Jeffrey A Fessler

Yixuan Jia, Siyi Chen, Yida Pan et al.

Data assimilation (DA) estimates the state of an evolving dynamical system from noisy, partial observations, and is widely used in scientific simulation as well as weather and climate science. In practice, filtering methods rely on frame-to-frame transition models. However, these models are fragile when observations are non-Markovian (when they form only a partial slice of a higher-dimensional latent state as in real-world weather data): they tend to accumulate errors over long horizons. At the same time, learned DA methods typically commit to a single regime, either filtering (nowcasting, real-time forecasting) or smoothing (retrospective reanalysis), which splits what should be a shared prior across application-specific pipelines. To address both issues, we introduce ForcingDAS, a unified and robust DA framework. Built on Diffusion Forcing with an independent noise level assigned to each frame, ForcingDAS learns a joint-trajectory prior instead of frame-to-frame transitions. This allows it to capture long-horizon temporal dependencies and reduce error accumulation. In addition, the same trained model spans the full filtering to smoothing spectrum at inference time. Specifically, nowcasting, fixed-lag smoothing, and batch reanalysis are selected through the inference schedule alone, without retraining. We evaluate ForcingDAS on 2D Navier-Stokes vorticity, precipitation nowcasting, and global atmospheric state estimation. Across all settings, a single model is competitive with or outperforms both learned and classical baselines that are specialized for individual regimes, with the largest gains observed on real-world weather benchmarks.

Javier Salazar Cavazos, Jeffrey A Fessler, Laura Balzano

Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction. Various methods have been proposed to extend PCA to the union of subspace (UoS) setting for clustering data that come from multiple subspaces like K-Subspaces (KSS). However, some applications involve heterogeneous data that vary in quality due to noise characteristics associated with each data sample. Heteroscedastic methods aim to deal with such mixed data quality. This paper develops a heteroscedastic-focused subspace clustering method, named ALPCAHUS, that can estimate the sample-wise noise variances and use this information to improve the estimate of the subspace bases associated with the low-rank structure of the data. This clustering algorithm builds on K-Subspaces (KSS) principles by extending the recently proposed heteroscedastic PCA method, named LR-ALPCAH, for clusters with heteroscedastic noise in the UoS setting. Simulations and real-data experiments show the effectiveness of accounting for data heteroscedasticity compared to existing clustering algorithms. Code available at https://github.com/javiersc1/ALPCAHUS.

Siyi Chen, Yixuan Jia, Qing Qu et al.

Data assimilation (DA) integrates observations with a dynamical model to estimate states of PDE-governed systems. Model-driven methods (e.g., Kalman, particle) presuppose full knowledge of the true dynamics, which is not always satisfied in practice, while purely data-driven solvers learn a deterministic mapping between observations and states and therefore miss the intrinsic stochasticity of real processes. Recently, score-based diffusion models learn a global diffusion prior and provide a good modeling of the stochastic dynamics, showing new potential for DA. However, their all-at-once generation rather than step-by-step transition limits their performance when dealing with highly complex stochastic processes and lacks physical interpretability. To tackle these drawbacks, we introduce FlowDAS, a generative DA framework that uses stochastic interpolants to directly learn state transition dynamics and achieve step-by-step transition to better model the real dynamics. We also improve the framework by combining the observation, better suiting the DA settings. Directly learning the underlying dynamics from collected data removes restrictive dynamical assumptions, and conditioning on observations at each interpolation step yields stable, measurement-consistent forecasts. Experiments on Lorenz-63, Navier-Stokes super-resolution/sparse-observation scenarios, and large-scale weather forecasting -- where dynamics are partly or wholly unknown -- show that FlowDAS surpasses model-driven methods, neural operators, and score-based baselines in accuracy and physical plausibility.

Anish Lahiri, Jeffrey A Fessler, Luis Hernandez-Garcia

Purpose: Arterial Spin Labeling (ASL) is a quantitative, non-invasive alternative to perfusion imaging with contrast agents. Fixing values of certain model parameters in traditional ASL, which actually vary from region to region, may introduce bias in perfusion estimates. Adopting Magnetic Resonance Fingerprinting (MRF) for ASL is an alternative where these parameters are estimated alongside perfusion, but multiparametric estimation can degrade precision. We aim to improve the sensitivity of ASL-MRF signals to underlying parameters to counter this problem, and provide precise estimates. We also propose a regression based estimation framework for MRF-ASL. Methods: To improve the sensitivity of MRF-ASL signals to underlying parameters, we optimize ASL labeling durations using the Cramer-Rao Lower Bound (CRLB). This paper also proposes a neural network regression based estimation framework trained using noisy synthetic signals generated from our ASL signal model. Results: We test our methods in silico and in vivo, and compare with multiple post labeling delay (multi-PLD) ASL and unoptimized MRF-ASL. We present comparisons of estimated maps for six parameters accounted for in our signal model. Conclusions: The scan design process facilitates precise estimates of multiple hemodynamic parameters and tissue properties from a single scan, in regions of gray and white matter, as well as regions with anomalous perfusion activity in the brain. The regression based estimation approach provides perfusion estimates rapidly, and bypasses problems with quantization error. Keywords: Arterial Spin Labeling, Magnetic Resonance Fingerprinting, Optimization, Cramer-Rao Bound, Scan Design, Regression, Neural Networks, Deep Learning, Precision, Estimation, Brain Hemodynamics.