differential equations - Fitting multiple data with model and NDSolve with different initial conditions, and other shared parameters

I know that there are already questions about fitting multiple datasets and about NDSolve and about shared and non shared parameters, but I tried to apply them and some things are still not clear.

Here is my equation :

l = 10^(-5)
k = 1/l
chic = 0.5
T = 100

eq = {R'[t] == -a[t]*R[t] + b[t], 
  b'[t] == beta/2*(Tanh[(chi[t] - chic)*k] - 1), 

  a'[t] == -alpha/2*(Tanh[(chi[t] - chic)*k] - 1), 
  chi'[t] == -kappa*R[t]*(chi[t] - 2*chic), a[0] == a0, b[0] == b0, 
  R[0] == R0, chi[0] == 0}

I want to fit with regards to the variables : $alpha, beta, kappa, a0, b0$ as shared parameters and $R0$ as non shared parameter, meaning it would be different fr each one.

The joined data is given as an appendix just afterwards.

The non-joined data (meaning the 5 data-sets separately) looks like that :

So I tried to change $R0$ as a variable, and I got inspired by the answer of @JimB in Finding NonlinearModelFit of multiple data sets with the same parameters and in two dimensions :

model[alpha_?NumberQ, beta_?NumberQ, kappa_?NumberQ, a0_?NumberQ, 

  b0_?NumberQ] :=  (model[alpha, beta, kappa, a0, b0] = 
   Module[{R, chi, b, a, t, R0}, 
    First[R /. 
      NDSolve[{D[R[t, R0], t] == -a[t, R0]*R[t, R0] + 
          b[t, R0], 
        D[b[t, R0], t] == beta/2*(Tanh[(chi[t, R0] - chic)*k] - 1), 
        D[a[t, R0], t] == -alpha/2*(Tanh[(chi[t, R0] - chic)*k] - 1), 
        D[chi[t, R0], t] == -kappa*(chi[t, R0] - 2*chic), 
        a[0, R0] == a0, b[0, R0] == b0, R[0, R0] == R0, chi[0,R0] == 0}, {R, b, 
        a, chi}, {t, 0, T}, {R0, 0, 300}]]]);

nlm = NonlinearModelFit[data, 
   {model[alpha, beta, kappa, a0, b0][t, 
    R0], alpha >= 0, beta >= 0, kappa >= 0, a0 >= 0, b0 >= 0}, {{alpha, 0.1}, { beta, 0.1}, { kappa, 0.05}, {a0, 0.01}, {b0,
      3}}, {t, R0}];
nlm["BestFitParameters"]

The parameters are believed to be around :

alpha = 0.1

beta= 0.1
kappa = 0.05
a0 = 0.01
b0 = 3

But it didn't work... :

NonlinearModelFit::nrnum: The function value 1/2 ((-22.6124+R$3721[3.,22.])^2+(-119.51+R$3721[3.,119.])^2+(-24.738+R$3721[6.,22.])^2+(-60.1536+R$3721[6.,60.])^2+(-126.123+R$3721[6.,119.])^2+(-16.8895+R$3721[9.,17.])^2+(-25.4959+R$3721[9.,22.])^2+(-57.9807+R$3721[9.,60.])^2+(-110.446+R$3721[9.,119.])^2+(-17.3404+R$3721[12.,17.])^2+(-26.1946+R$3721[12.,22.])^2+(-60.9089+R$3721[12.,60.])^2+(-110.332+R$3721[12.,119.])^2+<<25>>+(-200.187+R$3721[27.,185.])^2+(-20.6519+R$3721[30.,17.])^2+(-34.5678+R$3721[30.,22.])^2+(-68.705+R$3721[30.,60.])^2+(-111.198+R$3721[30.,119.])^2+(-199.25+R$3721[30.,185.])^2+(-19.4591+R$3721[33.,17.])^2+(-35.9263+R$3721[33.,22.])^2+(-68.2107+R$3721[33.,60.])^2+(-109.903+R$3721[33.,119.])^2+(-198.411+R$3721[33.,185.])^2+(-20.6855+R$3721[36.,17.])^2+<<819>>) is not a real number at {alpha,beta,kappa,a0,b0} = {0.1,0.1,0.05,0.01,3.}.

I assume there is an issue with $R0$, but I don't get where.

How could I proceed ?

Also, I don't know how I could fix a priori the initial conditions for each fit in order to extract only the shared parameters.

DATA

MathematicaStackExchange doesn't give the possibility to enter to much characters. I can give only the joined data.

