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