  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 AI represented by an expert system is limited by the
expressive ability of the model, and data-driven connectionism AI represented
by neural networks is prone to produce predictions that violate physical
mechanisms. In order to fully integrate domain knowledge with observations, and
make full use of the prior information and the strong fitting ability of neural
networks, this study proposes theory-guided hard constraint projection (HCP).
This 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. Based on rigorous mathematical
proofs, theory-guided HCP can ensure that model predictions strictly conform to
physical mechanisms in the constraint patch. The performance of the
theory-guided HCP is verified by experiments based on the heterogeneous
subsurface flow problem. Due to the application of hard constraints, compared
with fully connected neural networks and soft constraint models, such as
theory-guided neural networks and physics-informed neural networks,
theory-guided HCP requires fewer data, and achieves higher prediction accuracy
and stronger robustness to noisy observations.
