I want to use a Genetic Algorithm to predict the thickness value. Currently, my code runs and produces results where it hits upper bound and lower bound.

clear; clc;

%% 1) Experimental Data
time = [7200 14400 21600];
temp = [1123 1173 1223];

thick_exp = [16.83 24.51 31.09;
             26.28 37.98 49.91;
             41.56 59.12 69.01];

R = 8.314;   % Gas constant (J/mol·K)

%% 2) Convert matrix to vector
[temp_grid,time_grid] = ndgrid(temp,time);


time_vector = time_grid(:);
temp_vector = temp_grid(:);
thick_exp_vector = thick_exp(:);


%% 3) Define Kinetic Model
% thickness = k * sqrt(t)
% k = k0 * exp(-Q/RT)

kineticModel = @(params,t,T) ...
    sqrt( t .* params(1) .* exp(-params(2).*1000 ./ (8.314 .* T)) );



%% 4) Define RMSE Fitness Function
fitnessFcn = @(params) ...
    sqrt(mean((thick_exp_vector - kineticModel(params,time_vector,temp_vector)).^2));

%% 5) Define Bounds [k0 Q]
lb = [1e2  50000];
ub = [1e8  300000];


%% 6) GA Options
options = optimoptions('ga',...
    'PopulationSize',100,...
    'MaxGenerations',200,...
    'Display','iter');

%% 7) Run Genetic Algorithm
[nParams,fval] = ga(fitnessFcn,2,[],[],[],[],lb,ub,[],options);

Single objective optimization:
2 Variables

Options:
CreationFcn:       @gacreationuniform
CrossoverFcn:      @crossoverscattered
SelectionFcn:      @selectionstochunif
MutationFcn:       @mutationadaptfeasible

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
    1              200           42.76           42.76        0
    2              295           42.76           42.76        1
    3              390           42.76           42.76        2
    4              485           42.76           42.76        3
    5              580           42.76           42.76        4
    6              675           42.76           42.76        5
    7              770           42.76           42.76        6
    8              865           42.76           42.76        7
    9              960           42.76           42.76        8
   10             1055           42.76           42.76        9
   11             1150           42.76           42.76       10
   12             1245           42.76           42.76       11
   13             1340           42.76           42.76       12
   14             1435           42.76           42.76       13
   15             1530           42.76           42.76       14
   16             1625           42.76           42.76       15
   17             1720           42.76           42.76       16
   18             1815           42.76           42.76       17
   19             1910           42.76           42.76       18
   20             2005           42.76           42.76       19
   21             2100           42.76           42.76       20
   22             2195           42.76           42.76       21
   23             2290           42.76           42.76       22
   24             2385           42.76           42.76       23
   25             2480           42.76           42.76       24
   26             2575           42.76           42.76       25
   27             2670           42.76           42.76       26
   28             2765           42.76           42.76       27
   29             2860           42.76           42.76       28
   30             2955           42.76           42.76       29

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
   31             3050           42.76           42.76       30
   32             3145           42.76           42.76       31
   33             3240           42.76           42.76       32
   34             3335           42.76           42.76       33
   35             3430           42.76           42.76       34
   36             3525           42.76           42.76       35
   37             3620           42.76           42.76       36
   38             3715           42.76           42.76       37
   39             3810           42.76           42.76       38
   40             3905           42.76           42.76       39
   41             4000           42.76           42.76       40
   42             4095           42.76           42.76       41
   43             4190           42.76           42.76       42
   44             4285           42.76           42.76       43
   45             4380           42.76           42.76       44
   46             4475           42.76           42.76       45
   47             4570           42.76           42.76       46
   48             4665           42.76           42.76       47
   49             4760           42.76           42.76       48
   50             4855           42.76           42.76       49
   51             4950           42.76           42.76       50
ga stopped because the average change in the fitness value is less than options.FunctionTolerance.

thick_pred = kineticModel(nParams,time_vector,temp_vector);
MAPE = mean(abs((thick_exp_vector - thick_pred) ./ thick_exp_vector)) * 100;


