Experimental Design for Missing Physics

Abstract

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 (5)

• Figure 1: First experiment. The three states of the bioreactor, and the missing physics $\mu$. 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 $C_s$. 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 For the state $V$, all these lines coincide, and have been replaced by a black line.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 $\mu$. 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 $C_s$. 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 For the state $V$, all these lines coincide, and have been replaced by a black line.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.
• Figure 5: Selected output of the third experiment: The state $C_s$, and the missing physics $\mu$. 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 $C_s$. 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.