Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks

The introduction of a novel projection method for Physics-Informed Neural Networks (PINNs) represents a significant advancement in ensuring the conservation of integral quantities, which is crucial for maintaining the integrity of physical laws in computational models. Traditional soft constraints used in PINNs often allow for solutions that deviate from these laws, leading to inaccuracies. The newly proposed method, known as PINN-Proj, effectively addresses this issue by guaranteeing the conservation of both linear and quadratic integrals. By solving constrained non-linear optimization problems, the researchers demonstrated that PINN-Proj reduces errors in conservation by three to four orders of magnitude compared to previous methods, while also marginally improving the accuracy of partial differential equation (PDE) solutions. Additionally, the projection method enhances convergence by improving the conditioning of the loss landscape, suggesting its potential as a general framework f…