Cellular Automata Models

🚨 New paper alert! 🚨 This one is different. Ever since childhood I have been enthralled by Conway’s Game of Life (GOL), the public’s first introduction to cellular automata (CA). The basic idea of GOL and CA: On a square lattice, a small set of rules govern how cells interact with each other in a close neighborhood, producing a rich variety of unpredictable nonlinear behaviors, simulations that seem alive! Pick your initial conditions and set it loose! I’m such a fan of CA that I even own a signed first edition of the 1200 page book “A New Kind of Science” by Mathematica creator Stephen Wolfram ! While teaching introductory CA and GOL in an elective course on cancer modeling, I assigned myself a final project in parallel with the students: I set out to try and modify the classic GOL in a way that could serve as a model of somatic evolution in tumors… Here is the basic ruleset for GOL👇: Rule 1: death by loneliness Rule 2: survival to next step Rule 3: death by overcrowding Rule 4: spontaneous birth It’s really 3 distinct thresholds, since survival is just above the lonely threshold, while below the overcrowding threshold… So I introduced rare, random (fractional) mutations to a randomly selected threshold during birth events. Thresholds are inherited from one randomly selected neighbor-parent. Mutations can accumulate, but are rounded before rule enforcement. Upon cell death, thresholds are reset So it remained to be seen: would a new cell phenotype with greater fitness spontaneously emerge, and overtake the population? Note that under “wildtype” GOL rules initial cells rapidly die out to leave a sparse population of survivors/oscillators… Yes! Indeed a cancer-like phenotype of uncontrolled growth spontaneously emerges across the lattice, once the mutation magnitude is increased to about 0.5. This parameter multiplies a normally distributed random variable with mean=0 and std=1. So which threshold is most important for conferring an evolutionary advantage to our mutant cells? Lonely, born, or crowded? It’s the overcrowding threshold by a mile! Much greater accumulated mutations there and distinct regions of subclone populations across the lattice… There’s more observations, shared files and movies in this #OpenAccess paper so check it out! It’s an honor to be the first person to simulate evolution using the GOL framework, especially for such a well studied and storied problem. Thanks for reading! https://t.co/cZtbOHL7gd

Quantum simulation through random probabilistic cellular automata Quantum systems often pose significant computational challenges due to the complex interplay between randomness, probabilities, and quantum dynamics. Kreuzkamp et al. recently explored this problem using random probabilistic cellular automata (PCA). These automata simulate quantum behavior by modeling discrete systems where elements, or bits, follow deterministic yet probabilistically initiated rules, evolving into quantum-like phenomena such as periodic energy states and conserved momentum. The researchers used a numerical machine learning approach, specifically employing discrete Fourier transforms and eigen-decomposition techniques, to analyze large data sets generated from PCA simulations. Their approach involved mapping bit configurations onto wavefunctions, enabling the introduction of statistical observables like momentum and energy without definite configuration-based values. By discretizing quantum mechanics, the authors formulated step evolution operators as unitary matrices, which were then diagonalized numerically to identify energy eigenstates and analyze system evolution. Key findings indicate the effectiveness of their method in capturing typical quantum behaviors, notably periodic recurrences in the probability distributions and conservation laws analogous to energy and momentum in classical mechanics. For instance, the machine learning approach quantitatively revealed that initial states with well-defined momenta evolve into superpositions of discrete momentum states conserved modulo periodic boundaries, thus highlighting the quantum characteristics encoded within these automata. This work significantly enhances the computational toolkit for analyzing complex quantum systems with discrete models. Paper: https://lnkd.in/drNgqW55 #QuantumPhysics #MachineLearning #QuantumSimulation #QuantumComputing #CellularAutomata #ProbabilisticModels #NumericalMethods #FourierTransform #QuantumMechanics #ComputationalPhysics #PhysicsResearch #QuantumTech #DataScience #QuantumSystems #ResearchHighlights #AIforScience

So… I’ve been playing around with predator-prey dynamics, and simulating it using cellular automata is just next-level fun! Especially when you throw in different spatial distributions of vegetation as refuges - the landscape changes, and the whole system behaves in fascinating ways! 🌿🦌🐺 For this, Python is obviously the way to go, and NumPy, well, it’s an absolute lifesaver for handling all those grids and arrays. The speed and flexibility it offers make experimenting with different scenarios super efficient. 🌱Scenarios? Imagine lush green patches as safe zones for prey, sparse vegetation increasing predation pressure, or fragmented habitats… each one leads to new dynamics and emergent behaviors! 🔁 The beauty of cellular automata is that even simple rules can create complex, lifelike patterns. I’m amazed by how much this simulation can teach us about real-world predator-prey interactions and habitat management. Curious about how NumPy made all of this possible? Let’s connect and exchange ideas! #Python #NumPy #Simulation #Ecology #SpatialAnalysis #DataScience #CellularAutomata #LandscapeModeling

