Predicting a complete spatially correlated field from sparse observations is a fundamental challenge in spatial statistics and environmental modelling. Classical interpolation methods such as Kriging rely on Gaussian process assumptions and variography, which can limit their effectiveness in non-stationary settings and require substantial domain expertise. In this work, we leverage an architecture based on convolutional neural networks (CNNs) for spatial interpolation that is trained and applied on a single partially observed field, without access to external data or prior fields. The model is supervised directly on the observed locations and learns to predict values at unobserved points on the user defined grid. Unlike Kriging, our method does not require explicit covariance modelling or variogram estimation, and it can flexibly capture local spatial patterns in a data-driven manner. This work demonstrates the potential of CNNs for single-instance spatial interpolation under sparse supervision, offering a practical alternative to classical geostatistical methods, and extending the use of CNNs to a new problem domain.

Numeric tabular datasets are the dominant data format in scientific practice, yet large language models lack native mechanisms for representing numeric datasets in a meaningful way across heterogeneous feature spaces. Existing approaches either target predictive modeling over individual datasets, which requires a shared set of variable definitions, or lack mechanisms for interpretable cross-dataset alignment. The proposed methodology characterizes numeric tabular datasets through structured exploratory data analysis descriptors, embeds those descriptors into a shared vector space using a pretrained sentence transformer, and quantifies cross-dataset similarity via Canonical Correlation Analysis (CCA). Furthermore, a penalized formulation of CCA is applied to recover sparse, interpretable variable-level correspondences between datasets, identifying which statistical descriptors or variable-level quantities drive cross-dataset alignment without requiring shared variable names or feature conventions. Differential privacy is optionally applied to the descriptor set prior to embedding, supporting deployment in sensitive data contexts without requiring access to raw observations at time of comparison. The methodology is evaluated across 15 datasets spanning general-purpose benchmarks, materials informatics, and nuclear-grade graphite characterization. Results demonstrate a total P@1 score of 0.9, with known nearest-neighbor retrieval and cluster structure remaining robust across embedding ablations and differential privacy budgets. The proposed framework provides a principled pathway for integrating heterogeneous numeric data into retrieval-augmented generation pipelines while preserving statistical context, with direct applications to data-driven algorithm selection and simulation model initialization for unknown datasets.

Flow and diffusion generative models have established themselves as widely adopted density estimators for simulation-based inference (SBI), extending naturally from neural posterior estimation to likelihood and joint density estimation. Their principled optimization objectives and freedom from architectural constraints have driven rapid adoption across the natural sciences. Yet the most widely used SBI libraries remain PyTorch-based, leaving researchers who develop their forward models and analysis pipelines in JAX without a native option. We present GenSBI, an open-source library that implements flow matching, score matching, and denoising diffusion entirely in JAX. The library offers three transformer-based architectures -- SimFormer, Flux1, and a novel Flux1Joint that extends gate-modulated transformer blocks to joint density estimation -- all interchangeable through a unified interface that decouples generative method, neural backbone, and inference mode. GenSBI provides an end-to-end workflow from training through posterior calibration (SBC, TARP, LC2ST) and supports custom architectures with domain-specific embedding networks.

Deep neural networks (DNNs) have achieved remarkable empirical success, yet their training dynamics remain understood mainly from optimization rather than statistical principles. Here we develop a statistical framework for DNN training in the over-parameterized regime by showing that the prediction induced by continuous-time neural tangent kernel (NTK) gradient flow is exactly equivalent to that from a classical random-effects model. In this framework, training time acts as a variance component, or equivalently an empirical Bayes covariance hyperparameter, governing the allocation of variation from noise to structured signal. This equivalence reveals an optimization-inference duality: the gradient-flow path is both an optimization trajectory and an empirical Bayes random-effects inference path. Conditional on training time, the network output is the posterior mean of the latent signal, and estimating training time by restricted maximum likelihood (REML) turns early stopping into likelihood-based empirical Bayes inference rather than external tuning. This perspective yields a two-stage inferential procedure. First, a variance-component test determines whether DNN training captures statistically significant structure beyond initialization. Second, conditional on training being warranted, REML provides a likelihood-based early stopping rule. The resulting stopping time admits a spectral interpretation in the NTK eigenbasis, where training proceeds until spectral loss decorrelation is achieved. We further establish that REML-guided early stopping achieves asymptotically optimal prediction error for fixed-design in-sample prediction and, under additional random-design regularity conditions, for out-of-sample prediction. This work reframes DNN training as statistical inference and provides a principled foundation for deciding whether and how long to train deep neural networks.

