The purpose of this paper is to answer a few open questions in the interface of kernel methods and PDE gradient flows. Motivated by recent advances in machine learning, particularly in generative modeling and sampling, we present a rigorous investigation of Fisher-Rao and Wasserstein type gradient flows concerning their gradient structures, flow equations, and their kernel approximations. Specifically, we focus on the Fisher-Rao (also known as Hellinger) geometry and its various kernel-based approximations, developing a principled theoretical framework using tools from PDE gradient flows and optimal transport theory. We also provide a complete characterization of gradient flows in the maximum-mean discrepancy (MMD) space, with connections to existing learning and inference algorithms. Our analysis reveals precise theoretical insights linking Fisher-Rao flows, Stein flows, kernel discrepancies, and nonparametric regression. We then rigorously prove evolutionary $\Gamma$-convergence for kernel-approximated Fisher-Rao flows, providing theoretical guarantees beyond pointwise convergence. Finally, we analyze energy dissipation using the Helmholtz-Rayleigh principle, establishing important connections between classical theory in mechanics and modern machine learning practice. Our results provide a unified theoretical foundation for understanding and analyzing approximations of gradient flows in machine learning applications through a rigorous gradient flow and variational method perspective.

Over the last 15-20 years we have been witnessing a rapid development of machine learning (ML) and neural networks (NN) concepts. As the need for efficient processing and interpretation of large data sets is estimated to further grow in the years to come, many fundamental algorithmic and theoretical NN breakthroughs are to be expected. To be able to adequately address upcoming challenges an excellent understanding of the ultimate limits of the employed technologies is needed. We in this paper study a mathematical problem that is directly connected to a notion of network capacity which is an example of such a limit. Characterizing presence or absence of injectivity as a property of random functions is the mathematical problem of our interest here. The mere definition of the functional injectivity implies its critical role in studying inverse problems. Namely, well-or ill-posedness of these problems is in a direct correspondence with the associated injectivity. Recent utilization of neural networks in studying (nonlinear) inverse problems therefore critically relies on their injectivity properties (see, e.g., [6,11,15,16,19,31,36,38]). Consequently, injectivity as a purely mathematical object is in these contexts transformed into a practically rather important NN architectures feature.

The growing integration of digitized infrastructure with Internet of Things (IoT) networks has transformed the management and optimization of building energy consumption. By leveraging IoT-based monitoring systems, stakeholders such as building managers, energy suppliers, and policymakers can make data-driven decisions to improve energy efficiency. However, accurate energy forecasting and analytics face persistent challenges, primarily due to the inherent physical constraints of buildings and the diverse, heterogeneous nature of IoT-generated data. In this study, we conduct a comprehensive benchmarking of two publicly available IoT datasets, evaluating the performance of time series foundation models in the context of building energy analytics. Our analysis shows that single-modal models demonstrate significant promise in overcoming the complexities of data variability and physical limitations in buildings, with future work focusing on optimizing multi-modal models for sustainable energy management.

Nonlocal, integral operators have become an efficient surrogate for bottom-up homogenization, due to their ability to represent long-range dependence and multiscale effects. However, the nonlocal homogenized model has unavoidable discrepancy from the microscale model. Such errors accumulate and propagate in long-term simulations, making the resultant prediction unreliable. To develop a robust and reliable bottom-up homogenization framework, we propose a new framework, which we coin Embedded Nonlocal Operator Regression (ENOR), to learn a nonlocal homogenized surrogate model and its structural model error. This framework provides discrepancy-adaptive uncertainty quantification for homogenized material response predictions in long-term simulations. The method is built on Nonlocal Operator Regression (NOR), an optimization-based nonlocal kernel learning approach, together with an embedded model error term in the trainable kernel. Then, Bayesian inference is employed to infer the model error term parameters together with the kernel parameters. To make the problem computationally feasible, we use a multilevel delayed acceptance Markov chain Monte Carlo (MLDA-MCMC) method, enabling efficient Bayesian model calibration and model error estimation. We apply this technique to predict long-term wave propagation in a heterogeneous one-dimensional bar, and compare its performance with additive noise models. Owing to its ability to capture model error, the learned ENOR achieves improved estimation of posterior predictive uncertainty.

A key step in any resonant anomaly detection search is accurate modeling of the background distribution in each signal region. Data-driven methods like CATHODE accomplish this by training separate generative models on the complement of each signal region, and interpolating them into their corresponding signal regions. Having to re-train the generative model on essentially the entire dataset for each signal region is a major computational cost in a typical sliding window search with many signal regions. Here, we present SIGMA, a new, fully data-driven, computationally-efficient method for estimating background distributions. The idea is to train a single generative model on all of the data and interpolate its parameters in sideband regions in order to obtain a model for the background in the signal region. The SIGMA method significantly reduces the computational cost compared to previous approaches, while retaining a similar high quality of background modeling and sensitivity to anomalous signals.