1. joined data with R0 as a variable

Here is the joined data.

data={{9., 17., 16.8895}, {12., 17., 17.3404}, {15., 17., 17.1633}, {18., 
  17., 19.3417}, {21., 17., 17.9899}, {24., 17., 19.9677}, {27., 17., 
  19.4362}, {30., 17., 20.6519}, {33., 17., 19.4591}, {36., 17., 
  20.6855}, {39., 17., 20.1952}, {42., 17., 21.9949}, {45., 17., 

  21.0234}, {48., 17., 22.7408}, {51., 17., 22.3908}, {54., 17., 
  25.0918}, {57., 17., 23.5989}, {60., 17., 26.0703}, {63., 17., 
  24.5605}, {66., 17., 27.2539}, {69., 17., 26.1619}, {72., 17., 
  28.4762}, {75., 17., 27.5854}, {78., 17., 29.8393}, {81., 17., 
  28.3553}, {84., 17., 30.3221}, {87., 17., 29.675}, {90., 17., 
  31.5653}, {93., 17., 30.5337}, {96., 17., 33.3734}, {99., 17., 
  31.6876}, {102., 17., 34.1503}, {105., 17., 33.3065}, {108., 17., 
  35.3291}, {111., 17., 33.9209}, {114., 17., 36.773}, {117., 17., 
  35.4094}, {120., 17., 41.5902}, {123., 17., 36.1305}, {126., 17., 
  37.971}, {129., 17., 36.402}, {132., 17., 39.1158}, {135., 17., 

  38.0177}, {138., 17., 40.8558}, {141., 17., 39.6065}, {144., 17., 
  40.9749}, {147., 17., 39.8896}, {150., 17., 41.8237}, {153., 17., 
  40.5802}, {156., 17., 42.3858}, {159., 17., 40.6619}, {162., 17., 
  44.4442}, {165., 17., 45.4162}, {168., 17., 46.1884}, {171., 17., 
  44.6008}, {174., 17., 47.1647}, {177., 17., 45.3808}, {180., 17., 
  46.5859}, {183., 17., 45.3035}, {186., 17., 47.6604}, {189., 17., 
  46.6771}, {192., 17., 45.9242}, {195., 17., 46.767}, {198., 17., 
  44.6899}, {201., 17., 46.6628}, {204., 17., 46.1571}, {207., 17., 
  46.5555}, {210., 17., 44.835}, {213., 17., 45.1423}, {216., 17., 
  45.1954}, {219., 17., 45.309}, {222., 17., 47.7791}, {225., 17., 

  46.7777}, {228., 17., 48.135}, {231., 17., 45.6493}, {234., 17., 
  45.8933}, {237., 17., 46.1803}, {240., 17., 46.7285}, {243., 17., 
  46.8063}, {246., 17., 47.1679}, {249., 17., 46.8787}, {252., 17., 
  47.2715}, {255., 17., 47.5362}, {258., 17., 48.9234}, {261., 17., 
  47.5456}, {264., 17., 53.5554}, {267., 17., 52.5704}, {270., 17., 
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  51.193}, {336., 22., 48.5713}, {339., 22., 49.5028}, {342., 22., 

  49.4166}, {345., 22., 50.0304}, {348., 22., 50.9326}, {351., 22., 
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   89.9111}, {279., 119., 90.5619}, {282., 119., 88.4012}, {285., 
  119., 85.5809}, {288., 119., 76.692}, {291., 119., 80.0753}, {294., 
  119., 90.1118}, {297., 119., 91.8565}, {300., 119., 85.0882}, {303.,
   119., 89.1269}, {306., 119., 96.8869}, {309., 119., 
  75.4618}, {312., 119., 96.3013}, {315., 119., 89.4435}, {318., 119.,
   103.21}, {321., 119., 94.6233}, {324., 119., 102.48}, {327., 119., 
  96.7664}, {330., 119., 84.2408}, {333., 119., 97.3822}, {336., 119.,
   74.2619}, {339., 119., 87.2886}, {342., 119., 118.024}, {345., 
  119., 113.648}, {348., 119., 112.4}, {351., 119., 107.295}, {354., 

