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42 recordsLinked to original sources
arXiv2026

Collision-based logic in Lenia and its composition boundary

Continuous cellular automata such as Lenia spontaneously produce lifelike, self-propelling patterns, including the Orbium glider, which travels in a straight line while pulsing through a fixed breathing cycle. Collision-based logic, where moving patterns compute by colliding, is established in discrete cellular automata and continuous physical media. Within continuous cellular automata, computation has so far been trained into the rule rather than emerging from collisions, and whether a fixed-rule automaton like Lenia can support general collision-based computation remains open. This paper constructs an INHIBIT gate from collisions of the Orbium glider. Of the patterns searched across four continuous-CA rule types, the Orbium glider is the only one shown to survive a collision with both copies intact. A control glider deflects a signal glider off its output line, so the output carries a signal only when no control is present. The gate blocks across all twenty-four phases of the breathing cycle and nine integer offsets of the control. Two such gates in series, with one signal line and two controls, compose into an AND-NOT chain, correct on all eight input combinations. By contrast, routing a signal beyond that single chain is undemonstrated. A deflected signal is not restored to a fixed landing position, and no reusable absorber for the surviving gliders was found. The immediate open question for collision-based computation in Lenia therefore narrows from whether a gate exists to whether a deflected signal can be delivered to a downstream gate, the next requirement for composing the gate beyond a single straight chain.

cs.ET↗
arXiv2026

Basin Geometry and Reliable Recall of Dynamical Memories in Reservoir Computing

Reliable attractor recall conventionally requires broad basins of attraction. However, in reservoir-computing based associative memory, temporal cues reliably recover dynamical memories despite basins dominated by unpredictable, riddled-like regions. We reveal that memory basins exhibit an ``octopus-like'' structure: a robust ``head'' near the attractor and thin, intertwined ``tentacles'' spanning state space. Initial states in tentacular regions yield near-zero uncertainty exponents, making the recalled memory effectively unpredictable at finite precision. Yet, cue-driven generalized synchronization bypasses this unpredictability, driving the system into the robust basin head. This mechanism yields a quantitative relation linking minimum cue duration, synchronization rate, and basin-head radius. Trained recurrent neural networks exhibit similar geometry, suggesting this phenomenon extends beyond reservoir computing.

nlin.CD↗
arXiv2026

Emergent aggregation from collective foraging

Collective behaviour in living systems is usually modelled as the outcome of a \emph{direct} social drive: agents are rewarded, or hard-wired, to align with or approach their neighbours. Here we show that aggregation can instead emerge from an \emph{indirect} objective. We let reinforcement learning foragers, initially performing a random walk, optimize their dynamics from a purely individual reward for finding replenishable targets, while perceiving only their conspecifics and never the targets themselves. As the visual range grows, the agents undergo a sharp crossover from an environment-tuned individual search to a scale-agnostic collective one, and this crossover coincides with the onset of spatial aggregation. Thus a collective phase arises as a by-product of optimal foraging, without any direct reward for grouping. A minimal analytical first-passage model reproduces the transition as a crossover between the two search strategies. Our results identify indirect, resource-driven reward as a generic route to emergent collective phenomena.

cond-mat.stat-mech↗
arXiv2026

Reproducible macroscopic dynamics in a closed-loop human-AI learning system

Closed-loop human-AI systems generate high-dimensional behavioural trajectories whose collective dynamics remain obscure. Using 297,915 learners' adaptive-tutoring histories, we define semantic order variables before model fitting and test them in user-disjoint cohorts. The state exhibits reproducible basin-like flow and operationally defined, state-heterogeneous metastable-like kinetics. A construction-matched null distinguishes normalised-memory relaxation from a reproducible excess field. A four-term conditional mechanism recovers population drift (r = 0.946; learner-bootstrap 95% CI, 0.935-0.955). Predictive event-level self-supervised learning recovers the state and learned-plane flow; null-referenced corrections retain directional, partial-amplitude excess-field structure without full calibration. Shuffled-order training reverses learned-plane flow on ordered trajectories; support-alignment randomisation selectively reduces inward transport. Both axes remain linearly accessible without state supervision. Without cross-model fitting, the models share leading population drift (r = 0.866; learner-bootstrap 95% CI, 0.857-0.875) and persistence ordering; residual directions remain model-specific. These results identify an externally anchored leading-order effective field linking empirical dynamics, an interpretable mechanism and neural computation.

cs.LG↗
arXiv2026

Spatial symmetry invariance of solution of Kolmogorov flow

We prove a mathematical theorem that solution for all $t > 0$ of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition keeps the same spatial symmetry as its smooth initial condition. The proof of a similar theorem for the three-dimensional NS equations is given in the appendix. These mathematical theorems can be used to check the correctness and reliability of numerical simulations of NS turbulence. For example, they support the corresponding CNS (clean numerical simulation) results of the 2D and 3D turbulent Kolmogorov flows [1-3] that remain the same spatial symmetry in the whole time interval of simulation, but do not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate these mathematical theorems. Thus, these mathematical theorems rigorously confirm that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. All of these indicate that CNS can indeed provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.

physics.flu-dyn↗
arXiv2026

An Energy-Based Mechanism for Compositional Behavior

Flexible intelligence relies on the ability to reuse previously acquired behaviors and combine them differently as circumstances change. In biological and artificial systems, this ability is often attributed to gating mechanisms that determine how much each available behavior should contribute at a given time. Yet these gating rules, the dynamics that compute them, and the neural circuits that may implement them are usually introduced separately, leaving unclear whether they reflect a common underlying principle. Here, we show that they can all be derived from a single variational principle for behavioral composition. The resulting mechanism naturally gives rise to softmax gating, evolves as an energy-based dynamical system with guaranteed convergence, and admits a recurrent neural network instantiation featuring context-dependent and local interactions. Across collective behavior, human decision-making, and layered control, the same mechanism reproduces characteristic behavioral patterns, provides interpretable accounts of how different behaviors are combined, and matches or outperforms established approaches. These results provide a unified account of how behavioral composition can emerge from a common principle, with implications for understanding flexible behavior in natural systems and for designing artificial agents that can adapt by recombining existing capabilities.

math.OC↗
arXiv2026

Structure-Preserving Physics-Informed Neural Network for the Korteweg--de Vries (KdV) Equation

Physics-Informed Neural Networks (PINNs) offer a flexible framework for solving nonlinear partial differential equations (PDEs), yet conventional implementations often fail to preserve key physical invariants during long-term integration. This paper introduces a \emph{structure-preserving PINN} framework for the nonlinear Korteweg--de Vries (KdV) equation, a prototypical model for nonlinear and dispersive wave propagation. The proposed method embeds the conservation of mass and Hamiltonian energy directly into the loss function, ensuring physically consistent and energy-stable evolution throughout training and prediction. Unlike standard \texttt{tanh}-based PINNs~\cite{raissi2019pinn,wang2022modifiedpinn}, our approach employs sinusoidal activation functions that enhance spectral expressiveness and accurately capture the oscillatory and dispersive nature of KdV solitons. Through representative case studies -- including single-soliton propagation (shape-preserving translation), two-soliton interaction (elastic collision with phase shift), and cosine-pulse initialization (nonlinear dispersive breakup) -- the model successfully reproduces hallmark behaviors of KdV dynamics while maintaining conserved invariants. Ablation studies demonstrate that combining invariant-constrained optimization with sinusoidal feature mappings accelerates convergence, improves long-term stability, and mitigates drift without multi-stage pretraining. These results highlight that computationally efficient, invariant-aware regularization coupled with sinusoidal representations yields robust, energy-consistent PINNs for Hamiltonian partial differential equations such as the KdV equation.

cs.LG↗
arXiv2026

Reciprocity Separates Gradient Flow from Rotation in Conservative Physical Learning

Physical learning lets a trainable material or network use its own physical response to carry error signals, reducing the need for a separately programmed backward computation. We ask what determines whether such a system follows conventional gradient descent or evolves along a genuinely different learning trajectory. Our canonical model is a directed layered transport network in which every node redistributes a fixed amount of flow, so learning preserves positivity and total mass. In this model, conservation constrains only the allowable learning directions. Within the matched response class studied here, adjoint matching gives the physical output response a symmetric form. Non-negative mode-wise feedback then produces a reciprocal closed-loop response and a reweighted gradient flow. Adding an antisymmetric boundary component makes the closed-loop response rotational: the learning path can turn while the error driving that update still decreases at that moment. Turning is not automatically beneficial. Its finite-step effect is set by local curvature, and its accumulated effect also depends on step selection and on the new states visited along the path. Numerical consistency checks reproduce the exact response structure, predict the sign of the local effect across new network families, and show how trajectory drift can negate a local advantage. These results separate the roles of conservation, reciprocity, and nonreciprocity in physical learning.

cs.LG↗
arXiv2026

Unfolding Overlaps of the Exceptional Regular Polytopes

We find explicit ridge unfoldings of the three exceptional 4D polytopes (24-cell, 120-cell, 600-cell) that result in overlaps of their facets. These failures bring an end to the full classification of regular polytopes with the all-net property.

cs.CG↗
arXiv2026

Memory as an Energy Landscape---Hopfield

This chapter reconstructs the Hopfield network as a physical theory of memory rather than merely an early neural-network algorithm. It begins with the problem as it stood before 1982-threshold logic, Hebbian association, correlation memories, and recurrent binary networks-and isolates what Hopfield's synthesis added: a dynamical definition of content-addressable memory, a symmetric recurrent architecture with a Lyapunov function, a Hebbian embedding of patterns in its couplings, and a physical account of basins, robustness, and graceful degradation. The binary and graded-response energy functions are derived in full, together with the signal-crosstalk decomposition governing pattern stability, the mean-field theory of retrieval at extensive load, and the zero-temperature retrieval spinodal at (alpha 0.138) established by Amit, Gutfreund, and Sompolinsky. The energy-based program is then followed through analog optimization networks, polynomial dense associative memories, exponential interactions, and modern continuous Hopfield updates, including the precise conditions under which the update becomes scaled dot-product attention. Throughout, capacity claims are tied to their disorder ensemble, scaling limit, and success criterion, showing why numerically different storage limits need not conflict. A closing assessment distinguishes established results from surviving principles, assumption-bound limitations, and open problems, treating the Hopfield network as an effective theory whose symmetry, locality, and point-neuron assumptions delimit its biological reach. Fixed-seed numerical experiments expose the mechanisms discussed but do not substitute for analytical results.

cs.NE↗
arXiv2026

Sustained Heterogeneity: an emergent collective mechanism in LLM-driven traffic

Large language models (LLMs) are increasingly adopted as closed-loop controllers in physical multi-agent systems, yet their emergent collective dynamics remain incompletely characterised. We deploy 22 LLM agents as direct, real-time target-speed controllers (per 0.5 s cycle, with IDM as collision-avoidance clamp) on a 230 m ring road under the Sugiyama 2008 paradigm, reproducing human-like stop-and-go waves. Six matched controls spanning stochasticity (white noise, OU noise, temperature), population variance, and dynamical instability (delay, OV model) are systematically excluded. The surviving phenomenon, termed Sustained Heterogeneity (SH), is the persistent, approximately temperature-insensitive (approx. 8 percent across a 6x T sweep), per-cycle divergence in LLM-chosen target-speed adjustments, propagating through a three-stage cascade of drift, gap erosion, and nonlinear braking. Across four traffic densities, the critical LLM penetration fraction p_c decreases monotonically from no transition at density 43.5 veh/km to p_c approx 0.23 at density 95.7 veh/km, consistent with an initiation-threshold model governed by trigger distance, stochasticity, and fleet size. Chain-of-thought analysis of 39,600 decisions across three seeds shows agents engage in multi-factor safety reasoning, yet systematic divergence persists, implying stability must be enforced at the dynamics layer. This is the first study to identify a previously uncharacterised collective mechanism in LLM-controlled traffic and map a density-dependent phase boundary p_c(rho).

physics.soc-ph↗
arXiv2024

Predicting unobserved climate time series data at distant areas via spatial correlation using reservoir computing

Collecting time series data spatially distributed in many locations is often important for analyzing climate change and its impacts on ecosystems. However, comprehensive spatial data collection is not always feasible, requiring us to predict climate variables at some locations. This study focuses on a prediction of climatic elements, specifically near-surface temperature and pressure, at a target location apart from a data observation point. Our approach uses two prediction methods: reservoir computing (RC), known as a machine learning framework with low computational requirements, and vector autoregression models (VAR), recognized as a statistical method for analyzing time series data. Our results show that the accuracy of the predictions degrades with the distance between the observation and target locations. We quantitatively estimate the distance in which effective predictions are possible. We also find that in the context of climate data, a geographical distance is associated with data correlation, and a strong data correlation significantly improves the prediction accuracy with RC. In particular, RC outperforms VAR in predicting highly correlated data within the predictive range. These findings suggest that machine learning-based methods can be used more effectively to predict climatic elements in remote locations by assessing the distance to them from the data observation point in advance. Our study on low-cost and accurate prediction of climate variables has significant value for climate change strategies.

cs.LG↗
arXiv2026

Column Number of Delta-modular matrices: Refined Analysis via Sauer Matrices

In this paper, we build upon the analysis initiated by Gennadiy Averkov and Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \in \mathbb{Z}^{m \times n}$ of rank $m$ is $O(m^3 Δ)$. This upper bound was previously known only for odd values of $Δ$. Recall that a matrix is called $Δ$-modular if the maximum of the absolute values of its $m \times m$ minors equals $Δ$.