Introducing LifeGPT, showing that LLMs can simulate complex, Turing-complete systems like Conway's Game of Life with near-perfect accuracy—no prior topology needed. 🌐 This unlocks new potential for AI in modeling self-organizing systems in biology, materials science, & beyond. 🔬🤖 #AI #LifeGPT. Cellular Automata (CA), like Conway's Game of Life ("Life"), are computationally irreducible, meaning their evolution is difficult to predict without an a-priori understanding of the rules of the game, including the topology on which it is played. LifeGPT is a topology-agnostic generative model that learns the rules of Life without prior knowledge of its grid structure or boundary conditions, from only a tiny number of game states. This work suggests promising avenues for scientific discovery, particularly in bridging the gap between AI, artificial life & real-world biological systems, for both forward and inverse problems. The potential for universal computation within generative AI, including LLMs, through approaches like LifeGPT, represents an exciting area for future research, especially when combined with reinforcement learning. Accuracy & Temperature: LifeGPT achieves near-perfect accuracy, particularly at lower sampling temperatures, but can be continually tuned towards higher creativity to discover patterns that the original ruleset would not be able to produce. This finding highlights the trade-off between model creativity (higher temperature) and accuracy in deterministic predictions, with high relevance to model real-world dynamical systems for which no closed-form rulesets exist. Zero/Few-Shot Learning: Trained on a small fraction of possible initial conditions, LifeGPT demonstrates strong zero/few-shot learning, accurately simulating Life for unseen initial conditions. However, rare prediction errors highlight that LifeGPT approximates rather than perfectly replicates the Life algorithm. Autoregressive Autoregressor: A recursive implementation of LifeGPT demonstrates the model's ability to simulate Life over multiple timesteps. LifeGPT is topology-agnostic with respect to its training data and our results show that a GPT model is capable of capturing the deterministic rules of a Turing-complete system with near-perfect accuracy, given sufficiently diverse training data. The work showcases the possibility for future models to synthesize stochastic generative capabilities with deterministic computational capabilities. Paper 📰: Jaime Berkovich, Markus J. Buehler, LifeGPT: Topology-Agnostic Generative Pretrained Transformer Model for Cellular Automata, https://lnkd.in/e25S4dza, 2024 Code: https://lnkd.in/eC6JQARY Weights 🤗: https://lnkd.in/eziqNA7m Audio: Podcast generated using #NotebookLM

Graph Neural Cellular Automata oor GNCA, might sound like a heavy term at first... but if we break it down from the start, it’s actually a mix of two ideas that people in science have been working with for decades. One is Cellular Automata (CA) ... those little systems where you’ve got cells sitting on a grid, each cell has a state, and at every tick of time they update based on their neighbors. It’s simple, local, and yet out of those local rules you suddenly see these global patterns, almost like life emerging from nothing. Conway’s “Game of Life” is the most famous example, right? Then you bring in the second piece is ..Graph Neural Networks (GNNs). these are deep learning models that don’t just work on images or sequences but on graphs — meaning nodes and edges, relationships, networks of all kinds. GNNs have their mathematics built on message passing and permutation-equivariant functions, which basically means they don’t care about how you order the nodes, the result will be consistent. Now... GNCA are what happens when you fuse the two. Instead of a grid with fixed neighbors, you’ve got a graph — so every node is like a cell, and its neighbors are just the connected nodes in the graph. Each node has a state vector, something sitting in a high-dimensional space, and at each step it updates itself by looking at what its neighbors are saying. This update is done through functions that resemble those in GNNs ... aggregation functions like sum, mean, attention — and then a nonlinear update rule, often with residual mappings so the model doesn’t collapse. If you look at it mathematically, GNCA form a discrete dynamical system. Terms like fixed points, attractors, and stability come in, because we want to know, after many iterations, does the system calm down, repeat itself, or keep producing new complexity ?..... GNCA don’t need us to handcraft the rules , they learn the rules. And what they learn can look like growth, repair, self-organization, things that you usually model with partial differential equations or reaction-diffusion systems in physics and chemistry. Except here, it’s happening on graphs, which makes it useful for molecules, social networks, swarm robotics, even infrastructure networks. The system keeps this very local perspective — “I only talk to my neighbors” .... but the global behavior can be astonishingly rich. That’s why people see GNCA not just as another deep learning tool, but as a step toward machines that capture the same kind of distributed intelligence that nature itself seems to use.... Vikas Choudhary

The design of complex self-organizing systems producing life-like phenomena, such as the open-ended evolution of virtual creatures, is one of the main goals of artificial life. Lenia, a family of cellular automata (CA) generalizing Conway's Game of Life to continuous space, time, and states, has attracted a lot of attention because of the wide diversity of self-organizing patterns it can generate. Among those, some spatially localized patterns (SLPs) resemble life-like artificial creatures and display complex behaviors. However, those creatures are found in only a small subspace of the Lenia parameter space and are not trivial to discover, necessitating advanced search algorithms. Furthermore, each of these creatures exists only in worlds governed by specific update rules and thus cannot interact in the same one. This paper proposes a mass-conservative extension of Lenia, called Flow Lenia, that solves both of these issues. We present experiments demonstrating its effectiveness in generating SLPs with complex behaviors and show that the update rule parameters can be optimized to generate SLPs showing behaviors of interest. Finally, we show that Flow Lenia enables the integration of the parameters of the CA update rules within the CA dynamics, making them dynamic and localized, allowing for multi-species simulations, with locally coherent update rules that define properties of the emerging creatures, and that can be mixed with neighboring rules. We argue that this paves the way for the intrinsic evolution of self-organized artificial life forms within continuous CAs. #machinelearning #innovation #research