Mitigating Numerical Stiffness in Least-Squares Formulations of Elliptic PDEs for Physics-Informed Neural Networks

What the math gives to ML

The paper identifies a concrete conditioning problem in residual-based PINNs: an $L^2$ interior residual and pointwise boundary MSE can produce badly imbalanced gradient scales for elliptic PDEs. Its transferable mathematical asset is replacing the interior residual norm by an $H^{-1}$ norm, which smooths high-frequency residual components through the inverse elliptic operator and is therefore a principled residual preconditioner rather than an arbitrary loss weight. A practical implementation is to compute the residual on a fixed mesh or collocation grid, apply one differentiable Poisson solve to obtain the inverse-Laplacian filtered residual, and combine this with strongly imposed or appropriately preconditioned boundary conditions.