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

ICLR 2026 Conference Submission15884 Authors

Keywords: physics-informed neural network, pde, conservation, PINN, constraint, integral

Abstract: We propose a novel projection method that guarantees the conservation of integral quantities in Physics-Informed Neural Networks (PINNs). While the soft constraint PINNs use to enforce the structure of partial differential equations (PDEs) enables necessary flexibility during training, it also permits the discovered solution to violate physical laws. To address this, we introduce a projection method that guarantees the conservation of the linear and quadratic integrals, both separately and jointly. We derived the projection formulae by solving constrained non-linear optimization problems and found that our PINN modified with the projection, which we call PINN-Proj, reduced the error in the conservation of these quantities by three to four orders of magnitude compared to the soft constraint and marginally reduced the error in the PDE solution. We also found evidence that the projection improved convergence through improving the conditioning of the loss landscape. Our method holds promise as a general framework to guarantee the conservation of any integral quantity in a PINN if a projection is found and is tractable in neural network training.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 15884