Data Driven Equation Discovery for Phase-Ordering Dynamics : From Allen Cahn to the Ising Model

Data-driven discovery of governing equations from spatiotemporal data offers a promising route to obtaining coarse-grained descriptions of complex dynamical systems. Here, we investigate the performance of PDE-SINDy for discovering phase-ordering dynamics using the Allen--Cahn equation as a benchmark and the Ising model with Glauber spin-flip dynamics as a microscopic system. We systematically analyze the effects of data availability, size of the candidate library, and noise on the efficiency of the equation discovery. We find that stability-selection PDE-SINDy can robustly identify the relevant terms in the governing dynamics even under limited or noisy data, while the recovered coefficient values are substantially more sensitive to these factors. We further show that enlarging the candidate library can strongly affect both term identification and coefficient recovery. Incorporating library bagging with stability selection reduces this sensitivity and improves the efficiency of equation discovery. For the Glauber spin flip Ising model dynamics, the resulting coarse-grained equation reproduces the characteristic phase-separation and coarsening dynamics of the underlying microscopic system. Overall, our results demonstrate the potential of PDE-SINDy for phase-ordering systems while highlighting the importance of carefully assessing the factors that influence the efficiency of equation discovery.

Spatiotemporal Proximal Causal Inference under Hidden Confounding and Interference

Estimating causal effects from real-world spatiotemporal data is challenging due to hidden confounders and interference. Standard causal identification methods assume conditional exchangeability given observed covariates, which fails whenever hidden confounders affect both treatment and outcomes - a common setting in domains such as climate, environmental policy, epidemiology, and regional economics. In this paper, we propose a novel spatiotemporal proximal causal inference framework that extends proximal identification theory to spatiotemporal settings. The proposed method jointly captures local and neighborhood-level confounding information by introducing treatment- and outcome-inducing proxies, and we derive a spatiotemporal outcome confounding bridge function that identifies the potential outcome without requiring direct recovery of the hidden confounder. We establish the identifiability of this bridge function under proxy exclusion restrictions and a spatiotemporal completeness condition, and show that the resulting estimator recovers the outcome through a proximal generalization of the g-computation formula. To operationalize this identification result, we propose a neural architecture that learns proxies via transformer-based spatiotemporal encoders - coupled with a conditional mutual information critic to enforce exclusion restrictions and a moment-matching network to guarantee that the learned bridge function satisfies the underlying identifying equation. We further introduce a stabilized weighting scheme to address treatment support imbalance. Experiments on synthetic datasets demonstrate that our approach achieves comparable performance to baseline causal inference methods, while providing, to our knowledge, the first theoretically grounded outcomes for the hidden confounding in the presence of spatiotemporal interference through a proximal causal inference framework.

Variational Low-rank Tensor Decomposition for Multisubject Spatiotemporal Data Analysis

Modeling shared and subject-specific structure in multisubject spatiotemporal data remains challenging, particularly in neuroimaging, where both spatial and temporal patterns exhibit rich variability across subjects. Existing matrix and tensor decompositions provide interpretable factorizations, but rely on fixed multilinear structures or coupling schemes that may limit their flexibility in capturing complex variability. In this work, we introduce a spatiotemporal variational tensor decomposition (ST-VTD) framework that combines a tensor factorization generative model with structured priors to jointly represent spatial maps and temporal dynamics. Spatial factors are regularized to promote a low-rank structure inspired by the LL1 decomposition, while temporal factors are modeled using a learned Long short-term memory (LSTM)-based prior, enabling flexible and adaptive dynamics. Posterior inference is performed using an amortized variational formulation by unrolling iterations of an optimization algorithm, leading to an interpretable and parameter-efficient architecture. The proposed inference framework employs a warm-start strategy based on group independent component analysis, which we found to improve optimization performance. Experiments on a realistic synthetic functional MRI (fMRI) dataset demonstrate that the proposed approach significantly improves latent factor recovery compared with representative classical and probabilistic decomposition benchmarks.

CASC: Causal Adversarial Subspace Clustering for Multivariate Spatiotemporal Data

Deep subspace clustering plays a critical role in applications involving multivariate spatiotemporal data, such as sea ice monitoring, disease spread analysis, and tracking neuro-degeneration over time. Despite recent advances, existing methods primarily rely on geometric self-expressiveness, assume static subspace structures, and often fail to capture causal dependencies, local spatial interactions, and long-range temporal dynamics inherent in complex spatiotemporal systems. To address these limitations, we propose a novel Causal Adversarial Subspace Clustering (CASC) framework for discovering evolving latent regimes in high-dimensional spatiotemporal data. CASC integrates a U-Net-inspired deep adversarial clustering architecture with stacked FAConvLSTM layers to preserve spatial and temporal structure while learning robust latent representations. A graph attention transformer-based self-expressive network is introduced to jointly model local spatial relationships, global dependencies, and long-range temporal interactions. Furthermore, we propose two new learning objectives: (1) a Causal Subspace Preservation Loss that aligns self-expression coefficients with latent causal relationships, encouraging clusters to reflect underlying causal processes rather than simple feature similarity, and (2) a Dynamic Temporal Subspace Evolution Loss that captures evolving subspace structures and temporal regime transitions in nonstationary environments. Together, these components transform deep subspace clustering from a correlation-driven paradigm into a causal-temporal regime discovery framework.

Beyond the Performance Illusion: Structure-Aware Stratified Partitioning and Curriculum Distributionally Robust Optimization for Spatially Correlated Domains

Performance evaluation in AI systems commonly assumes that random dataset splits produce independent and identically distributed (i.i.d.) subsets. We show that this assumption often breaks down in spatiotemporally correlated domains such as aerial surveillance, precision agriculture, and medical imaging, leading to two systematic failures: data leakage, where correlated samples span training and validation splits and inflate performance estimates, and hidden stratification, where errors on minority subpopulations are obscured by aggregate metrics. To address these issues, we propose a unified evaluation and training framework for spatially correlated data. We introduce Structure-Aware Stratified Partitioning (SASP), which constructs validation splits that reduce spatiotemporal leakage while preserving meaningful class balance, and Curriculum Distributionally Robust Optimization (CDRO), a curriculum-based relaxation of distributionally robust training that stabilizes optimization under these stricter splits. Across multiple benchmarks, this combination yields consistently improved generalization, more reliable confidence calibration, and exposes failure modes that remain hidden under conventional random-split evaluation.