An extended physics informed neural network for preliminary analysis of parametric optimal control problems

TL;DR

This work extends Physics Informed Neural Networks (PINNs) to parametric PDEs and PDE-constrained optimal control problems, addressing real-time and many-query needs. It combines physics-driven loss terms, augmented inputs via extra features, and a physics-informed architectural strategy (PI-Arch) to improve training efficiency and predictive accuracy for multi-equation systems. Numerical experiments on Poisson and Stokes optimal control problems demonstrate that extra features accelerate convergence and that PI-Arch aligns state, control, and adjoint variables more faithfully than standard PINNs, especially under challenging parameter regimes. The approach promises a nonintrusive, fast surrogate framework for parametric PDEs with potential extensions to inverse problems and data assimilation.

Abstract

In this work we propose an extension of physics informed supervised learning strategies to parametric partial differential equations. Indeed, even if the latter are indisputably useful in many applications, they can be computationally expensive most of all in a real-time and many-query setting. Thus, our main goal is to provide a physics informed learning paradigm to simulate parametrized phenomena in a small amount of time. The physics information will be exploited in many ways, in the loss function (standard physics informed neural networks), as an augmented input (extra feature employment) and as a guideline to build an effective structure for the neural network (physics informed architecture). These three aspects, combined together, will lead to a faster training phase and to a more accurate parametric prediction. The methodology has been tested for several equations and also in an optimal control framework.

Figures (16)

• Figure 1: PI-Arch example.
• Figure 2: PI-Arch used to solve problem \ref{['eq:ocp_Poisson_PINN']}.
• Figure 3: Prediction for $\mu_2 = 1, 0.1, 0.01$ with respect to $\mu_1$ in $(x_0, x_1) = (0, 0)$ compared to the FE approximation for $\mu_1 = 1, 2, 3$. Left. Control variable. Right. State variable.
• Figure 4: Parametric Optimal Control Problem. Top row. PINN approximation for $\boldsymbol{\mu}=(3, 1)$. Bottom row. PINN approximation for $\boldsymbol{\mu}=(3, 0.01)$. Left. The control variable $u$. Center. The adjoint variable $z$. Right. The state variable $y$.
• Figure 5: Parametric Poisson Optimal Control Problem. Top row. PI-Arch approximation for $\boldsymbol{\mu}=(3, 1)$. Bottom row. PI-Arch approximation for $\boldsymbol{\mu}=(3, 0.01)$. Left. The control variable $u$. Center. The adjoint variable $z$. Right. The state variable $y$.

Theorems & Definitions (1)

• Remark : The non-parametric case