Predicting critical transitions in complex systems, such as epileptic seizures in the brain, represents a major challenge in scientific research. The high-dimensional characteristics and hidden critical signals further complicate early-warning tasks. This study proposes a novel early-warning framework that integrates manifold learning with stochastic dynamical system modeling. Through systematic comparison, six methods including diffusion maps (DM) are selected to construct low-dimensional representations. Based on these, a data-driven stochastic differential equation model is established to robustly estimate the probability evolution scoring function of the system. Building on this, a new Score Function (SF) indicator is defined by incorporating Schrödinger bridge theory to quantify the likelihood of significant state transitions in the system. Experiments demonstrate that this indicator exhibits higher sensitivity and robustness in epilepsy prediction, enables earlier identification of critical points, and clearly captures dynamic features across various stages before and after seizure onset. This work provides a systematic theoretical framework and practical methodology for extracting early-warning signals from high-dimensional data.

Deep learning offers powerful tools for anticipating tipping points in complex systems, yet its potential for detecting flickering (noise-driven switching between coexisting stable states) remains unexplored. Flickering is a hallmark of reduced resilience in climate systems, ecosystems, financial markets, and other systems. It can precede critical regime shifts that are highly impactful but difficult to predict. Here we show that convolutional-long short-term memory (CNN-LSTM) models, trained on synthetic time series generated from simple polynomial functions with additive noise, can accurately identify flickering patterns. Despite being trained on simplified dynamics, our models generalize to diverse stochastic systems and reliably detect flickering in empirical datasets, including dormouse body temperature records and palaeoclimate proxies from the African Humid Period. These findings demonstrate that deep learning can extract early warning signals from noisy, nonlinear time series, providing a flexible framework for identifying instability across a wide range of dynamical systems.

Dryland vegetation ecosystems are known to be susceptible to critical transitions between alternative stable states when subjected to external forcing. Such transitions are often discussed through the framework of bifurcation theory, but the spatial patterning of vegetation, which is characteristic of drylands, leads to dynamics that are much more complex and diverse than local bifurcations. Recent methodological developments in Early Warning Signal (EWS) detection have shown promise in identifying dynamical signatures of oncoming critical transitions, with particularly strong predictive capabilities being demonstrated by deep neural networks. However, a machine learning model trained on synthetic examples is only useful if it can effectively transfer to a test case of practical interest. These models' capacity to generalize in this manner has been demonstrated for bifurcation transitions, but it is not as well characterized for high-dimensional phase transitions. This paper explores the successes and shortcomings of neural EWS detection for spatially patterned phase transitions, and shows how these models can be used to gain insight into where and how EWS-relevant information is encoded in spatiotemporal dynamics. A few paradigmatic test systems are used to illustrate how the capabilities of such models can be probed in a number of ways, with particular attention to the performances of a number of proposed statistical indicators for EWS and to the supplementary task of distinguishing between abrupt and continuous transitions. Results reveal that model performance often changes dramatically when training and test data sources are interchanged, which offers new insight into the criteria for model generalization.

Discovering dominant patterns and exploring dynamic behaviors especially critical state transitions and tipping points in high-dimensional time-series data are challenging tasks in study of real-world complex systems, which demand interpretable data representations to facilitate comprehension of both spatial and temporal information within the original data space. Here, we proposed a general and analytical ultralow-dimensionality reduction method for dynamical systems named spatial-temporal principal component analysis (stPCA) to fully represent the dynamics of a high-dimensional time-series by only a single latent variable without distortion, which transforms high-dimensional spatial information into one-dimensional temporal information based on nonlinear delay-embedding theory. The dynamics of this single variable is analytically solved and theoretically preserves the temporal property of original high-dimensional time-series, thereby accurately and reliably identifying the tipping point before an upcoming critical transition. Its applications to real-world datasets such as individual-specific heterogeneous ICU records demonstrated the effectiveness of stPCA, which quantitatively and robustly provides the early-warning signals of the critical/tipping state on each patient.

For anticipating critical transitions in complex dynamical systems, the recent approach of parameter-driven reservoir computing requires explicit knowledge of the bifurcation parameter. We articulate a framework combining a variational autoencoder (VAE) and reservoir computing to address this challenge. In particular, the driving factor is detected from time series using the VAE in an unsupervised-learning fashion and the extracted information is then used as the parameter input to the reservoir computer for anticipating the critical transition. We demonstrate the power of the unsupervised learning scheme using prototypical dynamical systems including the spatiotemporal Kuramoto-Sivashinsky system. The scheme can also be extended to scenarios where the target system is driven by several independent parameters or with partial state observations.

Complex systems can undergo critical transitions, where slowly changing environmental conditions trigger a sudden shift to a new, potentially catastrophic state. Early warning signals for these events are crucial for decision-making in fields such as ecology, biology and climate science. Generic early warning signals motivated by dynamical systems theory have had mixed success on real noisy data. More recent studies found that deep learning classifiers trained on synthetic data could improve performance. However, neither of these methods take advantage of historical, system-specific data. Here, we introduce an approach that trains machine learning classifiers directly on surrogate data of past transitions, namely surrogate data-based machine learning (SDML). The approach provides early warning signals in empirical and experimental data from geology, climatology, sociology, and cardiology with higher sensitivity and specificity than two widely used generic early warning signals -- variance and lag-1 autocorrelation. Since the approach is trained directly on surrogates of historical data, it is not bound by the restricting assumption of a local bifurcation like previous methods. This system-specific approach can contribute to improved early warning signals to help humans better prepare for or avoid undesirable critical transitions.

We consider restricted Boltzmann machines with a binary visible layer and a Gaussian hidden layer trained by an unlabelled dataset composed of noisy realizations of a single ground pattern. We develop a statistical mechanics framework to describe the network generative capabilities, by exploiting the replica trick and assuming self-averaging of the underlying order parameters (i.e., replica symmetry). In particular, we outline the effective control parameters (e.g., the relative number of weights to be trained, the regularization parameter), whose tuning can yield qualitatively-different operative regimes. Further, we provide analytical and numerical evidence for the existence of a sub-region in the space of the hyperparameters where replica-symmetry breaking occurs.

Assessing acute brain dysfunction (ABD), including delirium and coma in the intensive care unit (ICU), is a critical challenge due to its prevalence and severe implications for patient outcomes. Current diagnostic methods rely on infrequent clinical observations, which can only determine a patient's ABD status after onset. Our research attempts to solve these problems by harnessing Electronic Health Records (EHR) data to develop automated methods for ABD prediction for patients in the ICU. Existing models solely predict a single state (e.g., either delirium or coma), require at least 24 hours of observation data to make predictions, do not dynamically predict fluctuating ABD conditions during ICU stay (typically a one-time prediction), and use small sample size, proprietary single-hospital datasets. Our research fills these gaps in the existing literature by dynamically predicting delirium, coma, and mortality for 12-hour intervals throughout an ICU stay and validating on two public datasets. Our research also introduces the concept of dynamically predicting critical transitions from non-ABD to ABD and between different ABD states in real time, which could be clinically more informative for the hospital staff. We compared the predictive performance of two state-of-the-art neural network models, the MAMBA selective state space model and the Longformer Transformer model. Using the MAMBA model, we achieved a mean area under the receiving operator characteristic curve (AUROC) of 0.95 on outcome prediction of ABD for 12-hour intervals. The model achieves a mean AUROC of 0.79 when predicting transitions between ABD states. Our study uses a curated dataset from the University of Florida Health Shands Hospital for internal validation and two publicly available datasets, MIMIC-IV and eICU, for external validation, demonstrating robustness across ICU stays from 203 hospitals and 140,945 patients.

Department of Physics, Arizona State University, Tempe, Arizona 85287, USA (Dated: February 26, 2024) Recent research on the Atlantic Meridional Overturning Circulation (AMOC) raised concern about its potential collapse through a tipping point due to the climate-change caused increase in the freshwater input into the North Atlantic. The predicted time window of collapse is centered about the middle of the century and the earliest possible start is approximately two years from now. More generally, anticipating a tipping point at which the system transitions from one stable steady state to another is relevant to a broad range of fields. We develop a machine-learning approach to predicting tipping in noisy dynamical systems with a time-varying parameter and test it on a number of systems including the AMOC, ecological networks, an electrical power system, and a climate model. For the AMOC, our prediction based on simulated fingerprint data and real data of the sea surface temperature places the time window of a potential collapse between the years 2040 and 2065.

Early warning for epilepsy patients is crucial for their safety and well-being, in particular to prevent or minimize the severity of seizures. Through the patients' EEG data, we propose a meta learning framework to improve the prediction of early ictal signals. The proposed bi-level optimization framework can help automatically label noisy data at the early ictal stage, as well as optimize the training accuracy of the backbone model. To validate our approach, we conduct a series of experiments to predict seizure onset in various long-term windows, with LSTM and ResNet implemented as the baseline models. Our study demonstrates that not only the ictal prediction accuracy obtained by meta learning is significantly improved, but also the resulting model captures some intrinsic patterns of the noisy data that a single backbone model could not learn. As a result, the predicted probability generated by the meta network serves as a highly effective early warning indicator.