Sparse ridge regression problems play a significant role across various domains.

Ensemble simulations of high-dimensional flow models (e.g., Navier-Stokes-type PDEs) are computationally prohibitive for real-time appli cations. Neural operators enable fast inference but are limited by costly data req uirements and poor generalization to 3D flows. We present a data-free operator n etwork for the Navier-Stokes equations that eliminates the need for paire d solution data and enables robust, real-time inference for large ensemble for ecasting. The physics-grounded architecture takes initial and boundary conditio ns as well as forcing functions, yielding solutions robust to high variability a nd perturbations. Across 2D benchmarks and 3D test cases, the method surpasses prior n eural operators in accuracy and, for ensembles, achieves greater efficie ncy than conventional numerical solvers. Notably, it delivers accurate solutions of the three-dimensional Navier-Stokes equations--a regime not previously demonstr ated for data-free neural operators. By uniting a numerically grounded archit ecture with the scalability of machine learning, this approach establishes a pra ctical pathway toward data-free, high-fidelity PDE surrogates for end-to-end sci entific simulation and prediction. Solving PDEs efficiently and accurately is one of the central interests for scienc e and engineering. In addition, when dealing with various boundary conditions, initial con ditions, or external forcing terms of PDEs in fields such as fluid mechanics [1-3], materials science [4, 5], weather forecasting [6, 7], and design optimization [8, 9], P DEs are often required to be solved repeatedly. However, conventional numeric al solvers become prohibitively expensive in such settings, particularly for three-dimensional incompressible Navier-Stokes equations (NSEs) [10, 11]. This is because these s olvers rely on spatial-temporal discretization and iterative treatment of nonline ar terms, while performing time marching that demands substantial memory and computation. Moreover, they are not well suited for solving large ensembles of scenarios simu ltaneously, such as those required for uncertainty quantification or design explora tion. The resulting computational time, coupled with the need for extensive sampling in e nsemble or probabilistic simulations, constitutes a critical bottleneck [7, 12].

Sparse ridge regression problems play a significant role across various domains.

We present Latent Diffeomorphic Dynamic Mode Decomposition (LDDMD), a new data reduction approach for the analysis of non-linear systems that combines the interpretability of Dynamic Mode Decomposition (DMD) with the predictive power of Recurrent Neural Networks (RNNs). Notably, LDDMD maintains simplicity, which enhances interpretability, while effectively modeling and learning complex non-linear systems with memory, enabling accurate predictions. This is exemplified by its successful application in streamflow prediction.

Learning from human feedback is a popular approach to train robots to adapt to user preferences and improve safety. Existing approaches typically consider a single querying (interaction) format when seeking human feedback and do not leverage multiple modes of user interaction with a robot. We examine how to learn a penalty function associated with unsafe behaviors, such as side effects, using multiple forms of human feedback, by optimizing the query state and feedback format. Our framework for adaptive feedback selection enables querying for feedback in critical states in the most informative format, while accounting for the cost and probability of receiving feedback in a certain format. We employ an iterative, two-phase approach which first selects critical states for querying, and then uses information gain to select a feedback format for querying across the sampled critical states. Our evaluation in simulation demonstrates the sample efficiency of our approach.

This work explores sequential model editing in large language models (LLMs), a critical task that involves modifying internal knowledge within LLMs continuously through multi-round editing, each incorporating updates or corrections to adjust the model outputs without the need for costly retraining. Existing model editing methods, especially those that alter model parameters, typically focus on single-round editing and often face significant challenges in sequential model editing-most notably issues of model forgetting and failure. To address these challenges, we introduce a new model editing method, namely \textbf{N}euron-level \textbf{S}equential \textbf{E}diting (NSE), tailored for supporting sequential model editing. Specifically, we optimize the target layer's hidden states using the model's original weights to prevent model failure. Furthermore, we iteratively select neurons in multiple layers for editing based on their activation values to mitigate model forgetting. Our empirical experiments demonstrate that NSE significantly outperforms current modifying parameters model editing methods, marking a substantial advancement in the field of sequential model editing. Our code is released on \url{https://github.com/jianghoucheng/NSE}.

We present a knowledge-guided machine learning (KGML) framework for modeling multi-scale processes, and study its performance in the context of streamflow forecasting in hydrology. Specifically, we propose a novel hierarchical recurrent neural architecture that factorizes the system dynamics at multiple temporal scales and captures their interactions. This framework consists of an inverse and a forward model. The inverse model is used to empirically resolve the system's temporal modes from data (physical model simulations, observed data, or a combination of them from the past), and these states are then used in the forward model to predict streamflow. In a hydrological system, these modes can represent different processes, evolving at different temporal scales (e.g., slow: groundwater recharge and baseflow vs. fast: surface runoff due to extreme rainfall). A key advantage of our framework is that once trained, it can incorporate new observations into the model's context (internal state) without expensive optimization approaches (e.g., EnKF) that are traditionally used in physical sciences for data assimilation. Experiments with several river catchments from the NWS NCRFC region show the efficacy of this ML-based data assimilation framework compared to standard baselines, especially for basins that have a long history of observations. Even for basins that have a shorter observation history, we present two orthogonal strategies of training our FHNN framework: (a) using simulation data from imperfect simulations and (b) using observation data from multiple basins to build a global model. We show that both of these strategies (that can be used individually or together) are highly effective in mitigating the lack of training data. The improvement in forecast accuracy is particularly noteworthy for basins where local models perform poorly because of data sparsity.

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new Spatiotemporal Fourier Neural Operator (SFNO) that learns maps between Bochner spaces, and a new learning framework to address these issues. This new paradigm leverages wisdom from traditional numerical PDE theory and techniques to refine the pipeline of commonly adopted end-to-end neural operator training and evaluations. Specifically, in the learning problems for the turbulent flow modeling by the Navier-Stokes Equations (NSE), the proposed architecture initiates the training with a few epochs for SFNO, concluding with the freezing of most model parameters. Then, the last linear spectral convolution layer is fine-tuned without the frequency truncation. The optimization uses a negative Sobolev norm for the first time as the loss in operator learning, defined through a reliable functional-type \emph{a posteriori} error estimator whose evaluation is almost exact thanks to the Parseval identity. This design allows the neural operators to effectively tackle low-frequency errors while the relief of the de-aliasing filter addresses high-frequency errors. Numerical experiments on commonly used benchmarks for the 2D NSE demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers.