Papers

Efforts to improve Kolmogorov–Arnold networks (KANs) with architectural enhancements have been stymied by the complexity those enhancements bring, undermining the interpretability that makes KANs attractive in the first place. Here we study overprovisioned architectures combined with sparsification, deep supervision, and depth selection, to learn compact, interpretable KANs without sacrificing accuracy. Crucially, we focus on differentiable mechanisms under a principled minimum description length objective, jointly optimizing activations, structure, and depth end-to-end. Experiments across function approximation benchmarks, dynamical systems forecasting, and real-world prediction tasks demonstrate that sparsification alone is insufficient, but the combination with depth selection achieves competitive or superior accuracy while discovering substantially smaller models. The result is a principled path toward models that are both more expressive and more interpretable, addressing a key tension in scientific machine learning.

Almost all scientific data have uncertainties originating from different sources. Gaussian process regression (GPR) models are a natural way to model data with Gaussian-distributed uncertainties. GPR also has the benefit of reducing I/O bandwidth and storage requirements for large scientific simulations. However, the reconstruction from the GPR models suffers from high computation complexity. To make the situation worse, classic approaches for visualizing the data uncertainties, like probabilistic marching cubes, are also computationally very expensive, especially for data of high resolutions. In this paper, we accelerate the level-crossing probability calculation efficiency on GPR models by subdividing the data spatially into a hierarchical data structure and only reconstructing values adaptively in the regions that have a non-zero probability. For each region, leveraging the known GPR kernel and the saved data observations, we propose a novel approach to efficiently calculate an upper bound for the level-crossing probability inside the region and use this upper bound to make the subdivision and reconstruction decisions. We demonstrate that our value occurrence probability estimation is accurate with a low computation cost by experiments that calculate the level-crossing probability fields on different datasets.

Identifying the graphical structure underlying the observed multivariate data is essential in numerous applications. Current methodologies are predominantly confined to deducing a singular graph under the presumption that the observed data are uniform. However, many contexts involve heterogeneous datasets that feature multiple closely related graphs, typically referred to as multiview graphs. Previous research on multiview graph learning promotes edge-based similarity across layers using pairwise or consensus-based regularizers. However, multiview graphs frequently exhibit a shared node-based architecture across different views, such as common hub nodes. Such commonalities can enhance the precision of learning and provide interpretive insight. In this paper, we propose a co-hub node model, positing that different views share a common group of hub nodes. The associated optimization framework is developed by enforcing structured sparsity on the connections of these co-hub nodes. Moreover, we present a theoretical examination of layer identifiability and determine bounds on estimation error. The proposed methodology is validated using both synthetic graph data and fMRI time series data from multiple subjects to discern several closely related graphs.

Understanding how habitats shape species distributions and abundances across river networks remains a longstanding and fundamental challenge in ecology, with direct implications for effective biodiversity management and conservation. We introduce a scalable spatial stream network (S3N) model that enables estimation, inference, and prediction with greater computational efficiency than previously possible. S3Ns extend nearest-neighbor Gaussian processes (NNGPs) to include ecologically salient stream network dependence structure. Additionally, S3Ns implement more efficient preprocessing than SSNs; while the computational cost of estimation is a function of the number of observation points and not of the number of reaches, the preprocessing is a function of both. We demonstrate that S3Ns accurately recover spatial and covariance parameters 2-3 orders of magnitude faster than existing spatial stream network models. We then apply S3Ns to estimate the population sizes and geographic distributions of 285 fish species in the entire Ohio River Basin (>4,000 river km, approximately 170,000 reaches and 9,000 observation points) on a laptop. These results indicate the promise of S3Ns for mapping freshwater variables and quantifying the influence of environmental drivers across extensive, complex river networks with many observation points.

We present a simple structure based model of how words are formed from morphemes. The model explains two major empirical facts: the typical distribution of word lengths and the appearance of Zipf like rank frequency curves. In contrast to classical explanations based on random text or communication efficiency, our approach uses only the combinatorial organization of prefixes, roots, suffixes and inflections. In this Morphemic Combinatorial Word Model, a word is created by activating several positional slots. Each slot turns on with a certain probability and selects one morpheme from its inventory. Morphemes are treated as stable building blocks that regularly appear in word formation and have characteristic positions. This mechanism produces realistic word length patterns with a concentrated middle zone and a thin long tail, closely matching real languages. Simulations with synthetic morpheme inventories also generate rank frequency curves with Zipf like exponents around 1.1-1.4, similar to English, Russian and Romance languages. The key result is that Zipf like behavior can emerge without meaning, communication pressure or optimization principles. The internal structure of morphology alone, combined with probabilistic activation of slots, is sufficient to create the robust statistical patterns observed across languages.

The designation “Bernstein-von Mises theorem” is apparently due to Lucien Le Cam. Roughly, the assertion of this theorem states that the posterior distribution of a parameter, conditioned on a large sample, is approximately normal, independent of a particular prior. The present paper discusses important steps in the development of this theorem and its applications, from Laplace in 1774 to Le Cam in 1953. Regarding Bernstein and his disciple Neyman, it thereby relies on sources which were widely unknown and hard to obtain until recently.

We provide an epsilon-delta interpretation of Chatterjee's rank correlation by tracing its origin to a notion of local dependence between random variables. Starting from a primitive epsilon-delta construction, we show that rank-based dependence measures arise naturally as epsilon to zero limits of local averaging procedures. Within this framework, Chatterjee's rank correlation admits a transparent interpretation as an empirical realization of a local L1 residual. We emphasize that the probability integral transform plays no structural role in the underlying epsilon-delta mechanism, and is introduced only as a normalization step that renders the final expression distribution-free. We further consider a moment-based analogue obtained by replacing the absolute deviation with a squared residual. This L2 formulation is independent of rank transformations and, under a Gaussian assumption, recovers Pearson's coefficient of determination.

The systematic collection of longitudinal data is very common in practice, making mixed models widely used. Most developments around these models focus on modeling the mean trajectory of repeated measurements, typically under the assumption of homoskedasticity. However, as data become increasingly rich through intensive collection over time, these models can become limiting and may introduce biases in analysis. In fact, such data are often heterogeneous, with the presence of outliers, heteroskedasticity, and asymmetry in the distribution of individual measurements. Therefore, ignoring these characteristics can lead to biased modeling results. In this work, we propose a mixed-effect distributional regression model based on the asymmetric Laplace distribution to: (1) address the presence of outliers, heteroskedasticity, and asymmetry in longitudinal measurements; (2) model the entire individual distribution of the heterogeneous longitudinal response over time, rather than just its conditional expectation; and (3) give a more comprehensive evaluation of the impact of covariates on the distribution of the responses through meaningful indicator. A Bayesian estimation procedure is presented. In order to choose between two distributional regression models, we also propose a new model selection criterion for longitudinal data. It measures the proximity between the individual distribution estimated by the model and the empirical individual distribution of the data over time, using a set of quantiles. The estimation procedure and the selection criterion are validated in a simulation study and the proposed model is compared to a distributional regression mixed model based on the Gaussian distribution and a location-scale linear quantile mixed model. Finally, the proposed model is applied to analyze blood pressure over time for hospitalized patients in the intensive care unit.

We develop a supervised deep-learning approach to estimate mutual information between two continuous random variables. As labels, we use the Linfoot informational correlation, a transformation of mutual information that has many important properties. Our method is based on ground truth labels for Gaussian and Clayton copulas. We compare our method with estimators based on kernel density, k-nearest neighbours and neural estimators. We show generally lower bias and lower variance. As a proof of principle, future research could look into training the model with a more diverse set of examples from other copulas for which ground truth labels are available.

Proper quantification of predictive uncertainty is essential for the use of machine learning in safety-critical applications. Various uncertainty measures have been proposed for this purpose, typically claiming superiority over other measures. In this paper, we argue that there is no single best measure. Instead, uncertainty quantification should be tailored to the specific application. To this end, we use a flexible family of uncertainty measures that distinguishes between total, aleatoric, and epistemic uncertainty of second-order distributions. These measures can be instantiated with specific loss functions, so-called proper scoring rules, to control their characteristics, and we show that different characteristics are useful for different tasks. In particular, we show that, for the task of selective prediction, the scoring rule should ideally match the task loss. On the other hand, for out-of-distribution detection, our results confirm that mutual information, a widely used measure of epistemic uncertainty, performs best. Furthermore, in an active learning setting, epistemic uncertainty based on zero-one loss is shown to consistently outperform other uncertainty measures.

Quantile regression (QR) is now widely used to analyze the effect of covariates on the conditional distribution of a response variable. It provides a more comprehensive picture of the relationship between a response and covariates compared with classical least squares regression. However, the non-differentiability of the check loss function precludes the use of gradient-based methods to solve the optimization problem in quantile regression estimation. To this end, This paper constructs a smoothed loss function based on multiquadric (MQ) function. The proposed loss function leads to a globally convex optimization problem that can be efficiently solved via (stochastic) gradient descent methods. As an example, we apply the Barzilai-Borwein gradient descent method to obtain the estimation of quantile regression. We establish the theoretical results of the proposed estimator under some regularity conditions, and compare it with other estimation methods using Monte Carlo simulations.

We prove that a classic sub-Gaussian mixture proposed by Robbins in a stochastic setting actually satisfies a path-wise (deterministic) regret bound. For every path in a natural “Ville event” $\mathcal E_\alpha$, this regret till time $T$ is bounded by $\ln^2(1/\alpha)/V_T + \ln (1/\alpha) + \ln \ln V_T$ up to universal constants, where $V_T$ is a nonnegative, nondecreasing, cumulative variance process. (The bound reduces to $\ln(1/\alpha) + \ln \ln V_T$ if $V_T \geq \ln(1/\alpha)$.) If the data were stochastic, then one can show that $\mathcal E_\alpha$ has probability at least $1-\alpha$ under a wide class of distributions (eg: sub-Gaussian, symmetric, variance-bounded, etc.). In fact, we show that on the Ville event $\mathcal E_0$ of probability one, the regret on every path in $\mathcal E_0$ is eventually bounded by $\ln \ln V_T$ (up to constants). We explain how this work helps bridge the world of adversarial online learning (which usually deals with regret bounds for bounded data), with game-theoretic statistics (which can handle unbounded data, albeit using stochastic assumptions). In short, conditional regret bounds serve as a bridge between stochastic and adversarial betting.

In this paper, we derive an integral representation for the distribution of the number of types $K_n$ in the Ewens-Pitman model. Based on this representation, we also establish precise large deviations and precise moderate deviations for $K_n$. After careful examination, we find that the rate function exhibits a second-order phase transition and the critical point is $\alpha=\frac{1}{2}$.

Outlier detection and concept drift detection represent two challenges in data analysis. Most studies address these issues separately. However, joint detection mechanisms in regression remain underexplored, where the continuous nature of output spaces makes distinguishing drifts from outliers inherently challenging. To address this, we propose a novel robust regression framework for joint outlier and concept drift detection. Specifically, we introduce a dual-channel decision process that orchestrates prediction residuals into two coupled logic flows: a rapid response channel for filtering point outliers and a deep analysis channel for diagnosing drifts. We further develop the Exponentially Weighted Moving Absolute Deviation with Distinguishable Types (EWMAD-DT) detector to autonomously differentiate between abrupt and incremental drifts via dynamic thresholding. Comprehensive experiments on both synthetic and real-world datasets demonstrate that our unified framework, enhanced by EWMAD-DT, exhibits superior detection performance even when point outliers and concept drifts coexist.

Currently, more and more machine learning (ML) surrogates are being developed for computationally expensive physical models. In this work we investigate the use of a Multi-Fidelity Ensemble Kalman Filter (MF-EnKF) in which the low-fidelity model is such a machine learning surrogate model, instead of a traditional low-resolution or reduced-order model. The idea behind this is to use an ensemble of a few expensive full model runs, together with an ensemble of many cheap but less accurate ML model runs. In this way we hope to reach increased accuracy within the same computational budget. We investigate the performance by testing the approach on two common test problems, namely the Lorenz-2005 model and the Quasi-Geostrophic model. By keeping the original physical model in place, we obtain a higher accuracy than when we completely replace it by the ML model. Furthermore, the MF-EnKF reaches improved accuracy within the same computational budget. The ML surrogate has similar or improved accuracy compared to the low-resolution one, but it can provide a larger speed-up. Our method contributes to increasing the effective ensemble size in the EnKF, which improves the estimation of the initial condition and hence accuracy of the predictions in fields such as meteorology and oceanography.

We propose Hellinger-type loss functions for training Generative Adversarial Networks (GANs), motivated by the boundedness, symmetry, and robustness properties of the Hellinger distance. We define an adversarial objective based on this divergence and study its statistical properties within a general parametric framework. We establish the existence, uniqueness, consistency, and joint asymptotic normality of the estimators obtained from the adversarial training procedure. In particular, we analyze the joint estimation of both generator and discriminator parameters, offering a comprehensive asymptotic characterization of the resulting estimators. We introduce two implementations of the Hellinger-type loss and we evaluate their empirical behavior in comparison with the classic (Maximum Likelihood-type) GAN loss. Through a controlled simulation study, we demonstrate that both proposed losses yield improved estimation accuracy and robustness under increasing levels of data contamination.

