Computational Control of Nonlinear Partial Differential Equations Using Machine Learning

Abstract: The numerical reconstruction of controls for nonlinear partial differential equations remains a challenging and relatively underdeveloped problem, despite the extensive literature on control theory. While recent works have introduced constructive approaches for semilinear wave and heat equations, the design of reliable computational methods for approximating control functions continues to raise significant analytical and numerical difficulties. In this work, we propose a novel framework based on physics-informed neural networks (PINNs) for the approximation of controls in nonlinear PDE settings. We develop an approach that incorporates the governing equations, boundary conditions, and control mechanisms directly into the learning process. In addition, we provide a convergence analysis of the proposed method and support the theoretical findings with numerical experiments demonstrating good performance. The resulting framework offers a flexible computational tool for approximating control functions from partial observations and provides a promising direction for the computational treatment of control reconstruction problems. Moreover, it can be applied to a broader class of problems, beyond the control of nonlinear PDEs.