2*x3+(1.622536988804513e-003)*x1*x2*x4+(11.72316720890722)*x1*x3*x3+(-0.1463931085480807)*x1*x3*x4+(-5.872822055720774e-004)*x1*x4*x4+(1.205698763703183e-006)*x2*x2*x2+(1.286795200473773e-004)*x2*x2*x3+(6.839371475233182e-006)*x2*x2*x4+(3.721430144278198e-002)*x2*x3*x3+(-4.335815325906642e-004)*x2*x3*x4+(-1.157491548269355e-007)*x2*x4*x4+(9.860323047148855)*x3*x3*x3+(-1.120116777723885e-002)*x3*x3*x4+(-1.625346889836123e-004)*x3*x4*x4+(-1.228545092598462e-005)*x4*x4*x4

有了公式，拟合y=f(y0,y1,y2,y3,y4)的最优值将比较方便，有兴趣的可自行研究。

将本例五元函数泰勒四级展开，不知精度将提高多少，有兴趣的可自行研究。

以上讨论，仅供参考。

补充说明：因xx1：Co/SiO2（实验数据提供了0.5、1、2、5共4个参数）、xx2：Co/SiO2 和 HAP 的质量比（实验数据提供了0.4925、0.5556、1、2.0303共4个参数）、xx3：催化剂总量（实验数据提供了100、140、400共3个参数）、xx4：乙醇浓度（实验数据提供了0.3、0.9、1.68、2.1共4个参数）数据量少，而拟合参数多，会存在两个问题导致模型失真：

1、过拟合。

2、龙格震荡现象。

今天的文章2021 年高教社杯全国大学生数学建模竞赛 B题 乙醇偶合制备 C4 烯烃 求解思路2分享到此就结束了，感谢您的阅读，如果确实帮到您，您可以动动手指转发给其他人。