Input-convex neural networks (ICNNs) are widely used for log-concave density estimation, convex-potential normalizing flows, optimal transport, and transport-map inversion for high-dimensional Bayesian posteriors. These tasks share a structural constraint: the inter-layer weights of the ICNN must remain non-negative. The standard recipe, projected gradient descent (PGD) onto the non-negative cone, applies a hard, non-smooth projection -- the stiff-penalty limit of an ADMM-style constraint splitting -- and its classical convergence guarantees do not transfer to the non-smooth ICNN training landscape; the differentiable alternative, softplus reparametrization, attenuates the gradient exponentially in the weight magnitude, stalling training with dead inter-layer weights and plateaued loss. Inspired by parameter-extension lifts of PDE-constrained inverse problems, we propose the lift: instead of constraining the inter-layer weights directly, we train an unconstrained hypernetwork that emits them from a permutation-invariant summary of the input batch. This adds stochasticity to the training dynamics that softens the loss landscape, letting the iterates escape the gradient-attenuated region where direct softplus stalls. We trace this softening to three structural ingredients -- a learnable bias acting as slack, a hypernetwork body that conditions on the target batch, and a cross-covariance coupling the two through batch stochasticity -- and prove each one necessary: deleting any single ingredient collapses the cross-covariance that carries the softening. On log-concave energy-based modeling from one-dimensional toy targets to image-flavored latents, and convex-potential normalizing flows on a 21-dimensional tabular benchmark, we show that the lift reaches a lower test loss than both PGD and direct softplus, and turns a plateau-bounded training trajectory into a valley-descending one.

We establish four structural results for feature learning in wide two-layer neural networks under the Maximal Update Parametrization ($μ$P). First, we prove global existence and uniqueness of the mean-field limit of noisy gradient descent under $μ$P, identifying the maximal admissible weight $w^*$ on the moment sequence of the initialization as the reciprocal parameter-moment-growth boundary, and hence the largest weighted moment class propagated by the flow. The finite-particle approximation has uniform-in-time squared-Wasserstein rate $O(N^{-1})$. Second, we characterize identifiability of the mean-field limit: two admissible parameter measures induce the same network function in $L^2$ exactly when their active components agree modulo the finite-rank realization symmetry of the architecture. The orbit depth $D^*_{\mathrm{orb}}$ is separated from the moment-variety depth $D^*_{\mathrm{var}}$. Third, under the Barron-Hermite target condition the active support of the long-time limit measure admits a sparse-dictionary decomposition: it is supported on at most $S^*$ atoms modulo finite-rank realization symmetry, with $S^*$ bounded by an explicit coefficient-threshold number. Fourth, we derive the total feature-learning-error decomposition into statistical, optimization, propagation-of-chaos, and sparse-residual components, with a target-dependent Hermite/Barron tail replacing any initialization-only residual. The four results are tied together by an architectural identity: the triple $(w^*, D^*_{\mathrm{orb}}, S^*)$ -- the maximal admissible weight, the orbit identifiability depth, and the sparse-dictionary depth at which the target is realizable -- is the natural learning cell of the architecture-data pair $(σ, ρ)$. The proofs are self-contained except for standard results from $μ$P and mean-field Langevin theory.

