Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A control system framework for counterfactuals: an optimization based approach

Pierluigi Francesco De Paola, Jared Miller, Alessandro Borri, Alessia Paglialonga, Fabrizio Dabbene

TL;DR

This work targets the gap between AI counterfactuals and underlying system dynamics by introducing a physics-informed control framework. It recasts counterfactuals as minimum-effort trajectories steering from an unsafe initial state to a safe terminal set using occupation measures and the moment-SOS hierarchy, solved via LMI relaxations with convergence guarantees. The approach covers both fully parametrized and uncertain systems, and provides concrete counterfactual extraction procedures from the final measure or the dual value function, demonstrated on a glucose–insulin model with robust behavior to parameter uncertainty. By embedding physics through Liouville dynamics and SOS relaxations, the method yields dynamically feasible counterfactuals and offers a path to integrate physics knowledge into AI pipelines for personalized disease prevention and decision support.

Abstract

Counterfactuals are a concept inherited from the field of logic and in general attain to the existence of causal relations between sentences or events. In particular, this concept has been introduced also in the context of interpretability in artificial intelligence, where counterfactuals refer to the minimum change to the feature values that changes the prediction of a classification model. The artificial intelligence framework of counterfactuals is mostly focused on machine learning approaches, typically neglecting the physics of the variables that determine a change in class. However, a theoretical formulation of counterfactuals in a control system framework - i.e., able to account for the mechanisms underlying a change in class - is lacking. To fill this gap, in this work we propose an original control system, physics-informed, theoretical foundation for counterfactuals, by means of the formulation of an optimal control problem. We apply the proposed methodology to a general glucose-insulin regulation model and results appear promising and pave the way to the possible integration with artificial intelligence techniques, with the aim of feeding machine learning models with the physics knowledge acquired through the system framework.

A control system framework for counterfactuals: an optimization based approach

TL;DR

This work targets the gap between AI counterfactuals and underlying system dynamics by introducing a physics-informed control framework. It recasts counterfactuals as minimum-effort trajectories steering from an unsafe initial state to a safe terminal set using occupation measures and the moment-SOS hierarchy, solved via LMI relaxations with convergence guarantees. The approach covers both fully parametrized and uncertain systems, and provides concrete counterfactual extraction procedures from the final measure or the dual value function, demonstrated on a glucose–insulin model with robust behavior to parameter uncertainty. By embedding physics through Liouville dynamics and SOS relaxations, the method yields dynamically feasible counterfactuals and offers a path to integrate physics knowledge into AI pipelines for personalized disease prevention and decision support.

Abstract

Counterfactuals are a concept inherited from the field of logic and in general attain to the existence of causal relations between sentences or events. In particular, this concept has been introduced also in the context of interpretability in artificial intelligence, where counterfactuals refer to the minimum change to the feature values that changes the prediction of a classification model. The artificial intelligence framework of counterfactuals is mostly focused on machine learning approaches, typically neglecting the physics of the variables that determine a change in class. However, a theoretical formulation of counterfactuals in a control system framework - i.e., able to account for the mechanisms underlying a change in class - is lacking. To fill this gap, in this work we propose an original control system, physics-informed, theoretical foundation for counterfactuals, by means of the formulation of an optimal control problem. We apply the proposed methodology to a general glucose-insulin regulation model and results appear promising and pave the way to the possible integration with artificial intelligence techniques, with the aim of feeding machine learning models with the physics knowledge acquired through the system framework.
Paper Structure (19 sections, 34 equations, 5 figures, 1 table, 3 algorithms)

This paper contains 19 sections, 34 equations, 5 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. METHODOLOGY
  3. Mathematical background
  4. Occupation Measures
  5. Moment-SOS hierarchy
  6. Fully parametrized systems
  7. Problem statement
  8. Problem solution in measure space
  9. LMI formulation
  10. Counterfactual Extraction
  11. RESULTS
  12. Uncertain systems
  13. Problem definition
  14. Problem solution in measures
  15. LMI formulation
  16. ...and 4 more sections

Figures (5)

  • Figure 1: Factual-counterfactual pairs as obtained by solving the optimization problem in the moment space with relaxation order $d=4$.
  • Figure 2: Factuals (red dots) and counterfactuals (blue dots) obtained by solving the optimization problem in the moment space with relaxation order $d=4$ (top panel) and $d=6$ (bottom panel).
  • Figure 3: Example of counterfactual trajectory as retrieved for the factual (250,0,0) by applying Algorithm 2.
  • Figure 4: Factual-counterfactual pairs as obtained by solving the optimization problem in the moment space with relaxation order $d=4$ (top panel) and $d=6$ (bottom panel). Robust counterfactuals lie at lower values of $G$ with increased average distance from the boundary.
  • Figure 5: Factuals vs Counterfactuals obtained for $d=4$ (top panel) and $d=6$ (bottom panel). Both the plots highlight the same dense region of counterfactuals within the interval of values of $I$ between $0$ and $10$