Journal Article 65K99 68T07 68W10 Allen-Cahn equation Backward Euler method Convergence analysis Crank–Nicolson scheme Gauss-Newton method Homotopy Homotopy continuation method Kronecker product Machine learning Multiple solutions Neural network discretization Neural networks Nonlinear differential equations Numerical approximation Optimization Partial differential equations Physics-informed neural networks Quasi-Linear Elliptic equations and systems Quasi-Newton method Relaxation Runge-Kutta method Stability Tensor product Variational form