IS-PINN: A Fast and Efficient Training of Physics-Informed Neural Networks Using Invariant Subspace Reductions for Solving Nonlinear Diffusion Equations

We propose the invariant subspace physics-informed neural network (IS-PINN), a novel approach that combines the invariant subspace method (ISM) with deep learning to solve nonlinear diffusion equations. By restricting the solution to a prescribed invariant subspace spanned by exact basis functions, IS-PINN eliminates spatial dependencies and transforms the nonlinear diffusion equation into a dynamical system. The temporal coefficients are learned by a neural network with hard-constrained initial conditions via least-squares projection. We validate IS-PINN on five problems: the semilinear heat equation, the Riabouchinsky–Proudman–Johnson equation (RPJE), the porous medium equation, the quasilinear parabolic equation, and the (1+2)D reaction–diffusion equation. Numerical results demonstrate that IS-PINN outperforms standard PINNs, achieving mean relative $L_2$ errors of $1.43\times {10}^{-5}$–$1.26\times {10}^{-4}$ over five random seeds, with mean training time reduced by approximately 74%–98%.