A PINN approach for the online identification and control of unknown PDEs

TL;DR

This work addresses online identification and control of PDEs when information is incomplete, by extending Physics-Informed Neural Networks (PINNs) to PDE-constrained optimal control problems (OCPs) through a Lagrangian-based optimality system. It introduces an OCP-PINN architecture with two PINNs in series: the first recovers unknown parameters from the uncontrolled solution, and the second enforces the state, adjoint, and optimality residuals to obtain the controlled state $y$, adjoint $p$, and control $u$ while updating the parameters online via the data. Numerical tests on viscous Burgers', Allen–Cahn, and KdV equations demonstrate accurate state/control reconstruction and robust online parameter identification under sparse data, with errors typically in the $O(10^{-2})$–$O(10^{-3})$ range. The results indicate that OCP-PINNs can effectively handle partially known dynamics and data scarcity, offering a flexible alternative to traditional methods for online discovery and control of PDEs, with potential extensions to multiscale and high-dimensional settings.

Abstract

Physics-Informed Neural Networks (PINNs) have revolutionized solving differential equations by integrating physical laws into neural networks training. This paper explores PINNs for open-loop optimal control problems (OCPs) with incomplete information, such as sparse initial and boundary data and partially unknown system parameters. We derive optimality conditions from the Lagrangian multipliers and use PINNs to predict the state, adjoint, and control variables. In contrast with previous methods, our approach integrates these elements into a single neural network and addresses scenarios with consistently limited data. In addition, we address the study of partially unknown equations identifying underlying parameters online by searching for the optimal solution recurring to a 2-in-series architecture of PINNs, in which scattered data of the uncontrolled solution is used. Numerical examples show the effectiveness of the proposed method even in scenarios characterized by a considerable lack of information.

Figures (21)

• Figure 1: OCP-PINN schematic workflow. We consider an architecture composed by 2 PINNs in series: the first PINN only deals with the resolution of the inverse problem of estimating possible unknown parameters through the learning of the uncontrolled solution, while the second PINN accounts for the control problem, taking as input also the parameters discovered from the first neural network. In the latter, the PINN architecture is integrated with the physical knowledge of the adjoint equation and the optimality condition, through the inclusion of the differential operators obtained following the Lagrange multipliers approach, as well as boundary conditions of the corresponding variables.
• Figure 2: Test 1(a): Burgers' equation with $\nu=0.5$. Data points of the uncontrolled (top) and controlled (bottom) state variables used for the training of the OCP-PINN.
• Figure 3: Test 1(a): Burgers' equation with $\nu=0.5$. Reference controlled solution of the state variable (top left), OCP-PINN controlled solution (top right), and reference uncontrolled solution (bottom).
• Figure 4: Test 1(a): Burgers' equation with $\nu=0.5$. Reference control variable (left) and control obtained by applying the OCP-PINN (right).
• Figure 5: Test 1(a): Burgers' equation with $\nu=0.5$. Comparison of the solutions at final time $T=4$ (left) and convergence history for discovering the unknown parameter $\nu$ (right).

Theorems & Definitions (2)

• Remark 1
• Remark 2