Modeling the thermal production of industrial and domestic heat producers presents a complex and challenging problem, primarily due to the inherent difficulties in obtaining accurate and comprehensive observational data. Heat production systems, whether on an industrial scale or within domestic settings, involve a multitude of dynamic processes that are influenced by a variety of factors. These factors include variations in fuel quality, changes in operational conditions, equipment efficiency, and external environmental influences such as ambient temperature and humidity. Each of these elements can significantly impact the thermal output and efficiency of the heat production system, making accurate modeling a formidable task.

One of the primary challenges in this domain is the scarcity of precise observational data. In many cases, the data available for modeling is either incomplete, inconsistent, or outdated, leading to models that may not accurately reflect the real-world behavior of heat production systems. Furthermore, the sensors and monitoring systems used to collect data often have limitations in terms of resolution, accuracy, and coverage, which further complicates the modeling process.

For illustration, we consider the following observations of the output temperature *T*_{out}:

Observation of the thermal output of the CHP (*T _{out}*).

which are related to the environmental temperature *T*_{env} and the energy input *Q*_{source}:

(Upper) time variation of input energy (*Q _{source}*) and (lower) time variation of the environmental temperature (

_{}

Our challenge is to develop a predictive model *M* that can predict the output temperature Tout considering the environment temperature *T*_{env} and energy input *Q*_{source} .

Considering that scarcity and low quality of the data, developing a predictive model based on the data alone turns out to be a difficult task. In fact, as illustrated by the Figure below, developing a model *M _{d}* based on these observations only, we find that the model is capable to perfectly describe the observations, however by no means is able to describe the overall true dynamics of the output thermal production. This is evident by comparing the true

Comparision between (i) the predicted values of *T*_{out} with model *M _{d}* that considers only the observations as input, (ii) scarce and noisy observations and (iii) the true thermal production (

Based on process knowledge (e.g. Van Riet [2019]), it is possible to describe the thermal production of a CHP with a mathematical model as follows:

Here, *α*, *β* and are engineering parameters related, a.o. to the thermal capacity of the CHP and enveloppe losses, respectively. Thermal efficiency of the CHP is captured with *η*_{th}, *Q*_{source} comprises the energy source, *Q ^{nom}* the CHP nominal load,

Following the concept of Physics Informed Neural Networks (as explained also here), we can add to the machine learning modelling concept also losses that represent the ability of the neural network to comply with the process based model described above: *M _{pinn}*.

As illustrated below, adding also losses that force the neural network to comply also to the process based knowledge mathematical model, we find that a neural network with the same structure as above (*M _{pinn}*), is able to closely describe the overall dynamics of the output thermal production.

This result clearly shows the advantage of this approach by exemplifying that by adding information of the (process-based) mathematical model describing the dynamics allows to predict the true dynamics, even in the case that only scare and low quality observations are present.

Comparision between (i) the predicted values of *T*_{out} with model *M _{pinn}* that considers both the observations as the governing equations, (ii) scarce and noisy observations and (iii) the true thermal production (

We showcase the advantages of Physics-Informed Neural Networks (PINNs) to model the output temperature of a CHP. We show that PINNs allow to accurately predict the output temperature T_{out} of a CHP hydronic system even when observations of this quantity of interest are scarce and noisy.

By leveraging Physics-Informed Neural Networks (PINNs), we can integrate fundamental physical laws with data-driven approaches, enhancing the accuracy and reliability of thermal production models even when observational data is limited. The adoption of this technology is paramount for our software product, *Foresight*, enabling it to provide more precise and actionable insights into heat production processes across diverse industrial and domestic applications.

Other common approaches which combine mathematical modelling with machine learning technology comprise __surrogate modelling__. An illustrative industry example on how PropheSea combined process-based models with machine learning technology considering this methodology is provided here.

Finally, the observant ready perhaps questions to which extent the presented results of *M*_{pinn} depend on the accurate values of the engineering parameters used while integrating the process based model into the data driven methodology. The cool thing about this methodology is that (i) it allows to estimate the values of the engineering parameters; and (2) the resulting model is reasonably robust to changes in parameter values. The illustration of these results is, however, for a different post.