Authors: Arno Strouwen, Sebastián Micluţa-Câmpeanu
Organizations: JuliaHub, MA 02111-1929 USA · Biosystems Department, KULeuven, Leuven, Belgium · Faculty of Physics, University of Bucharest, Bucharest, Romania
For most process systems, knowledge of the model structure is incomplete. This missing physics must then be learned from experimental data. Recently, a combination of universal differential equations and symbolic regression has become a popular tool to discover these missing physics. Universal differential equations employ neural networks to represent missing parts of the model structure, and symbolic regression aims to make these neural networks interpretable. These machine learning techniques require high-quality data to successfully recover the true model structure. To gather such informative data, a sequential experimental design technique is developed which is based on optimally discriminating between the plausible model structures suggested by symbolic regression. This technique is then applied to discovering the missing physics of a bioreactor.
Figures & tables
Figure 1: First experiment. The three states of the bioreactor, and the missing physics μ . The blue solid lines corresponds to the true system, while the blue dots correspond to measured values. The orange solid lines correspond to predictions made by the UDE, while the orange dots correspond to the predicted values at measured Cs . The dashed lines correspond to predictions made by the plausible model structures. For the state V , all these lines coincide, and have been replaced by a black line.
Figure 2: Top: Optimal control for the second experiment. Bottom: predictions made by the plausible model structures.
Figure 3: Second experiment. The three states of the bioreactor, and the missing physics μ . The blue solid lines corresponds to the true system, while the blue dots correspond to measured values. The orange solid lines correspond to predictions made by the UDE, while the orange dots correspond to the predicted values at measured Cs . The orange dots not only represent predictions for the second experiment, but also the first. The dashed lines correspond to predictions made by the plausible model structures. For the state V , all these lines coincide, and have been replaced by a black line.
Figure 4: Top: Optimal control for the third experiment. Bottom: predictions made by the plausible model structures.
Score
Equation
3.604e+01
0.23468
4.571e-01
Cs * 0.010807
5.788e-02
sin( Cs * 0.011293)
7.005e-01
Cs / ( Cs - -50.467)
2.787e+00
exp(-1.5505 / Cs ) * 0.37874
3.076e+00
Cs / (( Cs - -4.631) / 0.42347)
Table 1: Symbolic regression hall of fame after three experiments. Monod kinetics has the largest score.
Figure 5: Selected output of the third experiment: The state Cs , and the missing physics μ . The blue solid lines corresponds to the true system, while the blue dots correspond to measured values. The orange solid lines correspond to predictions made by the UDE, while the orange dots correspond to the predicted values at measured Cs . The orange dots not only represent predictions for the third experiment, but also the first and second. The dashed lines correspond to predictions made by the plausible model structures.
Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel · Department of Molecular Cell Biology, Weizmann Institute of Science, Rehovot, Israel · Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates +2