Computational Control of Nonlinear Partial Differential Equations Using Machine Learning

Abstract

The numerical reconstruction of controls for nonlinear partial differential equations (PDEs) remains a challenging and relatively underdeveloped problem, despite the extensive literature on controllability theory. In this work, we introduce an operator-decomposed physics-informed neural network framework, called WeightedPINN, for approximating controls in nonlinear PDE settings. The method is designed for both internal and bilinear control problems and incorporates the governing equation, boundary and initial conditions, and terminal control constraints directly into the training objective. The main feature of WeightedPINN is that the different components of the controlled PDE residual are weighted separately. In particular, the time derivative, directional diffusion terms, nonlinear response, and control term are assigned independent adaptive space--time weights, and the same weighted formulation is applied to the boundary, initial, and terminal constraints. This produces a control-aware residual metric that is more sensitive to operator-level imbalance and to the mechanism through which the unknown control enters the equation. We provide a convergence analysis for the proposed method and present numerical experiments for semilinear heat and wave equations with internal and bilinear controls. The high-dimensional experiments demonstrate improved residual-based testing errors compared with the standard PINN baseline, while lower-dimensional manufactured-solution benchmarks show improved direct reconstruction errors for both the state and the control against several adaptive and control-oriented PINN methods. The results suggest that WeightedPINN is particularly effective in regimes where componentwise residual imbalance, anisotropy, variable coefficients, or control-identification sensitivity play a significant role.