Almost seventy years old Marshall-Olkin copulas, then wider Marshall copulas, and finally even wider shock model (SM) copulas constitute a substantial part of nowadays copula theory due to numerous applications. Recently, Christian Genest with some coauthors introduced a new stochastic model for a special subclass of SM copulas which gives not only a new angle on these copulas but also widens the range of applications. In this paper we extend this type of stochastic model to all known subclasses of SM copulas. We also introduce a novel class of SM copulas and extend the new stochastic model to this class as well.

Bivariate copulas with prescribed diagonal section were first studied by Bertino. Their maximality was studied so far only from the point of view of upper bounds which brings quasi-copulas into the picture and limits the resulting set substantially. We propose to study maximality of these families in the order theoretic sense. A copula C with given diagonal section {\delta} is called undominated if there is no copula C' {\neq} C with the same diagonal section {\delta} such that C {\leq} C'. The main contribution of this paper is a new method that provides copulas of the kind. This method generates a much wider class that contains the known upper bounds as a very small subclass. There was a recent call for the study of asymmetry which is addressed by our class better than by the known ones. Corresponding quasi-copulas can be obtained from our copulas via splicing techniques. Most results are given on the level of tracks.

Dynamic decision-making in rapidly evolving research domains, including marketing, finance, and pharmaceutical development, presents a significant challenge. Researchers frequently confront the need for real-time action within a doubly sequential framework characterized by the continuous influx of high-volume data streams and the intermittent arrival of novel tasks. This calls for the development and implementation of new online inference protocols capable of handling both the continuous processing of incoming information and the efficient allocation of resources to address emerging priorities. We introduce a novel class of Safe and Always-Valid Alpha-investing (SAVA) rules that leverages powerful tools including always valid p-values, e-processes, and online false discovery rate methods. The SAVA algorithm effectively integrates information across all tasks, mitigates the alpha-death problem, and controls the false selection rate (FSR) at all decision points. We validate the efficacy of the SAVA framework through rigorous theoretical analysis and extensive numerical experiments. Our results demonstrate that SAVA not only offers effective control of the FSR but also significantly improves statistical power compared to traditional online testing approaches.

Model selection is a pivotal process in the quantitative sciences, where researchers must navigate between numerous candidate models of varying complexity. Traditional information criteria, such as the corrected Akaike Information Criterion (AICc), Bayesian Information Criterion (BIC), and Mallows's $\mathit{C_p}$, are valuable tools for identifying optimal models. However, the exponential increase in candidate models with each additional model parameter renders the evaluation of these criteria for all models – a strategy known as exhaustive, or brute-force, searches – computationally prohibitive. Consequently, heuristic approaches like stepwise regression are commonly employed, albeit without guarantees of finding the globally-optimal model. In this study, we challenge the prevailing notion that non-monotonicity in information criteria precludes bounds on the search space. We introduce a simple but novel bound that enables the development of branch-and-bound algorithms tailored for these non-monotonic functions. We demonstrate that our approach guarantees identification of the optimal model(s) across diverse model classes, sizes, and applications, often with orders of magnitude computational speedups. For instance, in one previously-published model selection task involving $2^{32}$ (approximately 4 billion) candidate models, our method achieves a computational speedup exceeding 6,000. These findings have broad implications for the scalability and effectiveness of model selection in complex scientific domains.

Financial exclusion remains a major barrier to digital public service delivery in resource-constrained and archipelagic nations. Traditional policy evaluations rely on retrospective data, limiting the ex-ante intelligence needed for agile resource allocation. This study introduces a predictive simulation framework to support anticipatory governance within government information systems. Using the UNCDF Pacific Digital Economy dataset of 10,108 respondents, we apply a three-stage pipeline: descriptive profiling, interpretable machine learning, and scenario simulation to forecast outcomes of digital financial literacy interventions before deployment. Leveraging cross-sectional structural associations, the framework projects intervention scenarios as prioritization heuristics rather than causal estimates. A transparent linear regression model with R-squared of 95.9 identifies modifiable policy levers. Simulations indicate that foundational digital capabilities such as device access and expense tracking yield the highest projected gains, up to 5.5 percent, outperforming attitudinal nudges. The model enables precision targeting, highlighting young female caregivers as high-leverage responders while flagging non-responders such as urban professionals to prevent resource misallocation. This research demonstrates how static survey data can be repurposed into actionable policy intelligence, offering a scalable and evidence-based blueprint for embedding predictive analytics into public-sector decision-support systems to advance equity-focused digital governance.