We consider graph diffusion processes constructed from finite i.i.d. samples drawn from an unknown manifold embedded in ambient Euclidean space, where the graph affinity is defined by an ambient Gaussian kernel matrix. We show that the manifold heat semigroup $Q_t = e^{tΔ}$ can be approximated directly by iterating the graph transition matrix $P$, under only low regularity assumptions on the test function $f$, including the case $f \in L^\infty$. We bound $\| P^n f - Q_t f \|$ in $\infty$-norm, with the operator application to $f$ properly defined, and we recover the classical graph-Laplacian pointwise rate $O(N^{-2/(d+6)})$ up to logarithmic factors, for diffusion times $t $ up to $O(1)$ and longer. The rate holds for in-sample error as well as out-of-sample generalization, where the estimator of $Q_t f$ at a new point is defined via kernel convolution. To handle non-uniform sampling densities on the manifold, we introduce a right-normalization of the graph transition matrix; under the assumption that the sampling density $p$ is $C^3$ and bounded away from zero, the same convergence rates hold. We numerically demonstrate the performance of the proposed estimator on simulated data.

Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we support deterministic, stochastic, and hybrid samplers. We provide error guarantees for these guided procedures. For the deterministic optimizer, a coercivity condition ensures trajectory boundedness and a phase-wise contraction yields logarithmic complexity in the target accuracy. For the stochastic sampler, we introduce adaptive guidance and assume dissipativity of the velocity field to obtain uniform moment bounds independent of the noise-floor parameter. This leads to polynomial-time error bounds, and a matching lower bound shows constant guidance induces an unavoidable positive bias, motivating adaptivity. A hybrid deterministic-stochastic analysis is also provided. Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.

Recently, Deng et al. (2026) proposed Generative Modeling via Drifting (GMD), a novel framework for generative tasks. This note presents an analysis of GMD through the lens of Wasserstein Gradient Flows (WGF), i.e., the path of steepest descent for a functional in the space of probability measures, equipped with the geometry of optimal transport. Unlike previous WGF-based contributions, GMD can be thought of as directly targeting a fixed point of a specific WGF flow. We demonstrate three main results: first, that one algorithm proposed by Deng et al. (2026) corresponds to finding the limiting point of a WGF on the KL divergence, with Parzen smoothing on the densities. Second, that the algorithm actually implemented by Deng et al. (2026) corresponds to a different procedure, which bears some resemblance to the fixed point of a WGF on the Sinkhorn divergence, but lacks certain desirable properties of the latter. Third, the same same idea can be extended to the limiting point of other WGFs, including the Maximum Mean Discrepancy (MMD), the sliced Wasserstein distance, and GAN critic functions.

Many bandit deployments (recommendation, clinical dosing, ad targeting) share two facts prior work handles only in isolation: rewards live on a low-dimensional latent subspace, and that subspace drifts. Stationary low-rank bandits exploit rank but break under subspace change; non-stationary linear bandits adapt to drift but pay ambient rate $\widetilde{O}(d\sqrt{T})$. We study piecewise-stationary low-rank linear contextual bandits with scalar feedback: $θ_t = B_k^\star w_t$ with rank-$r$ factor $B_k^\star\in\mathbb{R}^{d\times r}$ constant within each of $K$ unknown segments and able to shift at boundaries. Our results are tight along three axes. (i) Identification boundary. With single-play scalar rewards, the moving subspace is recoverable through quadratic functionals of rewards iff three probe-side conditions hold: known noise variance, bounded state-noise coupling, and full-dimensional probe support. Each is necessary in the unrestricted-second-moment problem, and jointly they are sufficient, characterizing the boundary of the solvable region. (ii) Algorithm and dynamic regret. SPSC interleaves isotropic probes with windowed projected ridge-UCB exploitation inside the learned $r$-dimensional subspace; a CUSUM-style variant discovers segment boundaries online. The costed dynamic regret is $\widetilde{O}(r\sqrt{T})+\widetilde{O}(T^{2/3})+O(W\,V_{\mathrm{in}})$, replacing the ambient $d\sqrt{T}$ rate with the intrinsic rank. (iii) Empirics. On eleven benchmarks spanning synthetic, UCI/MovieLens, semi-synthetic clinical, and ZOZOTOWN production-log data, SPSC outperforms non-stationary and low-rank baselines whenever $d-r\gtrsim T^{1/6}$, matching the analytical crossover. To our knowledge, this is the first work to characterize the identification boundary and attain the intrinsic-rank dynamic-regret rate in this setting.