  119., 111.618}, {357., 119., 112.181}, {360., 119., 112.119}, {363.,
   119., 90.6252}, {366., 119., 106.837}, {369., 119., 
  99.7227}, {372., 119., 97.5255}, {375., 119., 108.211}, {378., 119.,
   117.211}, {381., 119., 97.9301}, {384., 119., 104.567}, {387., 
  119., 117.343}, {390., 119., 121.622}, {393., 119., 106.117}, {396.,
   119., 116.022}, {399., 119., 118.856}, {402., 119., 
  106.854}, {405., 119., 112.418}, {408., 119., 112.79}, {411., 119., 
  112.225}, {414., 119., 116.686}, {417., 119., 111.297}, {420., 119.,
   115.404}, {423., 119., 117.563}, {426., 119., 116.243}, {429., 
  119., 119.805}, {432., 119., 112.863}, {435., 119., 103.505}, {438.,

   119., 116.846}, {441., 119., 115.508}, {444., 119., 
  115.579}, {447., 119., 101.756}, {450., 119., 102.848}, {453., 119.,
   112.506}, {456., 119., 113.93}, {459., 119., 116.386}, {462., 119.,
   108.138}, {465., 119., 108.635}, {468., 119., 110.514}, {471., 
  119., 108.217}, {474., 119., 110.008}, {477., 119., 95.7788}, {480.,
   119., 92.8073}, {483., 119., 104.382}, {486., 119., 98.77}, {489., 
  119., 112.527}, {492., 119., 94.6092}, {495., 119., 89.2861}, {498.,
   119., 92.0002}, {501., 119., 98.7618}, {504., 119., 
  105.274}, {507., 119., 96.7057}, {510., 119., 93.5207}, {513., 119.,
   90.5992}, {516., 119., 87.1486}, {519., 119., 103.466}, {522., 

  119., 100.133}, {525., 119., 120.605}, {528., 119., 125.717}, {12., 
  185., 185.791}, {15., 185., 199.035}, {18., 185., 197.796}, {21., 
  185., 185.256}, {24., 185., 199.576}, {27., 185., 200.187}, {30., 
  185., 199.25}, {33., 185., 198.411}, {36., 185., 198.288}, {39., 
  185., 194.506}, {42., 185., 189.658}, {45., 185., 191.203}, {48., 
  185., 185.757}, {51., 185., 183.642}, {54., 185., 183.513}, {57., 
  185., 186.524}, {60., 185., 182.793}, {63., 185., 182.218}, {66., 
  185., 182.045}, {69., 185., 176.614}, {72., 185., 182.432}, {75., 
  185., 181.409}, {78., 185., 182.438}, {81., 185., 179.939}, {84., 
  185., 182.435}, {87., 185., 181.521}, {90., 185., 176.654}, {93., 

  185., 175.39}, {96., 185., 179.446}, {99., 185., 173.541}, {102., 
  185., 176.645}, {105., 185., 176.715}, {108., 185., 173.915}, {111.,
   185., 173.14}, {114., 185., 173.045}, {117., 185., 160.089}, {120.,
   185., 165.306}, {123., 185., 165.906}, {126., 185., 
  165.712}, {129., 185., 159.285}, {132., 185., 163.219}, {135., 185.,
   156.287}, {138., 185., 150.445}, {141., 185., 153.388}, {144., 
  185., 138.083}, {147., 185., 137.152}, {150., 185., 133.003}, {153.,
   185., 130.634}, {156., 185., 131.832}, {159., 185., 
  136.142}, {162., 185., 133.906}, {165., 185., 130.929}, {168., 185.,
   136.717}, {171., 185., 129.749}, {174., 185., 148.377}, {177., 

  185., 133.068}, {180., 185., 149.921}, {183., 185., 134.802}, {186.,
   185., 150.543}, {189., 185., 138.678}, {192., 185., 147.06}, {195.,
   185., 143.604}, {198., 185., 143.368}, {201., 185., 
  140.587}, {204., 185., 138.171}, {207., 185., 140.699}, {210., 185.,
   137.346}, {213., 185., 126.241}, {216., 185., 131.743}, {219., 
  185., 134.835}, {222., 185., 134.086}, {225., 185., 137.185}, {228.,
   185., 135.892}, {231., 185., 141.62}, {234., 185., 135.963}, {237.,
   185., 133.382}, {240., 185., 134.258}, {243., 185., 
  141.568}, {246., 185., 137.642}, {249., 185., 131.681}, {252., 185.,
   132.635}, {255., 185., 134.506}, {258., 185., 136.089}, {261., 

