Philip E. Paré

Sebastian F. Ruf, Keith Paarporn, Philip E. Paré

The spread of new products in a networked population is often modeled as an epidemic. However, in the case of `complex' contagion, these models {do not capture nuanced, dynamic social reinforcement effects in adoption behavior}. In this paper, we investigate a model of complex contagion which allows a coevolutionary interplay between adoption, modeled as an SIS epidemic spreading process, and social reinforcement effects, modeled as consensus opinion dynamics. Asymptotic stability analysis of the all-adopt as well as the none-adopt equilibria of the combined opinion-adoption model is provided through the use of Lyapunov arguments. In doing so, sufficient conditions are provided which determine the stability of the `flop' state, where no one adopts the product and everyone's opinion of the product is least favorable, and the `hit' state, where everyone adopts and their opinions are most favorable. These conditions are shown to extend to the bounded confidence opinion dynamic under a stronger assumption on the model parameters. To conclude, numerical simulations demonstrate behavior of the model which reflect findings from the sociology literature on adoption behavior.

Chi Ho Leung, Philip E. Paré

We study the identification of continuous-time vector fields from irregularly sampled trajectories. We introduce spectral flow learning, which learns in a windowed flow space using a lag-linear label operator that aggregates lagged Koopman actions. We provide finite-sample, high-probability (FS-HP) guarantees for the class of variable-step linear multistep methods (vLMM). The FS-HP rates are constructed using spectral regularization with qualification-controlled filters for flow predictors under standard source and filter assumptions. A multistep observability inequality links flow error to vector-field error and yields two-term bounds that combine a statistical rate with an explicit discretization bias from vLMM theory. Simulations on a controlled mass-spring system corroborate the theory and clarify conditioning, step-sample tradeoffs, and practical implications.

José I. Caiza, Junjie Qin, Philip E. Paré

In this paper, we propose an SIR spread model in a population network coupled with an infrastructure network that has a pathogen spreading in it. We develop a threshold condition to characterize the monotonicity and peak time of a weighted average of the infection states in terms of the global (network-wide) effective reproduction number. We further define the distributed reproduction numbers (DRNs) of each node in the multilayer network which are used to provide local threshold conditions for the dynamical behavior of each entity. Furthermore, we leverage the DRNs to predict the global behavior based on the node-level assumptions. We use both analytical and simulation results to illustrate that the DRNs allow a more accurate analysis of the networked spreading process than the global effective reproduction number.

Chi Ho Leung, Philip E. Paré

We study port-Hamiltonian systems with energy functions that split into local storage terms. From the interconnection and dissipation structure, we construct a graph on the energy compartments. From this graph, we show that the shortest-path distance from a constrained compartment to the nearest actuated one gives a lower bound on the relative degree of the corresponding safety constraint. We also show that no smooth static feedback can reduce it when no path exists. When the relative degree exceeds one and the immediate graph neighbors of the constrained compartment is connected to at least one input port, we reshape the constraint by subtracting their shifted local storages, producing a candidate barrier function of relative degree one. We then identify sufficient regularity conditions that recover CBF feasibility under bounded inputs. We validate the framework on an LC ladder network, where the enforceability of a capacitor charge constraint depends only on the input topology.

Chi Ho Leung, Philip E. Paré

Control barrier functions for port-Hamiltonian systems inherit model uncertainty when the Hamiltonian is learned from data. We show how to propagate this uncertainty into a safety filter with independently tunable credibility budgets. To propagate this uncertainty, we employ a two-stage Bayesian approach. First, posterior prediction over the Hamiltonian yields credible bands for the energy storage, producing Bayesian barriers whose safe sets are high-probability inner approximations of the true allowable set with credibility $1 - (η_{\mathrm{ptB}})$. Independently, a drift credible ellipsoid accounts for vector field uncertainty in the CBF inequality with credibility $1 - (η_{\rm dr})$. Since energy and drift uncertainties enter through disjoint credible sets, the end-to-end safety guarantee is at least $1 - (η_{\rm dr} + η_{\mathrm{ptB}})$. Experiments on a mass-spring oscillator with a GP-learned Hamiltonian show that the proposed filter preserves safety despite limited and noisy observations. Moreover, we show that the proposed framework yields a larger safe set than an unstructured GP-CBF alternative on a planar manipulator.

Aiping Zhong, Baike She, Philip E. Paré

This work introduces a physics-informed neural networks (PINNs)-based model predictive control (MPC) framework for susceptible-infected-recovered ($SIR$) spreading models. Existing studies in MPC design for epidemic control often assume either 1) measurable states of the dynamics, where the parameters are learned, or 2) known parameters of the model, where the states are learned. In this work, we address the joint real-time estimation of states and parameters within the MPC framework using only noisy infected states, under the assumption that 1) only the recovery rate is known, or 2) only the basic reproduction number is known. Under the first assumption, we propose MPC-PINNs and two novel PINNs algorithms, all of which are integrated into the MPC framework. First, we introduce MPC-PINNs, which are designed for $SIR$ models with control. We then propose log-scaled PINNs (MPC-LS-PINNs), which incorporate a log-scaled loss function to improve robustness against noise. Next, we present split-integral PINNs (MPC-SI-PINNs), which leverage integral operators and state coupling in the neural network training process to effectively reconstruct the complete epidemic state information. Building upon these methods, we further extend our framework for the second assumption. We establish the necessary conditions and extend our PINNs algorithms, where MPC-SI-PINNs are simplified as split-PINNs (MPC-S-PINNs). By incorporating these algorithms into the MPC framework, we simultaneously estimate the epidemic states and parameters while generating optimal control strategies. Experiment results demonstrate the effectiveness of the proposed methods under different settings.