An alternating learning-based collocation method for solving inverse elliptic problems
We propose the Alternating Learning-Based Collocation (ALBC) method for solving inverse elliptic problems. Our approach employs sinusoidal shallow networks as adaptive basis generators. By alternately updating the state variable and the unknown parameter, we decompose the original nonconvex joint optimization problem into a sequence of tractable linear subproblems. This strategy effectively overcomes the fixed-basis limitations of classical collocation methods while avoiding the slow convergence typically encountered in deep learning approaches. Theoretically, we establish stability estimates and prove the convergence of the proposed algorithm. Numerical experiments on five benchmark problems demonstrate the efficacy of ALBC, which consistently outperforms the standard collocation method in accuracy. Furthermore, it achieves performance comparable to or better than that of physics-informed neural networks at a substantially lower computational cost. Finally, the method remains robust under noise levels of up to twenty percent.