Skip to main content

Table 1 Results for Case Study I

From: Modelling biochemical networks with intrinsic time delays: a hybrid semi-parametric approach

NN τ i   BIC    MSE   NN τ i   BIC    MSE  
   train valid test train valid test    train valid test train valid test
5 0 -12217 -5836 -5997 0.0141 0.0152 0.0210 6 0 -12220 -5869 -6039 0.0139 0.0157 0.0222
2 100 -13118 -6209 -6190 0.0368 0.0350 0.0337 2 110 -13058 -6150 -6157 0.0347 0.0310 0.0315
3 100 -13087 -6273 -6336 0.0350 0.0384 0.0437 3 110 -13043 -6269 -6275 0.0334 0.0381 0.0385
4 100 -11826 -5650 -5888 0.0096 0.0105 0.0170 4 110 -12273 -5805 -5832 0.0151 0.0144 0.0152
5 100 -11386 -5379 -5733 0.0060 0.0059 0.0120 5 110 -12302 -5864 -6008 0.0152 0.0156 0.0210
6 100 -12873 -6174 -6176 0.0265 0.0282 0.0284 6 110 -13162 -6336 -6329 0.0355 0.0392 0.0386
7 100 -13144 -6269 -6176 0.0342 0.0330 0.0273 7 110 -11516 -5572 -5731 0.0066 0.0081 0.0111
2 120 -13047 -6148 -6139 0.0343 0.0309 0.0303 2 130 -13242 -6332 -6371 0.0417 0.0449 0.0486
3 120 -12105 -5782 -5960 0.0130 0.0142 0.0204 3 130 -13076 -6173 -6203 0.0346 0.0314 0.0333
4 120 -11974 -5761 -5891 0.0111 0.0132 0.0171 4 130 -12652 -6087 -6090 0.0221 0.0254 0.0256
5 120 -11436 -5462 -5489 0.0062 0.0068 0.0071 5 130 -11823 -5604 -5676 0.0094 0.0092 0.0107
6 120 -10820 -5170 -5714 0.0033 0.0036 0.0108 6 130 -12679 -6093 -6108 0.0218 0.0240 0.0247
7 120 -12533 -6002 -5881 0.0184 0.0193 0.0151 7 130 -13269 -6384 -6393 0.0388 0.0417 0.0424
2 140 -13069 -6155 -6167 0.0351 0.0313 0.0321 2 160 -13195 -6295 -6257 0.0398 0.0416 0.0385
3 140 -12303 -5805 -5803 0.0158 0.0149 0.0149 3 160 -12252 -5823 -5771 0.0151 0.0155 0.0139
4 140 -13288 -6375 -6384 0.0420 0.0455 0.0464 4 160 -13063 -6186 -6241 0.0334 0.0311 0.0347
5 140 -12537 -6043 -6039 0.0193 0.0225 0.0223 5 160 -12022 -5716 -5909 0.0114 0.0116 0.0171
6 140 -12564 -6067 -6078 0.0194 0.0228 0.0233 6 160 -12052 -5800 -5995 0.0116 0.0133 0.0197
7 140 -11439 -5535 -5994 0.0061 0.0075 0.0189 7 160 -11466 -5431 -5441 0.0063 0.0061 0.0062
2 80, 120 -13016 -6146 -6079 0.0330 0.0305 0.0266 2 120, 160 -12984 -6137 -6027 0.0320 0.0299 0.0240
3 80, 120 -12334 -5860 -5968 0.0162 0.0164 0.0204 3 120, 160 -13115 -6296 -6163 0.0357 0.0397 0.0303
4 80, 120 -11221 -5276 -5566 0.0051 0.0048 0.0087 4 120, 160 -12250 -5872 -5934 0.0145 0.0162 0.0183
5 80, 120 -12780 -6221 -6207 0.0243 0.0314 0.0305 5 120, 160 -12293 -5872 -5984 0.0148 0.0155 0.0194
6 80, 120 -12233 -5837 -5944 0.0136 0.0139 0.0172 6 120, 160 -11240 -5352 -7991 0.0050 0.0052 1.0762
7 80, 120 -11688 -5663 -5630 0.0077 0.0094 0.0088 7 120, 160 -11703 -5623 -6004 0.0078 0.0086 0.0187
2 80, 120,160 -12994 -6144 -6034 0.0321 0.0300 0.0241 5 80, 120, 160 -12487 -5953 -6045 0.0178 0.0178 0.0215
3 80, 120, 160 -11855 -5641 -5937 0.0099 0.0104 0.0189 6 80, 120, 160 -12824 -6193 -6213 0.0244 0.0276 0.0288
4 80, 120, 160 -11879 -5605 -5734 0.0099 0.0092 0.0120 7 80, 120, 160 -12167 -5758 -5774 0.0122 0.0110 0.0113
  1. Effect of structure parameters (number of nodes in the hidden layer, NN, and number and values of time delays) on the performance of the structure displayed in Fig. 1C. For every structure incorporating delays two random initial weight sets were investigated. For those without delays four different random initial weight changes were investigated. At least 25 iterations were carried out for each set of weights. The number of iterations was expanded if network learning was observed during the last iterations. Integration of the material balances along with the differential equations resulting from the sensitivity method for parameter identification is carried out for this simulation case with the dde23 MATLAB function for the studies with delays, and with the ode23 MATLAB function for the ones without delays. This results in higher simulation times, but as the dimension of the set of equations is rather small, the total simulation time is maintainable.