We introduce FlowMixer, a single-layer neural architecture that leverages constrained matrix operations to model structured spatiotemporal patterns with enhanced interpretability. FlowMixer incorporates non-negative matrix mixing layers within a reversible mapping framework--applying transforms before mixing and their inverses afterward. This shape-preserving design enables a Kronecker-Koopman eigenmodes framework that bridges statistical learning with dynamical systems theory, providing interpretable spatiotemporal patterns and facilitating direct algebraic manipulation of prediction horizons without retraining. The architecture's semi-group property enables this single layer to mathematically represent any depth through composition, eliminating depth search entirely. Extensive experiments across diverse domains demonstrate FlowMixer's long-horizon forecasting capabilities while effectively modeling physical phenomena such as chaotic attractors and turbulent flows. Our results achieve performance matching state-of-the-art methods while offering superior interpretability through directly extractable eigenmodes. This work suggests that architectural constraints can simultaneously maintain competitive performance and enhance mathematical interpretability in neural forecasting systems.

Biological systems commonly exhibit complex spatiotemporal patterns whose underlying generative mechanisms pose a significant analytical challenge. Traditional approaches to spatiodynamic inference rely on dimensionality reduction through summary statistics, which sacrifice complexity and interdependent structure intrinsic to these data in favor of parameter identifiability. This imposes a fundamental constraint on reliably extracting mechanistic insights from spatiotemporal data, highlighting the need for analytical frameworks that preserve the full richness of these dynamical systems. To address this, we developed a simulation-based inference framework that employs vision transformer-driven variational encoding to generate compact representations of the data, exploiting the inherent contextual dependencies. These representations are subsequently integrated into a likelihood-free Bayesian approach for parameter inference. The central idea is to construct a fine-grained, structured mesh of latent representations from simulated dynamics through systematic exploration of the parameter space. This encoded mesh of latent embeddings then serves as a reference map for retrieving parameter values that correspond to observed data. By integrating generative modeling with Bayesian principles, our approach provides a unified inference framework to identify both spatial and temporal patterns that manifest in multivariate dynamical systems.

Forecasting violent conflict at high spatial and temporal resolution remains a central challenge for both researchers and policymakers. This study presents a novel neural network architecture for forecasting three distinct types of violence -- state-based, non-state, and one-sided -- at the subnational (priogrid-month) level, up to 36 months in advance. The model jointly performs classification and regression tasks, producing both probabilistic estimates and expected magnitudes of future events. It achieves state-of-the-art performance across all tasks and generates approximate predictive posterior distributions to quantify forecast uncertainty. The architecture is built on a Monte Carlo Dropout Long Short-Term Memory (LSTM) U-Net, integrating convolutional layers to capture spatial dependencies with recurrent structures to model temporal dynamics. Unlike many existing approaches, it requires no manual feature engineering and relies solely on historical conflict data. This design enables the model to autonomously learn complex spatiotemporal patterns underlying violent conflict. Beyond achieving state-of-the-art predictive performance, the model is also highly extensible: it can readily integrate additional data sources and jointly forecast auxiliary variables. These capabilities make it a promising tool for early warning systems, humanitarian response planning, and evidence-based peacebuilding initiatives.

Predicting human mobility is crucial for urban planning, traffic control, and emergency response. Mobility behaviors can be categorized into individual and collective, and these behaviors are recorded by diverse mobility data, such as individual trajectory and crowd flow. As different modalities of mobility data, individual trajectory and crowd flow have a close coupling relationship. Crowd flows originate from the bottom-up aggregation of individual trajectories, while the constraints imposed by crowd flows shape these individual trajectories. Existing mobility prediction methods are limited to single tasks due to modal gaps between individual trajectory and crowd flow. In this work, we aim to unify mobility prediction to break through the limitations of task-specific models. We propose a universal human mobility prediction model (named UniMob), which can be applied to both individual trajectory and crowd flow. UniMob leverages a multi-view mobility tokenizer that transforms both trajectory and flow data into spatiotemporal tokens, facilitating unified sequential modeling through a diffusion transformer architecture. To bridge the gap between the different characteristics of these two data modalities, we implement a novel bidirectional individual and collective alignment mechanism. This mechanism enables learning common spatiotemporal patterns from different mobility data, facilitating mutual enhancement of both trajectory and flow predictions. Extensive experiments on real-world datasets validate the superiority of our model over state-of-the-art baselines in trajectory and flow prediction. Especially in noisy and scarce data scenarios, our model achieves the highest performance improvement of more than 14% and 25% in MAPE and Accuracy@5.

