[Paper Review] An extended physics informed neural network for preliminary analysis of parametric optimal control problems

Nicola Demo, Maria Strazzullo+1 more|arXiv (Cornell University)|Oct 26, 2021

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes an extended physics-informed neural network (PINN) that integrates physics-based constraints not only in the loss function but also as extra input features and in the network architecture to accelerate training and improve accuracy for parametric optimal control problems (OCPs). The method achieves faster convergence—reducing training epochs by 33% in the Burgers problem—while maintaining high solution accuracy across parameterized PDEs, including parametric Poisson and OCPs.

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.

Motivation & Objective

• Address the high computational cost of solving parametric PDEs and optimal control problems (OCPs) in real-time and many-query applications.
• Overcome limitations of standard PINNs in parametric settings by incorporating physical knowledge beyond loss function regularization.
• Develop a unified, extendable PINN framework compatible with existing PINN codes that leverages physics for both input features and network structure.
• Accelerate training and improve generalization for OCPs with parameter-dependent equations and desired profiles.
• Demonstrate the method’s effectiveness on benchmark problems including Burgers’ equation and parametric Poisson equations.

Proposed method

• Integrate physics information into the PINN framework via three mechanisms: (1) loss function regularization, (2) extra input features (e.g., initial/boundary conditions, forcing terms), and (3) physics-informed neural network architecture design.
• Use the forcing term $ f(oldsymbol{x},m{ u}) = e^{-2((x_0- u_1)^2 + (x_1- u_1)^2)} $ as an extra input feature to improve training efficiency in parametric PDEs.
• Employ a 3-layer fully connected neural network with 20–10–5 neurons and hyperbolic tangent activation for the Burgers problem; use 20 neurons per layer with Softplus activation for the Poisson problem.
• Train using ADAM optimizer with learning rates of 0.006 (Burgers) and 0.03 (Poisson), with collocation points sampled via Latin hypercube (internal) and uniform (boundary) distributions.
• Define the loss function to include residual minimization of the PDE, boundary conditions, and, in the parametric case, parameter-dependent constraints.
• Use the parameter $ m{ u} $ as part of the input to the network to enable generalization across parameter instances.

Experimental results

Research questions

• RQ1 Can physics-informed neural networks be extended to accelerate training and improve accuracy in parametric optimal control problems?
• RQ2 How does incorporating physical quantities (e.g., initial conditions, forcing terms) as extra input features affect PINN convergence speed and solution accuracy?
• RQ3 To what extent does designing a physics-informed network architecture improve training efficiency compared to standard PINNs?
• RQ4 Can the proposed method maintain high accuracy across diverse parameter values while reducing training time in parametric PDEs?
• RQ5 How does the combination of extra features and physics-informed architecture compare to standard PINN in terms of loss convergence and pointwise error?

Key findings

• The inclusion of the forcing term as an extra input feature reduced the number of epochs required to reach a loss tolerance of 0.0001 from 180,000 to 120,000 in the Burgers problem, representing a 33% reduction in training time.
• The pointwise error between the PINN solution and the finite element reference solution remained below $ 3 \cdot 10^{-2} $ for the parametric Poisson problem across all tested parameter instances.
• The proposed method achieved accurate predictions for both $ m{ u} = (-0.8, -0.8) $ and $ m{ u} = (0.8, 0.8) $ in the parametric Poisson problem, with visual and quantitative agreement between PINN and FEM solutions.
• The use of extra features led to faster convergence even in complex problems like Burgers’ equation, where the solution develops shocks, though the improvement was less pronounced than in simpler cases.
• The physics-informed architecture and feature engineering approach were fully compatible with standard PINN implementations, enabling easy integration into existing PINN codebases.
• The method demonstrated robustness and scalability across different PDE types, including hyperbolic (Burgers), elliptic (Poisson), and optimal control problems, with consistent performance gains.