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

Jiwei Jia, Young Ju Lee, Ruitong Shan

TL;DR

The work introduces an unconstrained formulation that turns constrained PDEs into a sequence of unconstrained PDEs whose solutions converge to the constrained solution. This framework is applied to a finite neuron method for 2nd-order elliptic equations with Dirichlet BCs, and the authors prove that the method yields shallow neural-network solutions with an optimal $H^1$-norm error bound; notably, the first (penalized) element may fail to achieve this bound, while the second and subsequent elements do. Numerical experiments validate convergence and error performance, demonstrating the practical viability of solving constrained PDEs via a sequence of unconstrained problems solved by shallow neural networks. The approach advances the intersection of neural PDE solvers and constrained-constraint enforcement by providing both theoretical guarantees and empirical evidence of improved accuracy over penalized formulations.

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.

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

TL;DR

The work introduces an unconstrained formulation that turns constrained PDEs into a sequence of unconstrained PDEs whose solutions converge to the constrained solution. This framework is applied to a finite neuron method for 2nd-order elliptic equations with Dirichlet BCs, and the authors prove that the method yields shallow neural-network solutions with an optimal -norm error bound; notably, the first (penalized) element may fail to achieve this bound, while the second and subsequent elements do. Numerical experiments validate convergence and error performance, demonstrating the practical viability of solving constrained PDEs via a sequence of unconstrained problems solved by shallow neural networks. The approach advances the intersection of neural PDE solvers and constrained-constraint enforcement by providing both theoretical guarantees and empirical evidence of improved accuracy over penalized formulations.

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.
Paper Structure (29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 29 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 14 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem