There are currently two seemingly unrelated frameworks for understanding these patterns. Mechanistic models account for hexagonal firing fields as the result of pattern-forming dynamics in a recurrent neural network with hand-tuned center-surround connectivity. Normative models specify a neural architecture, a learning rule, and a navigational task, and observe that grid-like firing fields emerge due to the constraints of solving this task. Here we provide an analytic theory that unifies the two perspectives by casting the learning dynamics of neural networks trained on navigational tasks as a pattern forming dynamical system. This theory provides insight into the optimal solutions of diverse formulations of the normative task, and shows that symmetries in the representation of space correctly predict the structure of learned firing fields in trained neural networks. Further, our theory proves that a nonnegativity constraint on firing rates induces a symmetry-breaking mechanism which favors hexagonal firing fields. We extend this theory to the case of learning multiple grid maps and demonstrate that optimal solutions consist of a hierarchy of maps with increasing length scales. These results unify previous accounts of grid cell firing and provide a novel framework for predicting the learned representations of recurrent neural networks.

We investigate how to couple a learnable brain-like'' controller to a cell-like'' Gray--Scott substrate to steer pattern formation with minimal effort. A compact convolutional policy is embedded in a differentiable PyTorch reaction--diffusion simulator, producing spatially smooth, bounded modulations of the feed and kill parameters ($ΔF$, $ΔK$) under a warm--hold--decay gain schedule. Training optimizes Turing-band spectral targets (FFT-based) while penalizing control effort ($\ell_1/\ell_2$) and instability. We compare three regimes: pure reaction--diffusion, NN-dominant, and a hybrid coupling. The hybrid achieves reliable, fast formation of target textures: 100% strict convergence in $\sim 165$ steps, matching cell-only spectral selectivity (0.436 vs.\ 0.434) while using $\sim 15\times$ less $\ell_1$ effort and $>200\times$ less $\ell_2$ power than NN-dominant control. An amplitude sweep reveals a non-monotonic Goldilocks'' zone ($A \approx 0.03$--$0.045$) that yields 100\% quasi convergence in 94--96 steps, whereas weaker or stronger gains fail to converge or degrade selectivity. These results quantify morphological computation: the controller seeds then cedes,'' providing brief, sparse nudges that place the system in the correct basin of attraction, after which local physics maintains the pattern. The study offers a practical recipe for building steerable, robust, and energy-efficient embodied systems that exploit an optimal division of labor between centralized learning and distributed self-organization.

Multi-agent systems (MAS) have proven to be a versatile framework for studying diverse scalability problems in Science and Engineering, such as dynamic networks [35], autonomous vehicles [5], collective behaviour of humans or animals [42, 43], and many others [2, 6]. Mathematically, MAS are often modelled as large-scale dynamical systems where each agent can be considered as a subset of states, updated via interaction forces such as attraction, repulsion, alignment, etc., [27, 19] or through the optimization of a pay-off function in a control/game framework [32, 29]. In this work, we approach the study of MAS from a control viewpoint. We study a class of sparsely interconnected agents in one dimension, interacting through nonlinear couplings and a decentralized control law. The elementary building block of our approach is the celebrated Cucker-Smale model for consensus dynamics [19], which corresponds to a MAS where each agent is endowed with second-order nonlinear dynamics for velocity alignment, and where the influence of neighbouring agents decays with distance. The Cucker-Smale model and variants can represent the physical motion of agents on the real line, inspired by autonomous vehicle formations in platooning with a nearest-neighbour interaction scheme [41, 44].

Hydrodynamic pattern formation phenomena in printing and coating processes are still not fully understood. However, fundamental understanding is essential to achieve high-quality printed products and to tune printed patterns according to the needs of a specific application like printed electronics, graphical printing, or biomedical printing. The aim of the paper is to develop an automated pattern classification algorithm based on methods from supervised machine learning and reduced-order modeling. We use the HYPA-p dataset, a large image dataset of gravure-printed images, which shows various types of hydrodynamic pattern formation phenomena. It enables the correlation of printing process parameters and resulting printed patterns for the first time. 26880 images of the HYPA-p dataset have been labeled by a human observer as dot patterns, mixed patterns, or finger patterns; 864000 images (97%) are unlabeled. A singular value decomposition (SVD) is used to find the modes of the labeled images and to reduce the dimensionality of the full dataset by truncation and projection. Selected machine learning classification techniques are trained on the reduced-order data. We investigate the effect of several factors, including classifier choice, whether or not fast Fourier transform (FFT) is used to preprocess the labeled images, data balancing, and data normalization. The best performing model is a k-nearest neighbor (kNN) classifier trained on unbalanced, FFT-transformed data with a test error of 3%, which outperforms a human observer by 7%. Data balancing slightly increases the test error of the kNN-model to 5%, but also increases the recall of the mixed class from 90% to 94%. Finally, we demonstrate how the trained models can be used to predict the pattern class of unlabeled images and how the predictions can be correlated to the printing process parameters, in the form of regime maps.

