Lecture 5E/6A (2026-04-04): Parallel Tempering and Swarm Intelligence through Social Cohesion (Particle Swarm Optimization)

In this lecture, we finish our unit on physics-inspired ML and optimization by covering Parallel Tempering (PT), which combines multiple, parallel Metropolis–Hastings MCMC samplers each with different temperatures (rather than using an annealing schedule, as in Simulated Annealing (SA)). We then pivot toward motivating why certain problem sets, like optimizing high-dimensional weights of neural networks, may not be well suited by the optimization metaheuristics discussed so far in the course. We use this as an opportunity to introduce Swarm Intelligence and the Particle Swarm Optimization (PSO) algorithm, which is particularly good at finding and exploring local optima in spaces with many similarly performing local optima. We explore how PSO was inspired by the Boids Model from Craig Reynolds (in computer graphics) and how it overlaps with the Vicsek model (from statistical physics). We also show how PSO really depends on is social information but, under the influence of social information, tends to very quickly purge the diversity in its solution candidates. Online interactive demonstration modules associated with this lecture can be found at:

• Simulated Annealing: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/simulated_annealing/simulated_annealing_demo.html
• Parallel Tempering: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/parallel_tempering/parallel_tempering.html
• Reynolds' Boids Collective Motion Model: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/collective_motion/boids_explorer.html
• Vicsek Collective Motion Model: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/collective_motion/vicsek_explorer.html
• Particle Swarm Optimization (PSO): https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/particle_swarm_optimization/pso_explorer.html

Whiteboard notes for this lecture can be found at: https://www.dropbox.com/scl/fi/7jwuytadieywwilqazjq5/IEE598-Lecture5E_6A-2026-04-02-Parallel_Tempering_and_Particle_Swarm_Optimization-Notes.pdf?rlkey=p1pr7cs241okovkgjnevvhdp5&dl=0

Lecture 5D (2026-03-31): Metropolis–Hastings Markov Chain Monte Carlo and Simulated Annealing/Parallel Tempering

In this lecture, we start with a reminder that the Boltzmann–Gibbs distribution is the maximal entropy (MaxEnt) distribution of physical microstates when the average energy is fixed at a temperature at thermal equilibrium. We then move toward motivations where it would be useful to sample microstates from such a distribution. First, we introduce Monte Carlo methods for parameter estimation, and we pivot toward applications of Monte Carlo sampling for numerical integration. This leads us back to physics applications where integration using the Boltzmann–Gibbs is much more practical. This gives the opportunity to introduce Metropolis–Hastings Markov Chain Monte Carlo (MCMC) sampling, which allows for sampling from the Boltzmann–Gibbs and more. After discussing connections to importance sampling (from stochastic simulation) and Bayesian/MCMC statistics, we introduce Simulated Annealing, which combines Metropolis–Hastings sampling with an annealing schedule for temperature. We close with a very brief introduction to Parallel Tempering, which swaps out the annealing schedule for parallel MCMC samplers that periodically swap states based on their relative energies. We will pick up with Parallel Tempering in the next lecture.

On-line simulations referenced in this lecture can be found at:

• Boltzmann–Gibbs distribution: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/boltzmann_maxent/boltzmann_maxent_random_exchange.html
• SoftMax Explorer: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/softmax/softmax_temperature_explorer.html
• Monte Carlo Estimation/Integration Explorer: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/monte_carlo/mc_explorer.html
• Simulated Annealing Explorer: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/simulated_annealing/simulated_annealing_demo.html
• Parallel Tempering Explorer: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/parallel_tempering/parallel_tempering.html

Whiteboard notes for this lecture can be found at:
https://www.dropbox.com/scl/fi/s5dcgqrvm4qzz4y0fs64a/IEE598-Lecture5D-2026-03-31-Markov_Chain_Monte_Carlo_Metropolis_and_Simulated_Annealing_Parallel_Tempering-Notes.pdf?rlkey=v2m33lhh7sjhwogffotbyq3k7&dl=0

Lecture 5C (2026-03-26): Boltzmann–Gibbs and other Maximum Entropy Distributions

In this lecture, we start by reviewing the formal definition of Shannon entropy/information in both is discrete and continuous (differential entropy) forms. We then transition to discussing several different MaxEnt distributions and the constraints that they are associated with. Ultimately, this brings us to the Boltzmann–GIbbs distribution and several applications of it. Throughout the lecture, different interactive demonstrations were used (and can be accessed directly at the links below).

