Tao Zhou

Jizu Huang, Yue Qian, Tao Zhou

Evolution partial differential equations (PDEs) describe time-dependent physical systems governed by differential laws and arise widely across science and engineering. In recent years, operator learning has emerged as a powerful and efficient paradigm for solving evolution PDEs by learning mappings between infinite-dimensional function spaces, enabling solution prediction without explicit time-step integration. In this work, we propose PI-DOSnet, a physics-informed operator learning framework built upon DOSnet and operator splitting. Unlike purely data-driven operator learning methods, PI-DOSnet incorporates physical constraints during training, allowing it to operate even in the absence of paired input-output data. Once trained, PI-DOSnet performs long-time inference of PDE solutions through an iterative strategy. We analyze the linear stability and approximation error of PI-DOSnet and demonstrate its accuracy, efficiency, and robustness through multiple numerical experiments. Moreover, for the Allen--Cahn equation, PI-DOSnet achieves energy stable solutions even with a large time-step size.

Zhiyuan Hui, Changtao Sheng, Bihao Su et al.

In this paper, we propose a projected Walk on Spheres method (PWoS) for screened Poisson equations on embedded manifolds. The method employs local extensions together with the Green representation in local Euclidean balls, coupled with a closest-point projection that maps the boundary samples back to the manifold. This formulation yields a meshfree and highly parallelizable stochastic recursion in the ambient Euclidean space, rather than a direct discretization of the Laplace-Beltrami operator on the manifold. To recover the intrinsic geometric structure of the problem, we introduce a compensation term for the discrepancy between the ambient Laplacian of the closest-point extension and the Laplace-Beltrami operator on the manifold, together with an adaptive radius strategy determined by local geometric and boundary information. Under the geometric projection and prescribed compensation accuracy, we establish mean-square error estimates for the proposed Monte Carlo method in both the boundary and closed-manifold settings. Extensive numerical examples on parametrized, implicit, high-dimensional (up to 1000 dimensions), and point-cloud manifolds are presented to illustrate the convergence and efficiency of the proposed method across different geometries.