Physics-informed neural networks (PINNs) have recently gained attention as a powerful and efficient tool for solving partial differential equations (PDEs). Despite their empirical success, the theoretical understanding of PINNs, especially in the context of over-parameterization, remains incomplete. This paper presents a complete error analysis of over-parameterized PINNs for elliptic equations using projected stochastic gradient descent (PSGD) optimization. Our analysis rigorously examines the interplay of approximation error, statistical error, and optimization error, offering a unified framework for understanding the convergence behavior of PINNs. By leveraging the properties of PSGD, we establish convergence rates and derive conditions on neural network architecture, training sample requirements, and optimization parameters to ensure specified accuracy.

This work proposes deep nonparametric Instrumental variable quantile regression (IVQR), a two-stage estimator that combines conditional diffusion modeling with a kernel-smoothed conditional moment formulation. In the first stage, we estimate the joint conditional distribution of the outcome and endogenous covariates given the instrument using a variance-preserving conditional diffusion model. In the second stage, we approximate the conditional moment operator through Monte Carlo sampling and a kernel-smoothed surrogate for the indicator function, and then estimate the structural quantile function by empirical risk minimization over deep neural networks. We establish an excess-risk bound for the proposed estimator and derive end-to-end total variation guarantees for the conditional diffusion model under unbounded support, explicitly accounting for score estimation, early stopping, and discretization errors. Our theory is developed under a polynomial-tail envelope on the data distribution and degenerates continuously to the exponential setting: as the tail index grows, the obtained excess-risk rate converges to the minimax-optimal rate of nonparametric regression, thus our heavy-tailed theory covers the classical light-tailed nonparametric guarantees as a limiting case. Simulation studies and a real-data application demonstrate that the proposed method outperforms existing nonparametric IVQR approaches, with gains that become increasingly pronounced as the dimensionality of the covariates and instruments increases.

Sexual reproduction in plants is a tightly coordinated process that underpins biodiversity, drives speciation, and serves as a foundation for modern crop improvement. Here, we synthesize recent advances in elucidating the molecular mechanisms governing major reproductive events in flowering plants, including male and female gametogenesis, pollen-pistil communication, double fertilization, embryogenesis, and endosperm-mediated seed development. We also discuss the integration of asexual propagation through protoplast regeneration, highlighting its emerging importance in plant biotechnology. Together, these insights deepen our understanding of fundamental reproductive biology and open new opportunities for the rational design of hybrid breeding systems and stress-resilient crops, with broad implications for global food security under changing climatic conditions.

Vector-borne plant viruses depend on insect vectors for transmission and often suppress host defenses that limit vector survival and spread. However, their impact on volatile-mediated indirect defenses remains unclear. Here, we show that rice viruses inhibit methyl salicylate (MeSA) emission, impairing parasitoid recruitment and promoting vector persistence. Field experiments demonstrate that MeSA, a key herbivore-induced volatile, suppresses vector populations by attracting egg parasitoids. Viruses counter this by targeting basic-helix-loop-helix transcription factor OsMYC2, a jasmonic acid signaling hub, thereby down-regulating OsBSMT1 and MeSA biosynthesis, responses conserved across diverse rice viruses and vector species. MeSA applications in the field restore parasitoid-mediated vector suppression, highlighting its potential for sustainable disease control. MeSA is a central ecological signal in a previously unidentified viral strategy that enhances transmission.

Inference-time alignment for diffusion models aims to adapt a pre-trained reference diffusion model toward a target distribution without retraining the reference score network, thereby preserving the generative capacity of the reference model while enforcing desired properties at the inference time. A central mechanism for achieving such alignment is guidance, which modifies the sampling dynamics through an additional drift term. In this work, we introduce variationally stable Doob's matching, a novel framework for provable guidance estimation grounded in Doob's h-transform. Our approach formulates guidance as the gradient of logarithm of an underlying Doob's h-function and employs gradient-regularized regression to simultaneously estimate both the h-function and its gradient, resulting in a consistent estimator of the guidance. Theoretically, we establish non-asymptotic convergence rates for the estimated guidance. Moreover, we analyze the resulting controllable diffusion processes and prove non-asymptotic convergence guarantees for the generated distributions in the 2-Wasserstein distance. Finally, we show that variationally stable guidance estimators are adaptive to unknown low dimensionality, effectively mitigating the curse of dimensionality under low-dimensional subspace assumptions.

