My problem is to minimize a target function with differential equations in it. The target function is $ $ \min \mathrm{J}\left( p \right) =\sum_{i=1}^5{|x_i\left( p \right) -y_i|}\,\,$ $ where $ x_i(p)$ are differential equations like this:

`stateFunctionV[v1_, v2_, v3_, v4_, v5_, v6_, v7_, v8_] := { \[Mu] = (67/100)*(x2[t]/(x2[t] + 28/100))*(1 - x3[t]/940)*(1 - x4[t]/1050)^3*(1 - x5[t]/361); x1'[t] == \[Mu]*x1[t], x2'[t] == -(v1 + \[Mu]/v5)* x1[t], x3'[t] == (v2 + \[Mu]*v6)*x1[t], x4'[t] == (v3 + \[Mu]*v7)* x1[t], x5'[t] == (v4 + \[Mu]*v8)* x1[t], x1[0] == 0.2025, x2[0] == 441.337, x3[0] == 0, x4[0] == 0, x5[0] == 0} `

and $ y_i$ is the experiment data.

The code:

`PIP1[value_] := { Sum[Sum[Abs[ NDSolveValue[{ [Mu] = (67/100)*(x2[t]/(x2[t] + 28/100))*(1 - x3[t]/940)*(1 - x4[t]/1050)^3*(1 - x5[t]/361); x1'[t] == \[Mu]*x1[t], x2'[t] == -(value[[1]] + \[Mu]/value[[5]])* x1[t], x3'[t] == (value[[2]] + \[Mu]*value[[6]])*x1[t], x4'[t] == (value[[3]] + \[Mu]*value[[7]])* x1[t], x5'[t] == (value[[4]] + \[Mu]*value[[8]])* x1[t], x1[0] == 0.2025, x2[0] == 441.337, x3[0] == 0, x4[0] == 0, x5[0] == 0}, {x1, x2, x3, x4, x5}, {t, 0, 6.5}][[i]] [data2[[i, j, 1]]] - data2[[i, j, 2]] ], {j, 1, 7}], {i, 1, 5}]} NMinimize[PIP1[{v1, v2, v3, v4, v5, v6, v7, v8}], {v1, v2, v3, v4, v5, v6, v7, v8}] `

doesn’t work. I also tried to take the differential functions out as a new function and it also failed.

So I wonder if there’s a way to minimize this problem with NMinimize or not? I would be grateful for your advice on how to couple **NMinimize** and **NDSolveValue**.

The experiment data are below:

`data2 = {{{0, 0.2025}, {2, 0.76}, {3, 1.28}, {4, 2.2}, {5, 2.75}, {6, 2.95}, {6.5, 3.055}}, {{0, 441.337}, {2, 400.435}, {3, 300.326}, {4, 219.348}, {5, 120.109}, {6, 50}, {6.5, 25}}, {{0, 0}, {2, 42.2895}, {3, 78.6579}, {4, 117.026}, {5, 152.158}, {6, 229.553}, {6.5, 224.1205}}, {{0, 0}, {2, 1.66667}, {3, 13.55}, {4, 16.6667}, {5, 22.7333}, {6, 34.4333}, {6.5, 71.2333}}, {{0, 0}, {2, 6.73913}, {3, 7.86957}, {4, 9.3913}, {5, 8.34783}, {6, 10.9565}, {6.5, 11.8261}}}; `