Large-Scale Multi-Agent Systems (LS-MAS) consist of several autonomous components, interacting in a non-trivial way, so that the emerging behaviour of the ensemble depends on the individual dynamics of the components and their reciprocal interactions. These models can describe a rich variety of natural systems, as well as artificial ones, characterised by unparalleled scalability, robustness, and flexibility. Indeed, a crucial objective is devising efficient strategies to model and control the spatial behaviours of LS-MAS to achieve specific goals. However, the inherent complexity of these systems and the wide spectrum of their emerging behaviours pose significant challenges. The overarching goal of this thesis is, therefore, to advance methods for modelling, analyzing and controlling the spatial behaviours of LS-MAS, with applications to cellular populations and swarm robotics. The thesis begins with an overview of the existing Literature, and is then organized into two distinct parts. In the context of swarm robotics, Part I deals with distributed control algorithms to spatially organize agents on geometric patterns. The contribution is twofold, encompassing both the development of original control algorithms, and providing a novel formal analysis, which allows to guarantee the emergence of specific geometric patterns. In Part II, looking at the spatial behaviours of biological agents, experiments are carried out to study the movement of microorganisms and their response to light stimuli. This allows the derivation and parametrization of mathematical models that capture these behaviours, and pave the way for the development of innovative approaches for the spatial control of microorganisms. The results presented in the thesis were developed by leveraging formal analytical tools, simulations, and experiments, using innovative platforms and original computational frameworks.

There are currently two seemingly unrelated frameworks for understanding these patterns. Mechanistic models account for hexagonal firing fields as the result of pattern-forming dynamics in a recurrent neural network with hand-tuned center-surround connectivity. Normative models specify a neural architecture, a learning rule, and a navigational task, and observe that grid-like firing fields emerge due to the constraints of solving this task. Here we provide an analytic theory that unifies the two perspectives by casting the learning dynamics of neural networks trained on navigational tasks as a pattern forming dynamical system. This theory provides insight into the optimal solutions of diverse formulations of the normative task, and shows that symmetries in the representation of space correctly predict the structure of learned firing fields in trained neural networks.

In the general pattern formation (GPF) problem, a swarm of simple autonomous, disoriented robots must form a given pattern. The robots' simplicity imply a strong limitation: When the initial configuration is rotationally symmetric, only patterns with a similar symmetry can be formed [Yamashita, Suzyuki; TCS 2010]. The only known algorithm to form large patterns with limited visibility and without memory requires the robots to start in a near-gathering (a swarm of constant diameter) [Hahn et al.; SAND 2024]. However, not only do we not know any near-gathering algorithm guaranteed to preserve symmetry but most natural gathering strategies trivially increase symmetries [Castenow et al.; OPODIS 2022]. Thus, we study near-gathering without changing the swarm's rotational symmetry for disoriented, oblivious robots with limited visibility (the OBLOT-model, see [Flocchini et al.; 2019]). We introduce a technique based on the theory of dynamical systems to analyze how a given algorithm affects symmetry and provide sufficient conditions for symmetry preservation. Until now, it was unknown whether the considered OBLOT-model allows for any non-trivial algorithm that always preserves symmetry. Our first result shows that a variant of Go-to-the-Average always preserves symmetry but may sometimes lead to multiple, unconnected near-gathering clusters. Our second result is a symmetry-preserving near-gathering algorithm that works on swarms with a convex boundary (the outer boundary of the unit disc graph) and without holes (circles of diameter 1 inside the boundary without any robots).

Incorporating language comprehension into robotic operations unlocks significant advancements in robotics, but also presents distinct challenges, particularly in executing spatially oriented tasks like pattern formation. This paper introduces ZeroCAP, a novel system that integrates large language models with multi-robot systems for zero-shot context aware pattern formation. Grounded in the principles of language-conditioned robotics, ZeroCAP leverages the interpretative power of language models to translate natural language instructions into actionable robotic configurations. This approach combines the synergy of vision-language models, cutting-edge segmentation techniques and shape descriptors, enabling the realization of complex, context-driven pattern formations in the realm of multi robot coordination. Through extensive experiments, we demonstrate the systems proficiency in executing complex context aware pattern formations across a spectrum of tasks, from surrounding and caging objects to infilling regions. This not only validates the system's capability to interpret and implement intricate context-driven tasks but also underscores its adaptability and effectiveness across varied environments and scenarios. More details about this work are available at: https://sites.google.com/view/zerocap/home

It is promising but challenging to design flocking control for a robot swarm to autonomously follow changing patterns or shapes in a optimal distributed manner. The optimal flocking control with dynamic pattern formation is, therefore, investigated in this paper. A predictive flocking control algorithm is proposed based on a Gibbs random field (GRF), where bio-inspired potential energies are used to charaterize ``robot-robot'' and ``robot-environment'' interactions. Specialized performance-related energies, e.g., motion smoothness, are introduced in the proposed design to improve the flocking behaviors. The optimal control is obtained by maximizing a posterior distribution of a GRF. A region-based shape control is accomplished for pattern formation in light of a mean shift technique. The proposed algorithm is evaluated via the comparison with two state-of-the-art flocking control methods in an environment with obstacles. Both numerical simulations and real-world experiments are conducted to demonstrate the efficiency of the proposed design.

From spotty leopards to stripy zebras, nature has no shortage of distinct patterns on animals and plants. Now, the age-old question of how these patterns developed may have finally been solved. Scientists have shown that the same physical process that helps remove dirt from laundry could play a role in how tropical fish get their colourful spots and stripes. For their study, the team at the University of Colorado Boulder drew on the groundbreaking work of British computer pioneer Alan Turing, dating back more than 70 years. They believe their findings could help develop new materials and even new drugs.