The cooperation of a pair of robot manipulators is required to manipulate a target object without any fixtures. The conventional control methods coordinate the end-effector pose of each manipulator with that of the other using their kinematics and joint coordinate measurements. Yet, the manipulators' inaccurate kinematics and joint coordinate measurements can cause significant pose synchronization errors in practice. This paper thus proposes an image-based visual servoing approach for enhancing the cooperation of a dual-arm manipulation system. On top of the classical control, the visual servoing controller lets each manipulator use its carried camera to measure the image features of the other's marker and adapt its end-effector pose with the counterpart on the move. Because visual measurements are robust to kinematic errors, the proposed control can reduce the end-effector pose synchronization errors and the fluctuations of the interaction forces of the pair of manipulators on the move. Theoretical analyses have rigorously proven the stability of the closed-loop system. Comparative experiments on real robots have substantiated the effectiveness of the proposed control.

Cloud cover plays a critical role in weather prediction and impacts several sectors, including agriculture, solar power generation, and aviation. Despite advancements in numerical weather prediction (NWP) models, forecasting total cloud cover remains challenging due to the small-scale nature of cloud formation processes. In this study, we introduce CloudCast, a convolutional neural network (CNN) based on the U-Net architecture, designed to predict total cloud cover (TCC) up to five hours ahead. Trained on five years of satellite data, CloudCast significantly outperforms traditional NWP models and optical flow methods. Compared to a reference NWP model, CloudCast achieves a 24% lower mean absolute error and reduces multi-category prediction errors by 46%. The model demonstrates strong performance, particularly in capturing the large-scale structure of cloud cover in the first few forecast hours, though later predictions are subject to blurring and underestimation of cloud formation. An ablation study identified the optimal input features and loss functions, with MAE-based models performing the best. CloudCast has been integrated into the Finnish Meteorological Institute's operational nowcasting system, where it improves cloud cover forecasts used by public and private sector clients. While CloudCast is limited by a relatively short skillful lead time of about three hours, future work aims to extend this through more complex network architectures and higher-resolution data. CloudCast code is available at https://github.com/fmidev/cloudcast.

Seawater intrusion into coastal aquifers poses a significant threat to groundwater resources, especially with rising sea levels due to climate change. Accurate modeling and uncertainty quantification of this process are crucial but are often hindered by the high computational costs of traditional numerical simulations. In this work, we develop GeoFUSE, a novel deep-learning-based surrogate framework that integrates the U-Net Fourier Neural Operator (U-FNO) with Principal Component Analysis (PCA) and Ensemble Smoother with Multiple Data Assimilation (ESMDA). GeoFUSE enables fast and efficient simulation of seawater intrusion while significantly reducing uncertainty in model predictions. We apply GeoFUSE to a 2D cross-section of the Beaver Creek tidal stream-floodplain system in Washington State. Using 1,500 geological realizations, we train the U-FNO surrogate model to approximate salinity distribution and accumulation. The U-FNO model successfully reduces the computational time from hours (using PFLOTRAN simulations) to seconds, achieving a speedup of approximately 360,000 times while maintaining high accuracy. By integrating measurement data from monitoring wells, the framework significantly reduces geological uncertainty and improves the predictive accuracy of the salinity distribution over a 20-year period. Our results demonstrate that GeoFUSE improves computational efficiency and provides a robust tool for real-time uncertainty quantification and decision making in groundwater management. Future work will extend GeoFUSE to 3D models and incorporate additional factors such as sea-level rise and extreme weather events, making it applicable to a broader range of coastal and subsurface flow systems.

This paper presents a novel learning-based approach for online estimation of maximal safe sets for local motion planning tasks in mobile robotics. We leverage the idea of hypernetworks to achieve good generalization properties and real-time performance simultaneously. As the source of supervision, we employ the Hamilton-Jacobi (HJ) reachability analysis, allowing us to consider general nonlinear dynamics and arbitrary constraints. We integrate our model into a model predictive control (MPC) local planner as a safety constraint and compare the performance with relevant baselines in realistic 3D simulations for different environments and robot dynamics. The results show the advantages of our approach in terms of a significantly higher success rate: 2 to 18 percent over the best baseline, while achieving real-time performance.

Stochastic optimal control (SOC) aims to direct the behavior of noisy systems and has widespread applications in science, engineering, and artificial intelligence. In particular, reward fine-tuning of diffusion and flow matching models and sampling from unnormalized methods can be recast as SOC problems. A recent work has introduced Adjoint Matching (Domingo-Enrich et al., 2024), a loss function for SOC problems that vastly outperforms existing loss functions in the reward fine-tuning setup. The goal of this work is to clarify the connections between all the existing (and some new) SOC loss functions. Namely, we show that SOC loss functions can be grouped into classes that share the same gradient in expectation, which means that their optimization landscape is the same; they only differ in their gradient variance. We perform simple SOC experiments to understand the strengths and weaknesses of different loss functions.