Van den Akker, Werker, and Zhou (2025) showed that the limit experiment, in the sense of H\a'{a}jek-Le Cam, for (contextual) bandits whose arms' expected payoffs differ by $O(T^{-1/2})$, is Locally Asymptotically Quadratic (LAQ) but highly non-standard, being characterized by a system of coupled stochastic differential equations. The present paper considers the complementary case where the arms' expected payoffs are fixed with a unique optimal (in the sense of highest expected payoff) arm. It is shown that, under sampling schemes satisfying mild regularity conditions (including UCB and Thompson sampling), the model satisfies the standard Locally Asymptotically Normal (LAN) property.

Learned time-series models, whether continuous or discrete, are widely used for forecasting the states of dynamical systems but suffer from error accumulation in multi-step forecasts. To address this issue, we propose a Predictor-Corrector framework in which the Predictor is a learned time-series model that generates multi-step forecasts and the Corrector is a neural controlled differential equation that corrects the forecast errors. The Corrector works with irregularly sampled time series and is compatible with both continuous- and discrete-time Predictors. We further introduce two regularization strategies that improve the Corrector's extrapolation performance and accelerate its training. We also provide theoretical guarantees on the stability and convergence of the proposed framework. Experiments on synthetic, physics-based, and real-world datasets show that the proposed framework consistently improves forecasting performance across diverse Predictors, including neural ordinary differential equations, ContiFormer, and DLinear, demonstrating its predictor-agnostic nature.

This paper studies robust estimation in the dynamic Tobit model under local-to-unity (LUR) asymptotics. We show that both Gaussian maximum likelihood (ML) and censored least absolute deviations (CLAD) estimators are consistent, extending results from the stationary case where ordinary least squares (OLS) is inconsistent. The asymptotic distributions of MLE and CLAD are derived; for the short-run parameters they are shown to be Gaussian, yielding standard normal t-statistics. In contrast, although OLS remains consistent under LUR, its t-statistics are not standard normal. These results enable reliable model selection via sequential t-tests based on ML and CLAD, paralleling the linear autoregressive case. Applications to financial and epidemiological time series illustrate their practical relevance.

We present a comprehensive study on fully automated pollen recognition across both conventional optical and digital in-line holographic microscopy (DIHM) images of sample slides. Visually recognizing pollen in unreconstructed holographic images remains challenging due to speckle noise, twin-image artifacts and substantial divergence from bright-field appearances. We establish the performance baseline by training YOLOv8s for object detection and MobileNetV3L for classification on a dual-modality dataset of automatically annotated optical and affinely aligned DIHM images. On optical data, detection mAP50 reaches 91.3% and classification accuracy reaches 97%, whereas on DIHM data, we achieve only 8.15% for detection mAP50 and 50% for classification accuracy. Expanding the bounding boxes of pollens in DIHM images over those acquired in aligned optical images achieves 13.3% for detection mAP50 and 54% for classification accuracy. To improve object detection in DIHM images, we employ a Wasserstein GAN with spectral normalization (WGAN-SN) to create synthetic DIHM images, yielding an FID score of 58.246. Mixing real-world and synthetic data at the 1.0 : 1.5 ratio for DIHM images improves object detection up to 15.4%. These results demonstrate that GAN-based augmentation can reduce the performance divide, bringing fully automated DIHM workflows for veterinary imaging a small but important step closer to practice.

Data valuation is a natural framework for understanding which data sources matter most when aligning a Large Language Model (LLM) from multiple sources. The standard game-theoretic approach treats each source, or equivalently each preference dataset, as a player in a cooperative game and assigns it a contribution score through the Shapley value. In practice, however, Shapley-based valuation is computationally prohibitive because it requires aligning a separate model for every possible coalition of sources, i.e., an exponential number of alignments. We address this challenge for Direct Alignment Algorithms (DAAs), including IPO, which learn through log-policy ratios with respect to a reference policy. We show that, when a model is aligned sequentially source by source, exact optimization makes each stage contribute additively to the log-probability of a full response, up to a prompt-dependent normalization constant. This allows the log-probability assigned by any coalition to a fixed response to be reconstructed from the base policy and the policies trained on each source individually. This reduces the alignment cost of Shapley-based valuation from exponential to linear, since only one model per source needs to be trained to evaluate coalition scores. We test whether this theoretical property remains approximately valid under finite training across several base models and real-world data sources. We finally compute the Shapley values of these sources under multiple reward models, showing how their estimated contributions vary across evaluation criteria.