  185., 138.973}, {264., 185., 141.048}, {267., 185., 133.785}, {270.,
   185., 133.245}, {273., 185., 116.408}, {276., 185., 123.9}, {279., 
  185., 120.251}, {282., 185., 116.984}, {285., 185., 135.753}, {288.,
   185., 123.026}, {291., 185., 112.116}, {294., 185., 
  134.164}, {297., 185., 134.548}, {300., 185., 129.032}, {303., 185.,
   116.97}, {306., 185., 113.993}, {309., 185., 99.4695}, {312., 185.,
   97.4854}, {315., 185., 100.422}, {318., 185., 117.461}, {321., 
  185., 99.4758}, {324., 185., 106.366}, {327., 185., 108.271}, {330.,
   185., 104.738}, {333., 185., 117.487}, {336., 185., 
  101.704}, {339., 185., 101.32}, {342., 185., 112.97}, {345., 185., 

  96.6092}, {348., 185., 99.2531}, {351., 185., 120.19}, {354., 185., 
  124.284}, {357., 185., 130.082}, {360., 185., 121.699}, {363., 185.,
   108.539}, {366., 185., 103.98}, {369., 185., 100.293}, {372., 185.,
   94.7848}, {375., 185., 103.281}, {378., 185., 114.4}, {381., 185., 
  94.8752}, {384., 185., 101.51}, {387., 185., 104.285}, {390., 185., 
  107.424}, {393., 185., 112.506}, {396., 185., 104.061}, {399., 185.,
   113.713}, {402., 185., 136.378}, {405., 185., 134.92}, {408., 185.,
   139.111}, {411., 185., 143.397}, {414., 185., 139.998}, {417., 
  185., 137.19}, {420., 185., 143.812}, {423., 185., 133.346}, {426., 
  185., 141.8}, {429., 185., 136.171}, {432., 185., 137.842}, {435., 

  185., 147.509}, {438., 185., 140.488}, {441., 185., 142.855}, {444.,
   185., 151.992}, {447., 185., 145.348}, {450., 185., 
  138.757}, {453., 185., 135.964}, {456., 185., 140.381}, {459., 185.,
   143.697}, {462., 185., 136.854}, {465., 185., 129.477}, {468., 
  185., 138.181}, {471., 185., 142.726}, {474., 185., 143.633}, {477.,
   185., 133.913}, {480., 185., 157.635}, {483., 185., 
  147.941}, {486., 185., 142.015}, {489., 185., 130.545}, {492., 185.,
   141.941}, {495., 185., 142.863}, {498., 185., 135.462}, {501., 
  185., 139.637}, {504., 185., 128.002}, {507., 185., 140.211}, {510.,
   185., 140.209}, {513., 185., 132.36}, {516., 185., 141.088}, {519.,

   185., 142.756}, {522., 185., 152.256}, {525., 185., 
  164.725}, {528., 185., 153.737}}

Answer

The subject of parameter fitting comes up frequently on MSE. Parameter fitting is a difficult subject and will depend on your data quality, your model, and your intial guesses. I have been dabbling with StringTemplates as a potential way to encapsulate some of the basic parameter fitting work flow.

• Use ParametricNDSolveValue to create the model.


• Use StringTemplates to handle lists of parameters and variables.


• Generate a Manipulate slider model to debug model and understand the effects of parameter changes.


• Transfer initial guesses from manipulate to perform a fit.

I commented the code so I hope it self explanatory. First assign the constants and prep the data.

(* Evaluate data first *)
(* Constants *)
l = 10^(-5);
k = 1/l;
chic = 0.5;
T = 550;
(* Get unique R0s *)

R0s = Union@data[[All, 2]];
(* Subset Matching R0 and Delete 2nd Column *)
rdat = (Cases[data, {_, #, _}][[All, {1, 3}]] & /@ R0s);

Now, set up the equations and the Manipulate slider to view how the model behaves and try to improve initial parameters estimates.

(* Generate System of Differential Equations *)
e1 = R'[t] == -a[t]*R[t] + b[t];
e3 = b'[t] == beta/2*(Tanh[(chi[t] - chic)*k] - 1);
e2 = a'[t] == -alpha/2*(Tanh[(chi[t] - chic)*k] - 1);
e4 = chi'[t] == -kappa*R[t]*(chi[t] - 2*chic);

ics = {a[0] == a0, b[0] == b0, R[0] == R0, chi[0] == 0};
eqns = {e1, e2, e3, e4}~Join~ics;
(*Variables*)
vbles = {R, a, b, chi};
(*Parameters with target and desired ranges*)
mat = {
   {alpha, 0.1, 0.00025, 0.5},
   {beta, 0.1, 0.00025, 0.5},
   {kappa, 0.05, 0.0125, 0.1},
   {a0, 0.01, 0.00005, 0.1},

   {b0, 3, 1, 6},
   {R0, 17, 17, 185}
   };
(* reduce the matrix because R0 does not participate in parameter \
fits *)
rmat = mat[[1 ;; -2]];
(* Build Manipulate sliders *)
sfun =  StringRiffle[(StringTemplate[
         "{{`1`,`2`},`3`,`4`,Appearance\[Rule]\"Labeled\"}"] @@ #) & \
/@ #, ","] &;

sliders = sfun[rmat];
(* Extract Parameters from mat *)
parms = mat[[All, 1]];
rparms = rmat[[All, 1]];
(* Create String Representations of parms *)
sparms = StringRiffle[ToString[#] & /@ parms, ","];
rsparms = StringRiffle[ToString[#] & /@ rparms, ","];
(* Create patterns and string reps of parameters *)
pats = Pattern @@@ (#*_ & /@ parms);
spats = StringRiffle[ToString[#] & /@ pats, ","];

(* List Plot of the data *)
lp = Graphics[{Hue[#2/185], PointSize[0.01], Point[{#1, #3}]} & @@@ 
    data, Axes -> True];
(* ParametricNDSolveValue *)
pfun = ParametricNDSolveValue[eqns, vbles, {t, 0, T}, parms];
(*Create an appropriate model function to fit*)
modelstring = "(#[[1]])&";
(* Create some PlotLegends *)
pl = ",PlotLegends\[Rule]{" <> 
   StringRiffle["\"R0=" <> ToString[#] <> "\"" & /@ R0s, ","] <> "}";

(* Build the model expression *)
ToExpression[
 StringTemplate[
   "model[`pats`][t_]:=`ms`@Through[pfun[`params`][t],List]\
/;And@@NumericQ/@{`params`};"][<|"pats" -> spats, "params" -> sparms, 
   "ms" -> modelstring|>]]
(* Create slider model *)
globalstring = 
  StringTemplate["global={`params`};"][<|"params" -> rsparms|>];
mantemp = 

  "Manipulate[`g`\[IndentingNewLine]Show[lp,Plot[Evaluate@({model[\
alpha,beta,kappa,a0,b0,#][t]}&/@R0s),{t,0,T},PlotRange\[Rule]{0,200}`\
pl`],ImageSize->Large],`sliders`]";
ToExpression@
 StringTemplate[mantemp][<|"sliders" -> sliders, "params" -> rsparms, 
   "pl" -> pl, "g" -> globalstring|>]
(*Display global variable*)
Dynamic@global

Now set up to fit the funtions for each R0 value.

(* Grab The initial parameter guesses *)
initguess = MapThread[List, {rparms, First@Dynamic@global}];
(* Create a fit function to operate on different R0s *)
fitfn = FindFit[rdat[[#]], 
    model[alpha, beta, kappa, a0, b0, R0s[[#]]][t], initguess, t, 
    Method -> "Gradient"] &;
(* Perform Fits on R0s *)
fits = fitfn[#][[All, 2]] & /@ Range@Length@R0s;
(* Display Results *)

fits // MatrixForm
Mean@fits

The data is noisy leading to some dodgy results for the high R0. You can experiment with different fitting options, but you may need to improve your model and/or your data acquisition.

As requested, here is a way to fit per data set. I also allowed $R_0$ to be fit using the column value as an initial guess. In this case, each fitted row is plotted. A word of caution, some fitting methods will run forever, so you may need to experiment.

(* Grab The initial parameter guesses from dynamic variable of slider \
*)
initguess = 

  MapThread[List, {parms, (First@Dynamic@global)~Join~{R0s[[#]]}}] &;
(* Create a fit function to operate on different R0s *)
fitfn = FindFit[rdat[[#]], model[alpha, beta, kappa, a0, b0, R0][t], 
    initguess[#], t, Method -> "Gradient", WorkingPrecision -> 10] &;
(* Perform Fits on R0s *)
(*fits = fitfn[#][[All,2]]&/@Range@Length@R0s;*)
fits = fitfn[#][[All, 2]] & /@ {1, 2, 3, 4, 5};
(* Display Results *)
fits // MatrixForm
mfit = Mean@fits

mat2 = rmat;
mat2[[All, 2]] = mfit[[1 ;; -2]];
Show[{lp, 
  Plot[Evaluate@((model @@ #)[t] & /@ fits), {t, 0, T}, 
   PlotRange -> {0, 200}, 
   PlotLegends -> {"R0=17.", "R0=22.", "R0=60.", "R0=119.", 
     "R0=185."}]}, ImageSize -> Large]