Schrödinger-Föllmer sampler (SFS) is a novel and efficient approach for sampling from possibly unnormalized distributions without ergodicity. SFS is based on the Euler-Maruyama discretization of Schrödinger-Föllmer diffusion process $$\mathrm{d} X_{t}=-\nabla U\left(X_t, t\right) \mathrm{d} t+\mathrm{d} B_{t}, \quad t \in[0,1],\quad X_0=0$$ on the unit interval, which transports the degenerate distribution at time zero to the target distribution at time one. In \cite{sfs21}, the consistency of SFS is established under a restricted assumption that %the drift term $b(x,t)$ the potential $U(x,t)$ is uniformly (on $t$) strongly %concave convex (on $x$). In this paper we provide a nonasymptotic error bound of SFS in Wasserstein distance under some smooth and bounded conditions on the density ratio of the target distribution over the standard normal distribution, but without requiring the strongly convexity of the potential.

Site-directed mutagenesis of TaHOX2 homoeologs enhances floret fertility and grain number in wheat.

Genome editing via CRISPR/Cas9 has been widely adopted in cereal crops. In diploid species such as rice and barley, the generation of knockout mutants is relatively straightforward for functional characterization of the genes of interest due to their single-copy nature in the genome. In contrast, common wheat (Triticum aestivum L.) is a hexaploid species comprising three subgenomes (AABBDD); consequently, most genes are present as three homoeoalleles that retained substantial function redundancy during evolution. The generation of a complete set of single, double, and triple mutants is therefore essential for elucidating homoeoallele-specific functions and dissecting their contributions to the developmental and agronomic traits. Moreover, ensuring germplasm purity through the elimination of residual T-DNA is critical for maintaining stable mutation, particularly in single- and double-mutant lines. Here, we describe a hierarchical screening strategy for efficient identification of a comprehensive series of CRISPR/Cas9-induced mutants. This approach integrates high-throughput DNA isolation, selection of T-DNA-free mutants, maintenance of a uniform genetic background via backcrossing, systematic screening of all mutant combinations, and molecular confirmation of genome edits. This screening pipeline has proven effective in hexaploid common wheat and is readily adaptable to other polyploid species that are amenable to crossing. © 2026 Wiley Periodicals LLC. Basic Protocol 1: Wheat cultivation and leaf sample preparation Basic Protocol 2: High-throughput DNA isolation Basic Protocol 3: Genotyping using an optimized, cost-efficient T7E1 assay Alternate Protocol 1: Genotyping using the KASP assay Support Protocol 1: Screening of T-DNA-free triple mutants.

In offline RL, estimating the optimal action-value function Q^* can be formulated as solving the optimal Bellman equation based solely on offline observations. A fundamental challenge is that the reward function and transition kernel are unknown, so the optimal Bellman operator is not directly observable from data. To address this issue, we propose a novel framework that decouples operator estimation from value function learning. In this approach, we first formulate conditional diffusion models to estimate the reward law and transition kernel, which induces a data-driven approximation of the optimal Bellman operator. We then plug these estimators into the Bellman equation and obtain a deep estimator of Q^* by minimizing the empirical Bellman residual over a neural network function class. Theoretically, we first establish sharp nonasymptotic convergence rates for learning the optimal Bellman operator through an end-to-end analysis of conditional diffusion estimation in total variation distance. We then establish the oracle value-stage rate 𝒪(n^-2β/d_x+d_a+2β) for the excess Bellman residual risk. Finally, under a concentrability condition, we translate this residual bound into an L^2 convergence rate of 𝒪(n^-d_x+d_a+2β) for the resulting deep estimator of Q^*, where d_x and d_a denote the dimensions of the state and action spaces, respectively, and β denotes the Hölder smoothness index of Q^*. Importantly, our theoretical analysis does not rely on completeness assumptions commonly used in deep RL theory. Extensive numerical experiments demonstrate the effectiveness of the proposed method and its strong empirical performance.

