Critical Slowing Down

As modern societies rely more on autonomous systems to facilitate daily life, assuring their safe operation is paramount. Naturally, there are many techniques available to predict and prevent system failures. However, the safety afforded by such schemes may become misaligned with the true system, which can change in unexpected ways - from partial faults to natural wear-and-tear - that subtly degrade its stability. The implications that such subtle changes have on autonomous system stability can be observed through generic indicators of resilience derived from critical slowing down, popular for anticipating catastrophic tipping points in natural systems. Here, we show how one can systematically design these generic indicators for nonlinear control systems and show how these can reflect loss of stability though simulations of canonical robotic systems wherein their proximity to instability is manipulated directly. These results are affirmed through real-world flight experiments of a quadrotor that is nudged towards instability by progressively damaging its propeller blades. Our results show that the implications of degraded resilience on closed-loop stability are evident well before they appear, for which the indicators of resilience derived here can provide an early warning.

A looped transformer performs inference by iterating a weight-tied map, making its computation a dynamical process whose cost is set by the resulting inference dynamics. Here we show that networks with identical architecture and objective, trained to identical accuracy, nevertheless realize distinct dynamical phases depending strongly on initialization, and that the bifurcation defining each phase determines how test-time compute scales. The phases are distinguished by their bifurcation mechanisms, including a saddle-node fold and a Neimark-Sacker-type transition to bounded nonstationary motion. In the fold phase, a one-dimensional normal-form reduction predicts both the relaxation-time and spectral-gap amplitudes from local derivatives of the trained map, yielding the parameter-free relation $τ(\varepsilon)[1-λ_{\max}(-\varepsilon)]\toπ$. Composed with a regular distribution of problem difficulty, the same critical slowing down produces the workload-level tail $P(τ>N)\sim N^{-2}$. In the Neimark--Sacker phase, the fold scaling law disappears rather than merely changing its prefactor. Thus, test-time compute is not determined by architecture alone. It is governed by the dynamical phase of the solution found by training.

We develop a novel forecasting scheme to anticipate controller induced loss of control (LOC) in quadrotors and evaluate it on real LOC flight data from four different quadrotors. For this, early warning signals of LOC are derived using critical slowing down (CSD), a generic phenomenon shown to precede critical transitions across various complex ecological and biological systems. As such, our early warning indicators are generic in the sense that no system models are needed to facilitate forecasts of LOC. The approach is evaluated on real quadrotor flight data wherein LOC occurs due to unstable controller behavior arising from input-output delays. Our approach achieves a time-to-LOC forecast of up to 0.9 seconds before LOC occurs, outperforming state-of-the-art recurrent neural network quadrotor LOC forecasters in terms of detection accuracy and LOC data reliance. In particular, we leverage insights from CSD to accurately predict LOC without using data of the LOC event itself. Going further, we apply our forecasters without any re-parameterization to anticipate a different LOC scenario, quadrotor flyways, that occur on other quadrotors flying both indoors and outdoors. Despite these differences, our approach successfully detects LOC, demonstrating that it can generalize across controller architectures, quadrotors, and LOC scenarios.

Monte Carlo methods are essential across diverse scientific fields, yet their efficiency is frequently hampered by critical slowing down-a sharp increase in autocorrelation times near phase transitions. Although deep learning approaches, such as neural-network-based samplers, have been proposed to alleviate this issue, they face another serious problem: the difficulty of training the models. This difficulty partially stems from the overly general nature of original machine-learning architectures, which often ignore underlying physical symmetries and force networks to relearn them from scratch. In this paper, we demonstrate that incorporating physical priors into the model significantly enhances performance. Building upon existing strategies that integrate spin-spin interactions, we propose a framework that utilizes a prior probability distribution as a starting point for training. Our results for the Ising model, as well as for the Edwards-Anderson spin glass model, suggest that moving away from `blank slate' models in favor of physics-informed priors reduces the training burden and facilitates the simulation of larger system sizes in discrete spin models.

Computational sampling has been central to the sciences since the mid-20th century. While machine-learning-based approaches have recently enabled major advances, their behavior remains poorly understood, with limited theoretical control over when and why they succeed. Here we provide such insight for diffusion models-a class of generative schemes highly effective in practice-by analyzing their application to the $O(n)$ model of statistical field theory in the Gaussian limit $n \to \infty$. In this analytically tractable setting, we show that training a score model with a one-layer network architecture matching the exact solution exhibits a form of critical slowing down in parameter learning. This slowing down also impacts the generation process, indicating that the well-known difficulties of sampling near criticality persist even for learned generative models. To overcome this bottleneck, we demonstrate the power of combining architectural depth with physical locality. We find that using a two-layer architecture drastically reduces the critical slowing down, with the training time scaling logarithmically rather than quadratically with system size. By introducing a local score approximation we show that this acceleration in training time can be achieved without increasing the number of neural network parameters. Taken together, these results demonstrate that diffusion models can overcome the critical slowing down through appropriate architectural design, and establish a controlled framework for understanding and improving learned sampling methods in statistical physics and beyond.

TokaMind is a multi-modal transformer (MMT) foundation model pre-trained on tokamak plasma diagnostics data from MAST, where it was shown to outperform CNN-based approaches on fusion benchmarks. We investigate whether its learned representations generalize to physically distinct but structurally analogous domains. Through systematic experimentation across four domains-industrial bearing degradation, NASA CMAPSS turbofan degradation, and two independent power grid PMU datasets-we identify four transfer-favoring characteristics that help explain where TokaMind's pretrained representations are most effective. Power grid synchrophasor data matches this target-domain profile most directly, while industrial degradation datasets demonstrate that TokaMind can still yield useful performance under partial alignment, especially when task design and feature construction expose physically meaningful degradation structure. On the GESL/PNNL 500-event benchmark with provider-aware evaluation, TokaMind achieves test $\text{F1} = 0.837 \pm 0.040$ (3seeds) for severe event classification. Our central finding, however, is not the aggregate score: classification difficulty is structurally determined by provider-level grid topology, not model capacity. In the single-window early-warning regime, TokaMind outperforms a CNN baseline (F10.889 vs.~0.878)--a reversal that disappears as more event windows are provided. Furthermore, Critical Slowing Down (CSD) indicators, used as a confidence gate rather than a classification label, improve F1 from 0.696 to 0.750 at 63% coverage-outperforming the CNN baseline (0.636) at any coverage level. These results establish the first cross-domain validation of TokaMind outside nuclear fusion and propose a transferability framework and revised evaluation protocol for multi-source PMU datasets.

One partner says "Fine" meaning "resolution"; the other hears "surrender." The word is shared; the affective uptake is not. We formalize this as affective meaning divergence (AMD), the total-variation distance between interlocutors' anchor-conditioned affect distributions. Building on speech-act theory, common-ground accumulation, and entropy-regularized game theory, we derive a logit best-response map whose dynamics undergo a saddle-node bifurcation: when $βα> 4$, a monotone increase in AMD-driven load produces an abrupt, hysteretic collapse of repair coordination. On Conversations Gone Awry (CGA-Wiki; $N = 652$), derailing conversations exhibit critical-slowing-down (CSD) signatures across multiple levels: lexical divergence variance ($p < 0.001$, $d = 0.36$), AMD variance ($p = 0.001$, $d = 0.26$), and dialog-act repair variance ($p = 0.016$, $d = 0.20$), all significant after correction and stronger than toxicity and sentiment baselines. AMD provides a distinct temporal signature, with retrospectively measured variance peaking at the bifurcation point while toxicity variance peaks earlier, and is the only indicator grounded in the theoretical framework. Boundary-condition analysis on CGA-CMV ($N = 1,169$) yields mixed but directionally consistent evidence.

This paper develops a dynamical framework for adaptive coordination in systems of interacting agents referred to here as Feedback-Coupled Memory Systems (FCMS). Instead of framing coordination as equilibrium optimization or agent-centric learning, the model describes a closed-loop interaction between agents, incentives, and a persistent environment. The environment stores accumulated coordination signals, a distributed incentive field transmits them locally, and agents update in response, generating a feedback-driven dynamical system. Three main results are established. First, under dissipativity, the closed-loop system admits a bounded forward-invariant region, ensuring dynamical viability independently of global optimality. Second, when incentives depend on persistent environmental memory, coordination cannot be reduced to a static optimization problem. Third, within the FCMS class, coordination requires a bidirectional coupling in which memory-dependent incentives influence agent updates, while agent behavior reshapes the environmental state. Numerical analysis of a minimal specification identifies a Neimark-Sacker bifurcation at a critical coupling threshold ($β_c$), providing a stability boundary for the system. Near the bifurcation threshold, recovery time diverges and variance increases, yielding a computable early warning signature of coordination breakdown in observable time series. Additional simulations confirm robustness under nonlinear saturation and scalability to populations of up to $N = 10^{6}$ agents making it more relevant for real-world applications. The proposed framework offers a dynamical perspective on coordination in complex systems, with potential extensions to multi-agent systems, networked interactions, and macro-level collective dynamics.