math.CO↗
arXiv2025

Complexity of Activity Patterns in a Bio-Inspired Hopfield-Type Network in Different Topologies

Neural network models capable of storing memory have been extensively studied in computer science and computational neuroscience. The Hopfield network is a prototypical example of a model designed for associative, or content-addressable, memory and has been analyzed in many forms. Further, ideas and methods from complex network theory have been incorporated into artificial neural networks and learning, emphasizing their structural properties. Nevertheless, the temporal dynamics also play a vital role in biological neural networks, whose temporal structure is a crucial feature to examine. Biological neural networks display complex intermittency and, thus, can be studied through the lens of the temporal complexity (TC) theory. The TC approach look at the metastability of self-organized states, characterized by a power-law decay in the inter-event time distribution and in the total activity distribution or a scaling behavior in the corresponding event-driven diffusion processes. In this study, we present a temporal complexity (TC) analysis of a biologically-inspired Hopfield-type neural network model. We conducted a comparative assessment between scale-free and random network topologies, with particular emphasis on their global activation patterns. Our parametric analysis revealed comparable dynamical behaviors across both neural network architectures. Furthermore, our investigation into temporal complexity characteristics uncovered that seemingly distinct dynamical patterns exhibit similar temporal complexity behaviors. In particular, similar power-law decay in the activity distribution and similar complexity levels are observed in both topologies, but with a much reduced noise in the scale-free topology. Notably, most of the complex dynamical profiles were consistently observed in scale-free network configurations, thus confirming the crucial role of hubs in neural network dynamics.

q-bio.NC↗
arXiv2026

"More Is Different'' in Neural Circuits: Algebraic Emergence of Effective Theories in Canonical Recurrent Motifs of Biological Neuronal Networks

Canonical neural circuit motifs are usually described functionally: divisive normalization rescales population activity by a pooled signal, and winner-take-all competition selects one pattern through recurrent excitation and shared inhibition. We represent them, and their compositions, algebraically as finite transformation systems and analyze the transition monoids generated by their input-conditioned updates, distinguishing structure already present in a generator from structure that appears only through composition, and, on a joint state space, structure inherited from one factor from structure that lives on a joint configuration. Individually aperiodic updates can generate non-aperiodic monoids. In the WTA, every frozen-drive generator collapses to fixed points, yet short input sequences create local cycles of winner-dependent inhibitory gating: globally dissipative dynamics with a reversible action. The strongest result arises in WTA-to-DN composition. The composed monoid then contains a genuinely composite local cycle in which normalization state and the winner's gating state change together, although every primitive generator is aperiodic. Holonomy analysis certifies this as a group component of the Krohn-Rhodes cascade rather than an incidental cycle, and finds most group-carrying image sets on joint configurations, whereas the uncoupled product has none. An exhaustive interface sweep shows that the composite cycle is a property of the coupling rather than of a chosen map. If motifs are building blocks of neural computation, composing them is a form of programming: one chooses primitives and interfaces so that the generated algebra has the intended repertoire. The transition monoid is that repertoire - what a primitive presents to any later construction. Recurrent circuits are compositional transformation systems; their algebra constrains what they can be programmed to compute.

q-bio.NC↗
arXiv2026

Distinguishing classes of intersection graphs of homothets or similarities of two convex disks

For smooth convex disks $A$, i.e., convex compact subsets of the plane with non-empty interior and with at most one tangent at every boundary point, we classify the classes $G^{\text{hom}}(A)$ and $G^{\text{sim}}(A)$ of intersection graphs that can be obtained from homothets and similarities of $A$, respectively. Namely, we prove that $G^{\text{hom}}(A)=G^{\text{hom}}(B)$ if and only if $A$ and $B$ are affine equivalent, and $G^{\text{sim}}(A)=G^{\text{sim}}(B)$ if and only if $A$ and $B$ are similar.

cs.CG↗
arXiv2026

Optimization of a Triangular Delaunay Mesh Generator using Reinforcement Learning

In this work we introduce a triangular Delaunay mesh generator that can be trained using reinforcement learning to maximize a given mesh quality metric. Our mesh generator consists of a graph neural network that distributes and modifies vertices, and a standard Delaunay algorithm to triangulate the vertices. We explore various design choices and evaluate our mesh generator on various tasks including mesh generation, mesh improvement, and producing variable resolution meshes. The learned mesh generator outputs meshes that are comparable to those produced by Triangle and DistMesh, two popular Delaunay-based mesh generators.

cs.CG↗