Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System

TL;DR

The paper develops OSNN, a neural solver for elliptic PDE-constrained optimization, by solving a reduced KKT system with two neural networks that approximate the state and adjoint. It provides an error analysis that combines weak coercivity bounds with offset Rademacher complexity to derive $L^2(\Omega)$ error bounds for the state, adjoint, and control in terms of network parameters and Monte Carlo sampling size. The approach is demonstrated on linear and semilinear elliptic problems, including box-constrained controls, and is shown to be competitive with existing NN-based methods while offering simpler tuning and convergence guarantees. This work advances neural PDE solvers for control problems by delivering theoretical convergence guarantees and practical, scalable performance for high-dimensional settings.

Abstract

In this work, we investigate a neural network based solver for optimal control problems (without / with box constraint) for linear and semilinear second-order elliptic problems. It utilizes a coupled system derived from the first-order optimality system of the optimal control problem, and employs deep neural networks to represent the solutions to the reduced system. We present an error analysis of the scheme, and provide $L^2(Ω)$ error bounds on the state, control and adjoint in terms of neural network parameters (e.g., depth, width, and parameter bounds) and the numbers of sampling points. The main tools in the analysis include offset Rademacher complexity and boundedness and Lipschitz continuity of neural network functions. We present several numerical examples to illustrate the method and compare it with two existing ones.

Figures (6)

• Figure 1: The approximate optimal control $u^*$ (top), pointwise error $|u^*-\bar{u}|$ (mid), and approximate state $y^*$ (bottom) obtained by three NN-based methods for Example \ref{['exam:hidim-unconstraint']}, cross section at $x_3=x_4=0.5$.
• Figure 2: The training dynamics of the three methods, for Example \ref{['exam:hidim-unconstraint']}. Ticks on the left y-axis refer to cost objective $J$, and that on the right to losses $\widehat{\mathcal{L}}_{\rm nn}(y)$ and $\widehat{\mathcal{L}}_{\rm nn}(p)$.
• Figure 3: The approximate control $u^*$ (top), its pointwise error $|u^*-\bar{u}|$ (mid) and approximate state $y^*$ (bottom) obtained by three NN-based methods for Example \ref{['exam:4d-constraint']}, cross section at $x_3=0,x_4=0$.
• Figure 4: The approximate optimal control $u^*$ (top), its pointwise error $|u^*-\bar{u}|$ (middle), and the state $y^*$ (bottom) obtained by three different methods for Example \ref{['exam:semilinear']}, cross section at $(x_3,x_4)=(0.5,0.5)$.
• Figure 5: The exact solutions (top), approximate solutions (middle) and pointwise errors (bottom) by OSNN for Example \ref{['exam:highdimcpinn']}, cross section at $x_3=x_4=x_5=x_6=0.5$.

Theorems & Definitions (34)

• Remark 2.1
• Lemma 3.1
• proof
• Remark 3.1
• Lemma 3.2
• proof
• Definition 3.1
• Theorem 3.2
• proof
• Proposition 3.1