Bangmin Wu

Anxiao Yu, Bangmin Wu, Zhengbang Zha et al.

The Monge-Ampère equation is a fundamental fully nonlinear elliptic partial differential equation that finds extensive applications across multiple disciplines. This study proposes a novel physics-informed neural network integrated with attention feature expansion (PINN-AFE) for its numerical solution. A multi-head attention enhanced feature pool is constructed to enable adaptive nonlinear feature representation, and input convex neural networks are adopted to impose strict convexity of solutions with rigorous theoretical guarantees. Meanwhile, a dynamically weighted loss function combined with hybrid optimization is formulated to accelerate training convergence. Comprehensive numerical experiments validate the accuracy and computational efficiency of the developed framework. The PINN-AFE paradigm is further extended to image processing tasks, delivering high-quality and physically consistent results in both image enhancement and medical image registration scenarios.

Tielei Zhu, Zhihao Ge, Bangmin Wu

The inverse scattering problem for biharmonic waves, governing flexural vibrations of elastic plates, presents fundamental analytical challenges distinct from acoustic inverse problems due to the fourth-order differential operator and higher-order boundary conditions. This paper addresses the reconstruction of impenetrable obstacles with Dirichlet or Neumann boundary conditions from far-field measurements. We establish new factorizations of the far-field operator by considering structures of the biharmonic fundamental solution and the boundary conditions. We rigorously prove that the factorizations satisfy the range identities and derive characterizations of the obstacle's support by the factorization methods, valid for all wavenumbers except the associated transmission eigenvalues. Furthermore, we establish a monotonicity relation for the eigenvalues of the far-field operator, which yields an alternative characterization of the obstacle's support that remains applicable for all wavenumbers. Numerical experiments for the Dirichlet obstacles with various shapes are presented to demonstrate the effectiveness and robustness of the proposed reconstruction schemes.

Ban Li, Bangmin Wu

This work investigates finite element approximations for a general class of elliptic hemivariational inequalities arising in semipermeable media. The proposed model incorporates non-isotropic and heterogeneous diffusion coefficients, alongside both interior and boundary semipermeability terms, extending the isotropic and homogeneous framework examined by Han (2019). The existence and uniqueness of solutions are rigorously established. An optimal a priori error estimate for the linear finite element approximation is derived under appropriate solution regularity assumptions. Numerical experiments are presented to corroborate the theoretical analysis and to confirm the optimal convergence rates for the non-isotropic and heterogeneous case.