| Abstract: |
| Physics-Informed Neural Networks (PINNs) offer a data-efficient approach to solving forward and inverse problems for nonlinear PDEs under sparse observations. In this talk, we introduce an OCP-PINN architecture for open-loop optimal control of time-dependent nonlinear evolution PDEs when initial/boundary data and physical parameters are only partially known.
Using the adjoint method via Lagrange multipliers, we encode the coupled optimality system (composed of state equation, adjoint equation, and optimality condition) directly into a single neural network loss function. The framework employs two PINNs in series: the first identifies unknown parameters online from scattered data of the uncontrolled nonlinear dynamics; the second uses these parameters to simultaneously reconstruct the controlled state, the adjoint, and the optimal control.
We demonstrate the approach on challenging nonlinear test cases: viscous Burgers (advection-diffusion balance), Allen--Cahn (phase-field instability), and Korteweg--de Vries (nonlinear dispersive waves). Despite severe data scarcity, the method achieves accurate parameter discovery and effective control toward target states, highlighting its robustness for nonlinear problems with incomplete information. |
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