Direct and Indirect Physics-Informed Neural Networks for Dirichlet Boundary Control of Semilinear Parabolic Equations: A Conditional Error Analysis
Abstract
We study physics-informed neural networks (PINNs) for the Dirichlet boundary control of a semilinear parabolic equation with Tikhonov regularization. Two approaches are considered. A direct PINN parameterizes the state and control by separate networks and minimizes a penalized form of the tracking objective. An indirect PINN instead represents the state, adjoint, and control by unconstrained networks trained jointly to satisfy the first-order optimality system, with the state-control coupling and the homogeneous adjoint boundary and terminal conditions imposed as soft penalty terms rather than enforced architecturally. For the indirect formulation we develop an error estimation framework that decomposes the total error into approximation, optimization, quadrature, and soft boundary/terminal-constraint contributions. Under standing assumptions on optimal-solution regularity and compatibility, network approximability, uniform Hölder control of the soft-constraint residuals, and a local neighborhood of the reference optimality-system solution, we derive a quantitative linearized stability estimate and a conditional local nonlinear residual-to-error estimate, and construct a computable residual indicator with a conditional reliability bound. Numerical experiments on two manufactured test problems - a cubic reactiondiffusion equation and a linear equation with a nontrivial boundary control and adjoint - illustrate the behavior of the direct and indirect formulations.
Related
- Interior regularity of mixed local-nonlocal parabolic semilinear equations
- Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term
- Concentration of measure-valued solutions for semilinear parabolic equations
- Rigorous integration of semilinear parabolic PDEs: forward evolution bounds and projected boundary value problems
- Stability of Solutions to Semilinear Exterior Dirichlet Problems under Convergence of Lévy Symbols
BibTeX
@misc{quang2026direct,
title = {{Direct and Indirect Physics-Informed Neural Networks for Dirichlet Boundary Control of Semilinear Parabolic Equations: A Conditional Error Analysis}},
author = {Nguyen Thanh Quang and Ta Thi Thanh Mai and Bui Xuan Dieu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14269},
eprint = {2609.14269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14269},
}
RIS
TY - UNPB
TI - Direct and Indirect Physics-Informed Neural Networks for Dirichlet Boundary Control of Semilinear Parabolic Equations: A Conditional Error Analysis
AU - Nguyen Thanh Quang
AU - Ta Thi Thanh Mai
AU - Bui Xuan Dieu
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14269
UR - https://arxiv.org/abs/2609.14269
AB - We study physics-informed neural networks (PINNs) for the Dirichlet boundary control of a semilinear parabolic equation with Tikhonov regularization. Two approaches are considered. A direct PINN parameterizes the state and control by separate networks and minimizes a penalized form of the tracking objective. An indirect PINN instead represents the state, adjoint, and control by unconstrained networks trained jointly to satisfy the first-order optimality system, with the state-control coupling and the homogeneous adjoint boundary and terminal conditions imposed as soft penalty terms rather than enforced architecturally. For the indirect formulation we develop an error estimation framework that decomposes the total error into approximation, optimization, quadrature, and soft boundary/terminal-constraint contributions. Under standing assumptions on optimal-solution regularity and compatibility, network approximability, uniform Hölder control of the soft-constraint residuals, and a local neighborhood of the reference optimality-system solution, we derive a quantitative linearized stability estimate and a conditional local nonlinear residual-to-error estimate, and construct a computable residual indicator with a conditional reliability bound. Numerical experiments on two manufactured test problems - a cubic reactiondiffusion equation and a linear equation with a nontrivial boundary control and adjoint - illustrate the behavior of the direct and indirect formulations.
ER -