# Data-Completion and Model Correction by Means of Evanescent Regularization
###### Abstract  
System components are often regarded as part of a whole system, especially when it comes to data-driven modeling. Thus, subsystem modeling is disregarded in general when building a data-driven response, especially since multiple subsystem outputs are never measured in real applications. However, subsystem knowledge and accurate modeling are of utmost importance when aiming to repair, tune or troubleshoot a system. This work proposes a holistic modeling of subsystems in an embedded system setting. A hybrid modeling starting from the physics-based model is proposed in this work, correcting or enhancing the model, and predicting output variables, even when a measurement is never available for some of those variables. The process relies on the variables' history, and employs an adjoint-free neural ordinary differential equation technique, along with evanescent regularization to enhance the convergence on the unmeasurable variables. The updated model converges to the exact measurements, for both the measurable and the unmeasurable variables. Multiple examples are presented using synthetic data, to allow an easy evaluation of the hidden or unmeasurable variables. The relative error offered by the updated model is around 0.001% for the measurable quantities and 0.1% for the unmeasurable ones.  
model reduction; machine learning; complex systems; partially observable variables; evanescent regularization +
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are never measured, and probably are unmeasurable. Unmeasurable variables have been recently considered by the scientific community, with several application in the engineering field [5]. In [5], the authors used physical modeling and simulation to complement data-driven approach with the missing unmeasurable data, however, there was no data-driven correction methodology to update these unmeasurable data. In [6], the authors aimed to build the most suitable differential equation of a physical domain, through using a pool of multiple linear and non-linear operators. Discovering physical equations can later be used to predict missing data in regions where the data are not measured in the same domain. This is suitable in the case of continuum mechanics and continuous quantities in a domain, obeying to a certain differential equation in space and time. However, this approach is not suitable when the learning parameters that are not correlated between each other through differentiation of the inputs. In another work [7], the authors removed or hid the unmeasurable quantities through substituting the equations of the unmeasurable parameters into the equations of the measurable ones, without a need of computing the unmeasurable values.  
Multiple unmeasurable variables are listed in the literature, and workarounds are being created to overcome limitations [8]. The present work aims at addressing the problem of non-observed quantities that are of interest in a dynamical system. The method relies on the use of [PERSON]'s theorem. In fact, [PERSON]'s theorem affirms that the state of a system's output can be determined from previous historical observation of the same outputs [9]. Thus, considering the historical evolution of a system, one can predict existing dependencies, even if not all the inputs are observable, using mutual information [10].  
This work proposes a holistic framework for model updating and correction through a data-driven methodology, while also updating the unmeasurable variables, even if they are never observed. This approach is novel and has never been addressed until now. The starting point of the proposed methodology is the existence of prior knowledge, a model, which requires enhancement or correction. In Section 2, the available simulation is improved based on measured real data, available only for the measurable quantities of interest. In Section 3, a hybrid modeling of the available simulation, enhancing the results, is built. The methodology is illustrated on an academic use case to clearly illustrate the adopted approach, without any loss of generality.