Surrogate modeling plays a crucial role in simplifying and approximating complex physical models, making them suitable for large-scale simulations and optimization studies of industrial relevance. Machine learning models, such as neural networks (NNs), are particularly well-suited for this purpose due to their simplicity and strong regression capabilities [1]. However, despite exceptional advancements in machine learning, issues and skepticism regarding the black-box nature and physical inconsistency of these models hinder the adoption of machine learning-based tools (and, more broadly, artificial intelligence) in industrial applications [2, 3]. To mitigate this limitation, significant research has been carried out to enforce known mechanistic relationships between inputs and predictions in NNs. Soft-constrained neural networks represent an approach in which physical equations are included as penalty terms in the loss function [4, 5].

Due to their high energy intensity, buildings play a major role in the current worldwide energy transition. Building models are ubiquitous since they are needed at each stage of the life of buildings, i.e. for design, retrofitting, and control operations. Classical white-box models, based on physical equations, are bound to follow the laws of physics but the specific design of their underlying structure might hinder their expressiveness and hence their accuracy. On the other hand, black-box models are better suited to capture nonlinear building dynamics and thus can often achieve better accuracy, but they require a lot of data and might not follow the laws of physics, a problem that is particularly common for neural network (NN) models. To counter this known generalization issue, physics-informed NNs have recently been introduced, where researchers introduce prior knowledge in the structure of NNs to ground them in known underlying physical laws and avoid classical NN generalization issues. In this work, we present a novel physics-informed NN architecture, dubbed Physically Consistent NN (PCNN), which only requires past operational data and no engineering overhead, including prior knowledge in a linear module running in parallel to a classical NN. We formally prove that such networks are physically consistent -- by design and even on unseen data -- with respect to different control inputs and temperatures outside and in neighboring zones. We demonstrate their performance on a case study, where the PCNN attains an accuracy up to $50\%$ better than a classical physics-based resistance-capacitance model on $3$-day long prediction horizons. Furthermore, despite their constrained structure, PCNNs attain similar performance to classical NNs on the validation data, overfitting the training data less and retaining high expressiveness to tackle the generalization issue.