Demonstrations referenced in this lecture can be found at:

Softmax Visualizer: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/softmax/softmax_temperature_explorer.html

MaxEnt Explorer (SDM and NLP): https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/maxent/maxent_demo.html

Boltzmann Distribution via Random Exchanges of Conserved Quantity: https://tpavlic.github.io/asu-b]]ioinspired-ai-and-optimization/boltzmann_maxent/boltzmann_maxent_random_exchange.html

Beta Distribution Explorer: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/boltzmann_maxent/beta_spacings.html

Whiteboard notes for this lecture can be found at:
https://www.dropbox.com/scl/fi/zwdrab929yg47jm67vope/IEE598-Lecture5C-2026-03-26-Boltzmann-Gibbs_and_other_MaxEnt_Distributions-Notes.pdf?rlkey=3zka62o08gnw8z38r7lknjsqf&dl=0

Lecture 5B (2026-03-24): From Entropy to Maximum Entropy (MaxEnt) Methods

In this lecture, we pivot from our motivation from the Simulated Annealing optimization metaheuristic to thinking about how to sample from microstates within the physically inspired search process. This requires us to introduce the concept of entropy, a quantity which measures the number of microstates in a coarse-grained "macrostate" description of a system. Within the constraints of a system, we seek a distribution of microstates that represents only those constraints and not any additional information. This is the maximal entropy distribution for those constraints. We provide a few formalities on how to make this a little more rigorous and then introduce Maximum Entropy (MaxEnt) methods once popular in NLP that remain to be popular in Species Distribution Modeling and archaeology. We will use MaxEnt to help us define the Boltzmann–Gibbs distribution (and Monte Carlo methods to sample from it) next time.

Whiteboard notes for this lecture can be found at:
https://www.dropbox.com/scl/fi/01pfdkj3d3ilk7wiyu79a/IEE598-Lecture5B-2026-03-24-From_Entropy_to_Maximum_Entropy_MaxEnt_Methods-Notes.pdf?rlkey=xfe1pie4sxu0qklg871czuc05&dl=0

Lecture 4D/5A (2026-03-19): Distributed and Parallel GA's and Introduction to Simulated Annealing (SA)

In this lecture, we wrap up our units on evolutionary algorithms, closing on Distributed (Island Model) and Parallel Genetic Algorithms. We describe the basic population structure and migration approaches in Distributed GA's and explore whether Sewall Wright's shifting-balance theory (SBT) can explain DGA's success on certain landscapes. We then pivot to a new unit on physics-inspired ML and optimization approaches, where Simulated Annealing (SA) is one of the key topics. We introduce Simulated Annealing and discuss how hardware annealers can solve a broad set of combinatorial problems that can be QUBO (Quadratic Unconstrained Binary Optimization) encoded. We setup the basic content grammar for the unit by introducing macrostate, microstate, temperature, and energy, and then we give an animated outline of how the basic SA algorithm works. We will use this SA to motivate our explorations into entropy, MaxEnt, Boltzmann sampling, and more in future lectures in this unit.

Shifting-Balance Theory visualizer: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/shifting_balance_theory/sbt_four_peaks.html

Simulated Annealing explorer: https://tpavlic.github.io/asu-bioinspired-ai-and-optimization/simulated_annealing/simulated_annealing_demo.html

Whiteboard notes for this lecture can be found at:
https://www.dropbox.com/scl/fi/b8v78jmem4j9spju7sa8k/IEE598-Lecture4D_5A-2026-03-19-Distributed_and_Parallel_GAs_and_Introduction_to_Simulated_Annealing_SA-Notes.pdf?rlkey=qfh29uk7ckfb8aphn1k645r9e&dl=0

Lecture 4C (2026-03-17): From Niches to Meta-Populations: Toward Distributed and Parallel Genetic Algorithms (DGA/PGA)

In this lecture, we close out our discussion of "niching" diversity-preservation approaches for multi-modal and multi-objective evolutionary algorithms. We had covered clearing/clustering algorithms in the past lecture (Lecture 4B), and so we start on crowding algorithms, including Restricted Tournament Selection (RTS), briefly the introduce Species Conserving Genetic Algorithm (SCGA), and then close with a discussion of islanding approaches. This sets up an introduction to distributed (and parallel) genetic algorithms, which we will start out with in the next lecture.

Whiteboard notes for this lecture can be found at:
https://www.dropbox.com/scl/fi/ngcurzxer85i4oft1qn68/IEE598-Lecture4C-2026-03-17-From_Niches_to_Meta_Populations-Distributed_and_Parallel_GA-Notes.pdf?rlkey=x8mb0bn5d56lhwtjftjx323u6&dl=0

Lecture 4B (2026-03-05): Niching Methods for Diversity Preservation in Multi-Objective and Multi-Modal Evolutionary Algorithms

In this lecture, we cover several of the different "niching methods" used for diversity preservation in both multi-objective and multi-modal evolutionary algorithms. We start with an overall goal to create "negative frequency-dependent selection" (or density dependence) that has the potential to be able to stabilize different subpopulations coexisting with each other. We start by discussing how evolutionary models like Hawk–Dove ("Chicken") have mixed Nash equilibria that can represent stable co-existence of discrete phenotypes (due to negative frequency dependence). But then we pivot to habitat selection models, with particular focus on the Ideal Free Distribution (IFD), as a better match for the diversity-preservation problem in MOEA's and MMEA's. That allows us to introduce "fitness sharing" (which matches very closely to the IFD) and various other fitness-modification methods that each have different computational costs and diversity benefits. We close by introduction selection-based approaches, such as breaking tournament-selection ties by crowding distance.

Whiteboard notes for this lecture can be found at:
https://www.dropbox.com/scl/fi/d2ucw5j4lqtlj6hzue2mk/IEE598-Lecture4B-2026-03-05-Niching_Methods_for_Diversity_Preservation_in_MOEA_and_MMO-Notes.pdf?rlkey=rvvs5xy2qmbva7xl1glsmhnja&dl=0