Locally Imposing Function for Generalized Constraint Neural Networks - A Study on Equality Constraints

TL;DR

This paper introduces a locally imposing function (LIF) scheme for generalized constraint neural networks, enabling local enforcement of equality constraints and improving upon traditional global methods like Lagrange multipliers.

Contribution

The study proposes a novel locally imposing function scheme for equality constraints in GCNNs, allowing local constraint enforcement within the network.

Findings

LIF enables local and explicit constraint enforcement.

GCNN with LIF achieves exact satisfaction of boundary constraints.

LIF offers advantages over traditional global methods like Lagrange multipliers.

Abstract

This work is a further study on the Generalized Constraint Neural Network (GCNN) model [1], [2]. Two challenges are encountered in the study, that is, to embed any type of prior information and to select its imposing schemes. The work focuses on the second challenge and studies a new constraint imposing scheme for equality constraints. A new method called locally imposing function (LIF) is proposed to provide a local correction to the GCNN prediction function, which therefore falls within Locally Imposing Scheme (LIS). In comparison, the conventional Lagrange multiplier method is considered as Globally Imposing Scheme (GIS) because its added constraint term exhibits a global impact to its objective function. Two advantages are gained from LIS over GIS. First, LIS enables constraints to fire locally and explicitly in the domain only where they need on the prediction function. Second, constraints can be implemented within a network setting directly. We attempt to interpret several constraint methods graphically from a viewpoint of the locality principle. Numerical examples confirm the advantages of the proposed method. In solving boundary value problems with Dirichlet and Neumann constraints, the GCNN model with LIF is possible to achieve an exact satisfaction of the constraints.