%% 8) Display Results
disp(['Q  = ',num2str(nParams(2)),' J/mol'])

Q  = 50000 J/mol

disp(['k0 = ',num2str(nParams(1))])

k0 = 100

disp(['RMSE = ',num2str(fval)])

RMSE = 42.7649

disp(['MAPE = ',num2str(MAPE),' %'])

MAPE = 100 %

Answer 1

[Star Strider]

The reason is that the optimal values for the 'k0' and 'Q' parameter values are far outside the bounds that you set for them. Unconstrained, 'k0' is a bit less than 80, and 'Q' is a bit less than 400.

I am not certain what you are doing, so I cannot comment on the accuracy of your 'kineticModel' function. However that is where I would look in order to understand the reason the optimal estimated parameters are far outside the bounds that you set for them.

Example --

clear; clc;

%% 1) Experimental Data
time = [7200 14400 21600];
temp = [1123 1173 1223];

thick_exp = [16.83 24.51 31.09;
             26.28 37.98 49.91;
             41.56 59.12 69.01];

R = 8.314;   % Gas constant (J/mol·K)

%% 2) Convert matrix to vector
[temp_grid,time_grid] = ndgrid(temp,time);


time_vector = time_grid(:);
temp_vector = temp_grid(:);
thick_exp_vector = thick_exp(:);


%% 3) Define Kinetic Model
% thickness = k * sqrt(t)
% k = k0 * exp(-Q/RT)

kineticModel = @(params,t,T) ...
    sqrt( t .* params(1) .* exp(-params(2).*1000 ./ (8.314 .* T)) );



%% 4) Define RMSE Fitness Function
fitnessFcn = @(params) ...
    sqrt(mean((thick_exp_vector - kineticModel(params,time_vector,temp_vector)).^2));

%% 5) Define Bounds [k0 Q]
lb = [1e2  50000];
ub = [1e8  300000];

lb = [];
ub = [];

%% 6) GA Options
options = optimoptions('ga',...
    'PopulationSize',100,...
    'MaxGenerations',200,...
    'Display','iter');

%% 7) Run Genetic Algorithm
[nParams,fval] = ga(fitnessFcn,2,[],[],[],[],lb,ub,[],options);

Single objective optimization:
2 Variables

Options:
CreationFcn:       @gacreationuniform
CrossoverFcn:      @crossoverscattered
SelectionFcn:      @selectionstochunif
MutationFcn:       @mutationgaussian

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
    1              200           14.07           156.7        0
    2              295           10.74           104.4        0
    3              390           10.74            94.5        1
    4              485           10.74             129        2
    5              580           10.52           67.99        0
    6              675           9.926           89.29        0
    7              770           9.926           96.72        1
    8              865           9.908           62.39        0
    9              960           9.806           44.84        0
   10             1055           9.791           48.85        0
   11             1150           9.791            23.5        1
   12             1245           9.767           25.91        0
   13             1340           9.767           26.63        1
   14             1435           9.767           23.21        2
   15             1530           9.767           19.18        3
   16             1625           9.767           24.26        4
   17             1720           9.767           26.56        5
   18             1815           9.767           21.31        6
   19             1910           9.297           19.13        0
   20             2005           9.297           25.05        1
   21             2100           9.297           29.21        2
   22             2195           9.297           35.17        3
   23             2290           9.297           27.73        4
   24             2385           9.025           23.29        0
   25             2480           9.025           22.72        1
   26             2575           9.025           27.08        2
   27             2670           9.025           24.73        3
   28             2765           9.025           21.41        4
   29             2860           9.025            18.7        5
   30             2955           8.996           18.01        0

