Theory-guided hard constraint projection (HCP): A knowledge-based data-driven scientific machine learning method

•Theory-guided HCP is proposed to introduce domain knowledge and prior information as hard constraints into neural networks through projection.•A projection matrix is constructed based on the governing equation, which can be regarded as a non-trainable activation function in the neural networks.•Experiment shows that hard constraint can effectively reduce data demand and increase prediction accuracy with sparse and partial observations. Machine learning models have been successfully used in many scientific and engineering fields. However, it remains difficult for a model to simultaneously utilize domain knowledge and experimental observation data. The application of knowledge-based symbolic artificial intelligence (AI) represented by expert systems is limited by the expressive ability of the model, and data-driven connectionism AI represented by neural networks is prone to produce predictions that might violate physical principles. In order to fully integrate domain knowledge with observations and make full use of the strong fitting ability of neural networks, this study proposes theory-guided hard constraint projection (HCP). This deep learning model converts physical constraints, such as governing equations, into a form that is easy to handle through discretization, and then implements hard constraint optimization through projection in a patch. Based on rigorous mathematical proofs, theory-guided HCP can ensure that model predictions strictly conform to physical mechanisms in the constraint patch. The training process of theory-guided HCP only needs a small amount of labeled data (sparse observation), and it can supervise the model by combining the coordinates (label-free data) with domain knowledge. The performance of the theory-guided HCP is verified by experiments based on a published heterogeneous subsurface flow problem. The experiments show that theory-guided HCP requires fewer data, and achieves higher prediction accuracy and stronger robustness to noisy observations, than the fully connected neural networks and soft constraint models. Furthermore, due to the application of domain knowledge, theory-guided HCP possesses the ability to extrapolate and can accurately predict points outside of the range of the training dataset.