We propose a novel method for sampling from unnormalized Boltzmann densities based on a probability-flow ordinary differential equation (ODE) derived from linear stochastic interpolants. The key innovation of our approach is the use of a sequence of Langevin samplers to enable efficient simulation of the flow. Specifically, these Langevin samplers are employed (i) to generate samples from the interpolant distribution at intermediate times and (ii) to construct, starting from these intermediate times, a robust estimator of the velocity field governing the flow ODE. For both applications of the Langevin diffusions, we establish convergence guarantees. Extensive numerical experiments demonstrate the efficiency of the proposed method on challenging multimodal distributions across a range of dimensions, as well as its effectiveness in Bayesian inference tasks.

This paper introduces score-based sequential Langevin sampling (SSLS), a novel approach to nonlinear data assimilation within a recursive Bayesian filtering framework. The proposed method decomposes the assimilation process into alternating prediction and update steps, using dynamic models for state prediction and incorporating observational data via score-based Langevin Monte Carlo during the updates. To overcome inherent challenges in highly non-log-concave posterior sampling, we integrate an annealing strategy into the update mechanism. Theoretically, we establish convergence guarantees for SSLS in total variation (TV) distance, yielding concrete insights into the algorithm's error behavior with respect to key hyperparameters. Crucially, our derived error bounds demonstrate the asymptotic stability of SSLS, guaranteeing that local posterior sampling errors do not accumulate indefinitely over time. Extensive numerical experiments across challenging scenarios, including high-dimensional systems, strong nonlinearity, and sparse observations, highlight the robust performance of the proposed method. Furthermore, SSLS effectively quantifies the uncertainty associated with state estimates, rendering it particularly valuable for reliable error calibration.

Grass inflorescence morphology displays remarkable diversity across species and is a key determinant of crop yield. Here, to elucidate how developmental morphodynamics shapes inflorescence architecture, we conducted a comparative analysis of early inflorescence development in bread wheat and rice. Computational modelling revealed that meristem fate transition and primordium initiation modes collectively contribute to the observed architecture diversity. Furthermore, the model elucidates the formation of distinct supernumerary spikelet types in wheat and predicts two independent developmental pathways for generating paired spikelets-a specialized form of inflorescence branching. We also identified a mutant allele, duo2, that results in accelerated developmental progression and demonstrated significant yield improvement in duo2 plants under field conditions. The causal gene RA2-D, an orthologue of maize RAMOSA2 (RA2), was found to regulate floral transition. This study elucidates how perturbations in developmental dynamics drive the diversification of grass inflorescence morphologies.

This paper investigates the off-policy evaluation (OPE) problem from a distributional perspective. Rather than focusing solely on the expectation of the total return, as in most existing OPE methods, we aim to estimate the entire return distribution. To this end, we introduce a quantile-based approach for OPE using deep quantile process regression, presenting a novel algorithm called Deep Quantile Process regression-based Off-Policy Evaluation (DQPOPE). We provide new theoretical insights into the deep quantile process regression technique, extending existing approaches that estimate discrete quantiles to estimate a continuous quantile function. A key contribution of our work is the rigorous sample complexity analysis for distributional OPE with deep neural networks, bridging theoretical analysis with practical algorithmic implementations. We show that DQPOPE achieves statistical advantages by estimating the full return distribution using the same sample size required to estimate a single policy value using conventional methods. Empirical studies further show that DQPOPE provides significantly more precise and robust policy value estimates than standard methods, thereby enhancing the practical applicability and effectiveness of distributional reinforcement learning approaches.

Conditional generative modeling remains a challenging problem in semi-supervised settings where labeled data is scarce but unlabeled samples are abundant. To effectively leverage structural information embedded within the unlabeled dataset and compensate for sparse conditioning signals, we propose a semi-supervised framework combining conditional stochastic interpolation with low-dimensional latent representations. RepG decomposes generation into two stages: label-dependent latent sampling and high-dimensional reconstruction. This isolates the supervised learning of conditional dependencies to a low-dimensional space, requiring few labels while utilizing the abundant unlabeled data purely for reconstruction. Theoretically, we establish an error decomposition showing that the Kullback-Leibler divergence of RepG comprises stage-wise estimation errors and a structural bias quantified by conditional mutual information. For deep neural network estimators, we derive non-asymptotic convergence rates proving that RepG significantly improves sample complexity. By confining the supervised estimation burden to the low intrinsic dimension of the latent representation, RepG achieves a strictly faster convergence rate. Complemented by a minimax lower bound, our theoretical results demonstrate that this method effectively mitigates the curse of dimensionality inherent in direct ambient-space generative modeling.