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
   31             3050           8.957           17.91        0
   32             3145           8.957           18.16        1
   33             3240           8.957           15.15        2
   34             3335           8.957            18.7        0
   35             3430           8.957           21.13        1
   36             3525           8.957           20.42        2
   37             3620           8.906           19.32        0
   38             3715           8.893           20.28        0
   39             3810           8.893           33.31        1
   40             3905           8.844           19.38        0
   41             4000           8.844           15.79        1
   42             4095           8.844           20.79        2
   43             4190           8.844           18.77        3
   44             4285           8.844           15.44        4
   45             4380           8.844           17.58        5
   46             4475           8.736           15.17        0
   47             4570           8.736           20.08        1
   48             4665           8.615           18.29        0
   49             4760           8.615           17.65        1
   50             4855           8.615           20.69        2
   51             4950           8.562           22.22        0
   52             5045           8.562           28.51        1
   53             5140           8.549           30.08        0
   54             5235           8.549           25.28        1
   55             5330           8.488           18.09        0
   56             5425           8.488           19.24        1
   57             5520           8.481           18.87        0
   58             5615           8.481           20.75        1
   59             5710            8.48           20.06        0
   60             5805            8.48           21.11        1

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
   61             5900            8.48           18.86        2
   62             5995            8.48           14.19        3
   63             6090            8.48           15.17        4
   64             6185           8.438           13.97        0
   65             6280           8.438           13.29        1
   66             6375            8.43           17.63        0
   67             6470            8.43           15.74        1
   68             6565           8.407           12.39        0
   69             6660           8.342            12.8        0
   70             6755           8.342            13.5        1
   71             6850           8.342           20.65        2
   72             6945           8.339           15.66        0
   73             7040           8.339           15.84        1
   74             7135           8.339           13.03        2
   75             7230           8.339           13.98        3
   76             7325           8.339           13.04        4
   77             7420           8.339           12.38        5
   78             7515           8.309           12.33        0
   79             7610           8.309           12.53        1
   80             7705           8.309           12.71        2
   81             7800           8.309            13.2        3
   82             7895           8.309           12.56        4
   83             7990           8.309            13.5        5
   84             8085           8.309           16.09        6
   85             8180           8.304           14.83        0
   86             8275           8.276           15.31        0
   87             8370           8.263           16.64        0
   88             8465            8.22           12.14        0
   89             8560            8.22           11.84        1
   90             8655            8.22           11.24        2

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
   91             8750            8.22           12.43        3
   92             8845           8.218           14.84        0
   93             8940           8.217           12.62        0
   94             9035           8.217           12.22        1
   95             9130           8.217           13.43        0
   96             9225           8.217           12.16        1
   97             9320           8.217           13.79        2
   98             9415           8.171           12.65        0
   99             9510           8.171           15.65        1
  100             9605           8.171           14.59        2
  101             9700            8.17           14.44        0
  102             9795            8.17           14.52        1
  103             9890           8.168           13.44        0
  104             9985           8.152           13.17        0
  105            10080           8.152           12.61        0
  106            10175           8.152           13.73        1
  107            10270           8.152           14.79        2
  108            10365           8.152            11.2        3
  109            10460            8.15           12.29        0
  110            10555           8.146            11.9        0
  111            10650           8.146           12.07        1
  112            10745           8.146           11.85        2
  113            10840           8.146           14.05        3
  114            10935           8.146           12.07        4
  115            11030           8.132           12.88        0
  116            11125           8.132           11.72        1
  117            11220           8.132           11.54        2
  118            11315           8.132           10.87        3
  119            11410           8.131           11.82        0
  120            11505            8.13           11.27        0

