Two-scale neural networks for optimal control of linear convection-dominated equations

TL;DR

This paper introduces a two-scale neural network approach for optimal control of convection-dominated equations, effectively capturing sharp layers by using specialized architectures and training strategies.

Contribution

It develops a novel two-scale neural network architecture tailored for convection-dominated PDE control problems, incorporating separate networks for state and adjoint variables.

Findings

The method accurately captures sharp layers in solutions.

Two formulations demonstrate flexibility and effectiveness.

Numerical experiments validate the approach on benchmark problems.

Abstract

We propose a two-scale neural network method for optimal control problems governed by convection-dominated convection-diffusion-reaction equations. Building on two-scale architectures developed for singularly perturbed forward problems, we augment the spatial input with suitably rescaled features that become increasingly important as the diffusion coefficient becomes small. The approach employs separate neural networks for the state and adjoint state variables of the optimality system, reflecting the fact that these quantities develop sharp layers in different parts of the domain due to opposite convection fields. By choosing different center points for the two networks, the architecture naturally aligns with the layer location of each variable. We present two formulations of the method, one based on the first-order optimality conditions and another using penalization of the PDE constraint, and combine them with a successive training strategy that gradually decreases the diffusion coefficient toward its target value. Numerical experiments on benchmark problems illustrate the effectiveness and behavior of the proposed approach.