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Predicting AI Agent Behavior through Approximation of the Perron-Frobenius Operator

Shiqi Zhang, Darshan Gadginmath, Fabio Pasqualetti

TL;DR

The paper tackles predicting AI-driven agent behavior by modeling state evolution as a nonlinear dynamical system and studying the probabilistic evolution of states via the Perron-Frobenius operator $P_\tau$. It introduces PISA, a scalable, data-driven method that learns a spectral-decomposition-inspired surrogate for $P_\tau$ using neural networks to represent basis components, enabling simultaneous short-horizon density propagation and long-horizon terminal density estimation $\rho^*$. The approach is validated on unicycle dynamics with NN controllers, score-based diffusion models, and the UCY pedestrian dataset, showing improved long-horizon predictive accuracy compared to baselines. Theoretical foundations rely on a finite-level spectral decomposition of $P_\tau$ and an invariant terminal density expressed as $\rho^* = (1/l) \sum_i g_i(x)$, with practical implications for alignment and reliability assessment of AI-driven systems.

Abstract

Predicting the behavior of AI-driven agents is particularly challenging without a preexisting model. In our paper, we address this by treating AI agents as nonlinear dynamical systems and adopting a probabilistic perspective to predict their statistical behavior using the Perron-Frobenius (PF) operator. We formulate the approximation of the PF operator as an entropy minimization problem, which can be solved by leveraging the Markovian property of the operator and decomposing its spectrum. Our data-driven methodology simultaneously approximates the PF operator to perform prediction of the evolution of the agents and also predicts the terminal probability density of AI agents, such as robotic systems and generative models. We demonstrate the effectiveness of our prediction model through extensive experiments on practical systems driven by AI algorithms.

Predicting AI Agent Behavior through Approximation of the Perron-Frobenius Operator

TL;DR

The paper tackles predicting AI-driven agent behavior by modeling state evolution as a nonlinear dynamical system and studying the probabilistic evolution of states via the Perron-Frobenius operator . It introduces PISA, a scalable, data-driven method that learns a spectral-decomposition-inspired surrogate for using neural networks to represent basis components, enabling simultaneous short-horizon density propagation and long-horizon terminal density estimation . The approach is validated on unicycle dynamics with NN controllers, score-based diffusion models, and the UCY pedestrian dataset, showing improved long-horizon predictive accuracy compared to baselines. Theoretical foundations rely on a finite-level spectral decomposition of and an invariant terminal density expressed as , with practical implications for alignment and reliability assessment of AI-driven systems.

Abstract

Predicting the behavior of AI-driven agents is particularly challenging without a preexisting model. In our paper, we address this by treating AI agents as nonlinear dynamical systems and adopting a probabilistic perspective to predict their statistical behavior using the Perron-Frobenius (PF) operator. We formulate the approximation of the PF operator as an entropy minimization problem, which can be solved by leveraging the Markovian property of the operator and decomposing its spectrum. Our data-driven methodology simultaneously approximates the PF operator to perform prediction of the evolution of the agents and also predicts the terminal probability density of AI agents, such as robotic systems and generative models. We demonstrate the effectiveness of our prediction model through extensive experiments on practical systems driven by AI algorithms.
Paper Structure (16 sections, 3 theorems, 27 equations, 4 figures, 1 algorithm)

This paper contains 16 sections, 3 theorems, 27 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Our Contributions
  4. Problem Formulation
  5. Prediction Informed by Spectral-decomposition Algorithm (PISA) for Learning Perron-Frobenius Operators
  6. Density Estimation using Kernel Density Estimation
  7. Proposed Algorithm
  8. Numerical Experiments
  9. Unicycle Model with an NN-Based Controller
  10. Predicting Behavior of Score-Based Generative Model
  11. UCY Pedestrian Dataset
  12. Theoretical Foundations of Spectral Decomposition
  13. Conclusion
  14. Supplemental Material
  15. Preliminaries and Mathematical Notations
  16. ...and 1 more sections

Key Result

Lemma 1

lasota2013chaosboyarsky1988spectral Let $P$ be a constrictive Markov operator. Then there exists an integer $l$, two sequences of non-negative functions $g_i(x)\in\mathcal{L}_1$ and $h_i(x)\in\mathcal{L}_{\infty}$, $i=1,2,\cdots,l,$ and an operator $Q:\mathcal{L}_1\mapsto\mathcal{L}_1$ such that for where The functions $g_i(x)$ and the operator $Q$ have the following properites: 1) Each $g_i(x)$

Figures (4)

  • Figure 1: Illustration of the relationship between the states and probability density of the Van der Pol oscillator in a bounded domain. The x-axis in the figures corresponds to $x_1$ and the y-axis in the figures corresponds to $x_2$. (a) Initial states of several agents driven by Van der Pol dynamics. (b) Initial density of state of agents. (c) States of the agent after time $t = 1500\tau$. (d) Density of the states at time $t = 1500\tau$. Brighter colors in (b) and (d) represent higher probability. The states are sampled with $t = 0.01$.
  • Figure 2: Experiments on the unicycle model data generated using controller from KE-DG-FP:2024. (a) Initial states are uniformly distributed, depicted in blue, final states are in red. (b) The estimated $\rho^*$ by PISA. (c) Performance comparison between PISA and YM-DS-etal:2022 on testing data of length $3$ seconds. Performance metric is the KL divergence between predicted and true densities. Our dataset contains $K= 800$ sampled instants, wherein we use the first $500$ samples of each trajectory as the training data set and the remaining $300$ as the testing set.
  • Figure 3: Experiments on the five dimensional score-based generative model YS-JSD-etal:2020. (a) Predicted terminal density $\rho^*$ projected on the first two dimension. Brighter colours indicate a higher probability. (b) Comparison of performance of PISA and YM-DS-etal:2022 on testing dataset.
  • Figure 4: Experiments on the UCY pedestrian dataset. (a) A snapshot from the dataset. (b) Corresponding estimated probability density. (c) Comparison between PISA and YM-DS-etal:2022 in the estimation of future probability densities.

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 1
  • Proof 1
  • Remark 1
  • Proposition 1