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
  121            11600           8.053           11.72        0
  122            11695           8.053           11.11        1
  123            11790           8.053           10.79        2
  124            11885           8.053           11.13        3
  125            11980           8.053           10.66        4
  126            12075           8.024           10.24        0
  127            12170           8.024           10.34        1
  128            12265           8.024           10.41        2
  129            12360           8.024           11.35        3
  130            12455           8.004           11.15        0
  131            12550           7.997            11.7        0
  132            12645           7.997           11.73        1
  133            12740           7.997           11.51        2
  134            12835           7.997           10.36        3
  135            12930           7.997           10.38        4
  136            13025           7.997           10.15        5
  137            13120           7.997           10.46        6
  138            13215           7.997           10.81        7
  139            13310           7.997           10.72        8
  140            13405           7.959            11.1        0
  141            13500           7.959           10.59        1
  142            13595           7.959           10.51        2
  143            13690           7.959           10.36        3
  144            13785           7.958           10.49        0
  145            13880           7.957            10.8        0
  146            13975           7.957           11.04        1
  147            14070           7.956            10.4        0
  148            14165           7.954           9.872        0
  149            14260           7.954           9.718        1
  150            14355           7.953           9.984        0

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
  151            14450           7.953           9.994        1
  152            14545           7.953           9.339        2
  153            14640           7.953           9.656        3
  154            14735           7.915           9.325        0
  155            14830           7.915           9.813        1
  156            14925           7.915           9.717        2
  157            15020           7.915           9.765        3
  158            15115           7.914           9.305        0
  159            15210           7.906           8.667        0
  160            15305           7.905           8.862        0
  161            15400           7.903           9.023        0
  162            15495           7.903           8.826        1
  163            15590           7.903           9.219        2
  164            15685           7.903           9.257        0
  165            15780           7.892           8.922        0
  166            15875           7.891             8.8        0
  167            15970           7.879           8.847        0
  168            16065           7.879           8.906        1
  169            16160           7.876           9.178        0
  170            16255           7.867           8.323        0
  171            16350           7.867           8.495        1
  172            16445           7.859           8.496        0
  173            16540           7.859           8.397        1
  174            16635           7.859           8.264        2
  175            16730           7.859           8.557        3
  176            16825           7.856           8.358        0
  177            16920           7.856           8.288        1
  178            17015           7.849           8.275        0
  179            17110           7.849           8.292        1
  180            17205           7.849           8.447        0

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
  181            17300           7.849           8.236        1
  182            17395           7.848           8.225        0
  183            17490           7.846             8.2        0
  184            17585           7.846           8.172        1
  185            17680           7.844           8.161        0
  186            17775           7.844            8.16        1
  187            17870           7.843           8.069        0
  188            17965           7.842           8.056        0
  189            18060           7.842           8.081        1
  190            18155           7.839           8.013        0
  191            18250           7.839           7.942        1
  192            18345           7.839           7.942        2
  193            18440           7.838           7.936        0
  194            18535           7.836             7.9        0
  195            18630           7.836           7.896        1
  196            18725           7.836           7.886        2
  197            18820           7.836            7.88        0
  198            18915           7.836            7.87        0
  199            19010           7.836           7.864        0
  200            19105           7.836           7.841        0
ga stopped because it exceeded options.MaxGenerations.

thick_pred = kineticModel(nParams,time_vector,temp_vector);
MAPE = mean(abs((thick_exp_vector - thick_pred) ./ thick_exp_vector)) * 100;

thick_pred_mtx = reshape(thick_pred,numel(time), numel(temp)).';        % Used in 'surf' Plot


%% 8) Display Results
disp(['Q  = ',num2str(nParams(2)),' J/mol'])

Q  = 78.3864 J/mol

disp(['k0 = ',num2str(nParams(1))])

k0 = 364.0751

disp(['RMSE = ',num2str(fval)])

RMSE = 7.8356

disp(['MAPE = ',num2str(MAPE),' %'])

MAPE = 21.107 %

figure
surf(time, temp, thick_exp, DisplayName='Data')
hold on
surf(time_grid, temp_grid, thick_pred_mtx, FaceAlpha=0.5, EdgeColor='r', DisplayName='Function Fit')
hold off
xlabel('Time')
ylabel('Temperature (K)')
zlabel('Thickness')
title('Unconstrained Parameter Fit')
legend(Location='best')
view(40,30)

●

.

EDIT -- (13 Feb 2026 at 18:25)

The transposed 'thick_pred_mtx' seems to more closely approximate the data than if it is not transposed.

.