Ruitong Shan | Searcharxiv

arXiv2024

fOGA: An Orthogonal Greedy Algorithm for Fractional Laplacian Problems

In this paper, we propose a numerical method for fractional Laplace equations that combines finite difference discretization with shallow neural network approximation. The fractional Laplace operator is discretized using a directional representation of Riemann--Liouville type, which leads to a finite difference approximation of the nonlocal operator. In two dimensions, the angular integral is approximated by a quadrature rule, and auxiliary points are introduced along each direction to facilitate the evaluation of the operator. Based on the resulting discrete system, the solution is then represented by a shallow neural network constructed through the orthogonal greedy algorithm (OGA).

math.NA↗

arXiv2024

An Unconstrained Formulation of Some Constrained Partial Differential Equations and its Application to Finite Neuron Methods

In this paper, we present a new framework how a PDE with constraints can be formulated into a sequence of PDEs with no constraints, whose solutions are convergent to the solution of the PDE with constraints. This framework is then used to build a novel finite neuron method to solve the 2nd order elliptic equations with the Dirichlet boundary condition. Our algorithm is the first algorithm, proven to lead to shallow neural network solutions with an optimal H1 norm error. We show that a widely used penalized PDE, which imposes the Dirichlet boundary condition weakly can be interpreted as the first element of the sequence of PDEs within our framework. Furthermore, numerically, we show that it may not lead to the solution with the optimal H1 norm error bound in general. On the other hand, we theoretically demonstrate that the second and later elements of a sequence of PDEs can lead to an adequate solution with the optimal H1 norm error bound. A number of sample tests are performed to confirm the effectiveness of the proposed algorithm and the relevant theory.

math.NA↗