PI-CoF: A Bilevel Optimization Framework for Solving Active Learning Problems using Physics-Information

TL;DR

PI-CoF tackles data scarcity and safety in active learning for ML-driven optimization by introducing physics-informed correction factors that reconcile ML predictions with governing equations through a regularized inner optimization, forming a bilevel problem that feeds corrected predictions into safe BO or real-time optimization. The approach is validated on a toy numerical example and a real-time fuel cell case, showing improved constraint satisfaction and reduced fuel consumption at the cost of additional computation. The contributions include (i) the PI-CoF framework with additive/multiplicative corrections, (ii) its bilevel integration with BO/RTO under uncertainty, and (iii) practical demonstrations that highlight safety and efficiency gains. The work provides a general, physics-informed route to make ML-informed optimization more reliable in engineering applications.

Abstract

Physics informed neural networks (PINNs) have recently been proposed as surrogate models for solving process optimization problems. However, in an active learning setting collecting enough data for reliably training PINNs poses a challenge. This study proposes a broadly applicable method for incorporating physics information into existing machine learning (ML) models of any type. The proposed method - referred to as PI-CoF for Physics-Informed Correction Factors - introduces additive or multiplicative correction factors for pointwise inference, which are identified by solving a regularized unconstrained optimization problem for reconciliation of physics information and ML model predictions. When ML models are used in an optimization context, using the proposed approach translates into a bilevel optimization problem, where the reconciliation problem is solved as an inner problem each time before evaluating the objective and constraint functions of the outer problem. The utility of the proposed approach is demonstrated through a numerical example, emphasizing constraint satisfaction in a safe Bayesian optimization (BO) setting. Furthermore, a simulation study is carried out by using PI-CoF for the real-time optimization of a fuel cell system. The results show reduced fuel consumption and better reference tracking performance when using the proposed PI-CoF approach in comparison to a constrained BO algorithm not using physics information.

Figures (8)

• Figure 1: PI-CoF corrected predictions for $p_2$ of the numerical example over the allowed range of the decision variable $x$ at the start of the algorithm.
• Figure 2: The evolution of the optimization objective for maximizing $p_1$ with PI-CoF and constrained BO
• Figure 3: The value of $p_2$ obtained throughout the BO trials and the limit set at 140. Results shown for PI-CoF and constrained BO
• Figure 4: PI-CoF corrected predictions for $p_2$ of the numerical example over the allowed range of the decision variable $x$ at the end of the algorithm
• Figure 5: Control system architecture for the case study problem