Understanding and distinguishing temporal patterns in time series data is essential for scientific discovery and decision-making. For example, in biomedical research, uncovering meaningful patterns in physiological signals can improve diagnosis, risk assessment, and patient outcomes. However, existing methods for time series pattern discovery face major challenges, including high computational complexity, limited interpretability, and difficulty in capturing meaningful temporal structures. To address these gaps, we introduce a novel learning framework that jointly trains two Transformer models using complementary time series representations: shapelet-based representations to capture localized temporal structures and traditional feature engineering to encode statistical properties. The learned shapelets serve as interpretable signatures that differentiate time series across classification labels. Additionally, we develop a visual analytics system – SigTIme – with coordinated views to facilitate exploration of time series signatures from multiple perspectives, aiding in useful insights generation. We quantitatively evaluate our learning framework on eight publicly available datasets and one proprietary clinical dataset. Additionally, we demonstrate the effectiveness of our system through two usage scenarios along with the domain experts: one involving public ECG data and the other focused on preterm labor analysis.

For many conditions, it is of clinical importance to know not just the ability of a test to distinguish between those with and without the disease, but also the sensitivity to detect disease at different stages: in particular, the test's ability to detect disease at a stage most amenable to treatment. In a systematic review of test accuracy, pooled stage-specific estimates can be produced using subgroup analysis or meta-regression. However, this requires stage-specific data from each study, which is often not reported. Studies may however report test sensitivity for merged stage categories (e.g. stages I-II) or merged across all stages, together with information on the proportion of patients with disease at each stage. We demonstrate how to incorporate studies reporting merged stage data alongside studies reporting stage-specific data, to allow the inclusion of more studies in the meta-analysis. We consider both meta-analysis of tests with binary results, and meta-analysis of tests with continuous results, where the sensitivity to detect disease of each stage across the whole range of observed thresholds is estimated. The methods are demonstrated using a series of simulated datasets and applied to data from a systematic review of the accuracy of tests used to screen for hepatocellular carcinoma in people with liver cirrhosis. We show that incorporating studies with merged stage data can lead to more precise estimates and, in some cases, corrects biologically implausible results that can arise when the availability of stage-specific data is limited.

stochtree is a C++ library for Bayesian tree ensemble models such as BART and Bayesian Causal Forests (BCF), as well as user-specified variations. Unlike previous BART packages, stochtree provides bindings to both R and Python for full interoperability. stochtree boasts a more comprehensive range of models relative to previous packages, including heteroskedastic forests, random effects, and treed linear models. Additionally, stochtree offers flexible handling of model fits: the ability to save model fits, reinitialize models from existing fits (facilitating improved model initialization heuristics), and pass fits between R and Python. On both platforms, stochtree exposes lower-level functionality, allowing users to specify models incorporating Bayesian tree ensembles without needing to modify C++ code. We illustrate the use of stochtree in three settings: i) straightfoward applications of existing models such as BART and BCF, ii) models that include more sophisticated components like heteroskedasticity and leaf-wise regression models, and iii) as a component of custom MCMC routines to fit nonstandard tree ensemble models.

Goal-Conditioned Reinforcement Learning (GCRL) mitigates the difficulty of reward design by framing tasks as goal reaching rather than maximizing hand-crafted reward signals. In this setting, the optimal goal-conditioned value function naturally forms a quasimetric, motivating Quasimetric RL (QRL), which constrains value learning to quasimetric mappings and enforces local consistency through discrete, trajectory-based constraints. We propose Eikonal-Constrained Quasimetric RL (Eik-QRL), a continuous-time reformulation of QRL based on the Eikonal Partial Differential Equation (PDE). This PDE-based structure makes Eik-QRL trajectory-free, requiring only sampled states and goals, while improving out-of-distribution generalization. We provide theoretical guarantees for Eik-QRL and identify limitations that arise under complex dynamics. To address these challenges, we introduce Eik-Hierarchical QRL (Eik-HiQRL), which integrates Eik-QRL into a hierarchical decomposition. Empirically, Eik-HiQRL achieves state-of-the-art performance in offline goal-conditioned navigation and yields consistent gains over QRL in manipulation tasks, matching temporal-difference methods.