Least-squares enhanced physics-informed learning for singular and ill-posed partial differential equations

• Proposes a least-squares enhanced PINN framework for singular and ill-posed PDEs • Embeds weighted residual norms into PINN loss to mitigate singularities • Incorporates null-space avoiding projection to suppress non-unique solution modes • Combines LSFEM theory with PINNs in a mesh-free deep learning framework • Provides improved accuracy and stability in challenging PDE settings Partial differential equations that exhibit reduced regularity due to geometric singularities or possess nontrivial null space components arising from intrinsic properties of the differential operator pose serious challenges for both conventional numerical methods and standard physics-informed neural networks (PINNs). Least-squares finite element methods (LSFEM) have long provided robust tools for addressing such issues through weighted norm formulations and null-space projection techniques. In this work, we propose a hybrid PINN framework that systematically incorporates key elements of LSFEM to overcome limitations of existing PINNs. By embedding weighted least-squares functionals or projection mechanisms into the PINN architecture, the proposed method effectively handles singularities and ill-posedness within a deep learning paradigm. This approach enhances solution stability, avoids pollution from singular modes, and improves accuracy in problems where standard PINNs struggle.