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

TL;DR

This paper introduces a new unconstrained PDE formulation framework that converts constrained PDEs into a sequence of unconstrained PDEs, enabling the development of a novel finite neuron method with proven optimal error bounds for solving elliptic equations.

Contribution

The paper presents a novel framework for unconstrained PDE formulation and introduces a finite neuron method with proven optimal H1 norm error bounds for elliptic equations.

Findings

The proposed algorithm achieves solutions with optimal H1 norm error.

Widely used penalized PDE methods may not always yield optimal error bounds.

Numerical tests confirm the effectiveness of the new approach.

Abstract

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.