A Neural Network Approach to Thermal Gray-Box Modeling

Abstract

Accurate thermal models are important in precision systems where temperature variations can affect performance, stability, and control. This report explores a gray-box modeling approach for thermal systems, where a lumped thermal network is combined with neural-network parameterizations for selected difficult-to-model model components. The aim is to retain the simplicity and physical interpretability of lumped thermal models while using neural networks to learn complex and operating-dependent relations from data. The proposed method uses a lumped thermal model, selects difficult-to-model and sensitive parameters using physical insight and sensitivity analysis, and represents these parameters using constant, linear, or neural-network functions. The approach is evaluated on a simulated thermal cooling system containing a Peltier element, heat pipe and heat exchangers with fluid-flow. The results show that learning the heat-pipe parameters reduces the prediction error substantially, with the constant-parameter model resulting in 4.2 and 3.1 times larger mean errors compared to the linear and neural-network heat-pipe models, respectively. The linear and neural-network heat-pipe parameterizations perform similarly, differing by only 11 mK, suggesting that neural networks can be used even if simpler models are sufficient. Furthermore, replacing the analytical Peltier equation with a neural network gives similar in-distribution performance, but weaker generalization. When evaluated 40% outside the training range, the Peltier neuralnetwork models increase in error by approximately 7 times, compared with about 3–4 times for the models that retain the analytical Peltier equation. Overall, the results support a constrained gray-box approach where known thermal physics is preserved and data-driven functions are applied only to uncertain, sensitive, or difficult-to-model components. This provides a compact and interpretable model structure that is relevant for applications such as state estimation, fault detection, digital twins, and model predictive control.