Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks
Anthony Baez, Wang Zhang, Ziwen Ma, Lam Nguyen, Subhro Das, Luca Daniel
TL;DR
This work tackles the challenge that standard PINNs enforce physics via soft constraints and may violate integral conservation. It proposes a projection-based hard constraint, PINN-Proj, that forces conservation of linear and quadratic integrals by projecting network outputs onto the conservation manifold with differentiable closed-form updates. The authors derive projection formulas for linear, quadratic, and combined constraints, demonstrate three- to four-order-magnitude reductions in conservation error, and show marginal improvements to the PDE solution and improved loss-landscape conditioning. They also discuss extension to non-conserved and higher-dimensional systems, and outline a general framework for enforcing conservation of any tractable integral in PINNs.
Abstract
We propose a novel projection method that guarantees the conservation of integral quantities in Physics-Informed Neural Networks (PINNs). While the soft constraint that 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 PDE solution error. 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 tractable solution exists.