The modal analysis techniques face difficulties in handling nonstationary phenomena. This paper presents a variational mode decomposition-based nonstationary coherent structure (VMD-NCS) analysis that enables the extraction and analysis of coherent structures in case of nonstationary phenomena from high-dimensional spatiotemporal data. The VMD-NCS analysis decomposes the input spatiotemporal data into intrinsic coherent structures (ICSs) that represent nonstationary spatiotemporal patterns and exhibit coherence in both the spatial and temporal directions. Furthermore, unlike many conventional modal analysis techniques, the proposed method accounts for the temporal changes in the spatial distribution with time. The performance of the VMD-NCS analysis was validated based on the transient growth phenomena in the flow around a cylinder. It was confirmed that the temporal changes in the spatial distribution, depicting the transient growth of vortex shedding where fluctuations arising in the far-wake region gradually approach the near-wake region, were represented as a single ICS. Further, in the analysis of the quasi-periodic flow field around a pitching airfoil, the temporal changes in the spatial distribution and the amplitude of vortex shedding behind the airfoil, influenced by the pitching motion of the airfoil, were captured as a single ICS. Additionally, the impact of two parameters, adjusting the number of ICSs ($K$) and the penalty factor related to the temporal coherence ($\alpha$), was investigated. The results revealed that $K$ has a significant impact on the VMD-NCS analysis results. In the case of a relatively high $K$, the VMD-NCS analysis tends to extract more periodic spatiotemporal patterns resembling the results of dynamic mode decomposition, whereas in the case of a small $K$, the analysis tends to extract more nonstationary spatiotemporal patterns.

We study image segmentation using spatiotemporal dynamics in a recurrent neural network where the state of each unit is given by a complex number. We show that this network generates sophisticated spatiotemporal dynamics that can effectively divide an image into groups according to a scene's structural characteristics. Using an exact solution of the recurrent network's dynamics, we present a precise description of the mechanism underlying object segmentation in this network, providing a clear mathematical interpretation of how the network performs this task. We then demonstrate a simple algorithm for object segmentation that generalizes across inputs ranging from simple geometric objects in grayscale images to natural images. Object segmentation across all images is accomplished with one recurrent neural network that has a single, fixed set of weights. This demonstrates the expressive potential of recurrent neural networks when constructed using a mathematical approach that brings together their structure, dynamics, and computation.

The precision of this chemical sensing is limited by fluctuations inherent in reaction and diffusion processes involving a Current address: Computational Neuroscience Graduate Program, The University of Chicago.

Discovering the underlying dynamics of complex systems from data is an important practical topic. Constrained optimization algorithms are widely utilized and lead to many successes. Yet, such purely data-driven methods may bring about incorrect physics in the presence of random noise and cannot easily handle the situation with incomplete data. In this paper, a new iterative learning algorithm for complex turbulent systems with partial observations is developed that alternates between identifying model structures, recovering unobserved variables, and estimating parameters. First, a causality-based learning approach is utilized for the sparse identification of model structures, which takes into account certain physics knowledge that is pre-learned from data. It has unique advantages in coping with indirect coupling between features and is robust to the stochastic noise. A practical algorithm is designed to facilitate the causal inference for high-dimensional systems. Next, a systematic nonlinear stochastic parameterization is built to characterize the time evolution of the unobserved variables. Closed analytic formula via an efficient nonlinear data assimilation is exploited to sample the trajectories of the unobserved variables, which are then treated as synthetic observations to advance a rapid parameter estimation. Furthermore, the localization of the state variable dependence and the physics constraints are incorporated into the learning procedure, which mitigate the curse of dimensionality and prevent the finite time blow-up issue. Numerical experiments show that the new algorithm succeeds in identifying the model structure and providing suitable stochastic parameterizations for many complex nonlinear systems with chaotic dynamics, spatiotemporal multiscale structures, intermittency, and extreme events.

A fundamental problem underlying location-based tasks is to construct a complete profile of users' spatiotemporal patterns. In many real-world settings, the sparsity of location data makes it difficult to construct such a profile. As a remedy, we describe a Bayesian probabilistic graphical model, called Collaborative Place Model (CPM), which infers similarities across users to construct complete and time-dependent profiles of users' whereabouts from unsupervised location data. We apply CPM to both sparse and dense datasets, and demonstrate how it both improves location prediction performance and provides new insights into users' spatiotemporal patterns.

User behavior modeling is important for industrial applications such as demographic attribute prediction, content recommendation, and target advertising. Existing methods represent behavior log as a sequence of adopted items and find sequential patterns; however, concrete location and time information in the behavior log, reflecting dynamic and periodic patterns, joint with the spatial dimension, can be useful for modeling users and predicting their characteristics. In this work, we propose a novel model based on graph neural networks for learning user representations from spatiotemporal behavior data. A behavior log comprises a sequence of sessions; and a session has a location, start time, end time, and a sequence of adopted items. Our model's architecture incorporates two networked structures. One is a tripartite network of items, sessions, and locations. The other is a hierarchical calendar network of hour, week, and weekday nodes. It first aggregates embeddings of location and items into session embeddings via the tripartite network, and then generates user embeddings from the session embeddings via the calendar structure. The user embeddings preserve spatial patterns and temporal patterns of a variety of periodicity (e.g., hourly, weekly, and weekday patterns). It adopts the attention mechanism to model complex interactions among the multiple patterns in user behaviors. Experiments on real datasets (i.e., clicks on news articles in a mobile app) show our approach outperforms strong baselines for predicting missing demographic attributes.