Clustering is a fundamental problem in many scientific applications. This paper introduces the concept of K-regression, which divides a random sample of size n into K clusters such that the observations within each cluster exhibit an identical linear pattern of dependence, and the observations in different clusters exhibit distinctive structures of linear dependence. We estimate the coefficients of the clustering regressions through minimizing the within cluster & ell;(1) and & ell;(2) loss functions. From the asymptotic perspective, the resulting estimates obtained with either the & ell;(1) or the & ell;(2) loss are strongly consistent and asymptotically normal. From the non-asymptotic perspective, we further explore the conditions under which the models are identifiable and the algorithms are convergent. Furthermore, we propose a tailored Bayesian Information Criterion (BIC) designed specifically for regression-based clustering. Through extensive simulations and an application to clinical trial subgroup analysis, we demonstrate the effectiveness of K-regression. Numerical results highlight that, in the presence of heterogeneity, & ell;K-1-regression outperforms alternative methods (including & ell;K-2-regression) in coefficient estimation, cluster number determination, and subgroup classification while maintaining computational efficiency. These advantages make & ell;K-1-regression particularly appealing for large-scale data analysis, especially when heterogeneous subpopulations are present.

Data is important in many deep learning-based inverse problem solvers. However, obtaining sufficient paired data in many scenarios remains highly challenging, while unpaired data is cheap. To maximize data utilization, this paper proposes LUD-DIF, a diffusion-based approach for solving inverse problems with unpaired data. Starting from the evidence lower bound (ELBO) of the joint distribution, we decouple it into two independent diffusion processes under the weak-coupling assumption. The method provides theoretical support from a variational inference perspective, derives the loss function, quantitatively analyzes the error bound introduced by the assumption, and offers a theorem-motivated heuristic for hyperparameter selection. Experimental results demonstrate that LUD-DIF achieves outstanding performance on multiple image inverse problems, validating its effectiveness and generalization capability in unpaired inverse problem settings.

In this work, we propose a novel deep bootstrap framework for nonparametric regression based on conditional diffusion models. Specifically, we construct a conditional diffusion model to learn the distribution of the response variable given the covariates. This model is then used to generate bootstrap samples by pairing the original covariates with newly synthesized responses. We reformulate nonparametric regression as conditional sample mean estimation, which is implemented directly via the learned conditional diffusion model. Unlike traditional bootstrap methods that decouple the estimation of the conditional distribution, sampling, and nonparametric regression, our approach integrates these components into a unified generative framework. With the expressive capacity of diffusion models, our method facilitates both efficient sampling from high-dimensional or multimodal distributions and accurate nonparametric estimation. We establish rigorous theoretical guarantees for the proposed method. In particular, we derive optimal end-to-end convergence rates in the Wasserstein distance between the learned and target conditional distributions. Building on this foundation, we further establish the convergence guarantees of the resulting bootstrap procedure. Numerical studies demonstrate the effectiveness and scalability of our approach for complex regression tasks.

Large Language Models (LLMs) have demonstrated remarkable proficiency across diverse tasks, exhibiting emergent properties such as semantic prompt comprehension, In-Context Learning (ICL), and Chain-of-Thought (CoT) reasoning. Despite their empirical success, the theoretical mechanisms driving these phenomena remain poorly understood. This study dives into the foundations of these observations by addressing three critical questions: (1) How do LLMs accurately decode prompt semantics despite being trained solely on a next-token prediction objective? (2) Through what mechanism does ICL facilitate performance gains without explicit parameter updates? and (3) Why do intermediate reasoning steps in CoT prompting effectively unlock capabilities for complex, multi-step problems? Our results demonstrate that, through the autoregressive process, LLMs are capable of exactly inferring the transition probabilities between tokens across distinct tasks using provided prompts. We show that ICL enhances performance by reducing prompt ambiguity and facilitating posterior concentration on the intended task. Furthermore, we find that CoT prompting activates the model's capacity for task decomposition, breaking complex problems into a sequence of simpler sub-tasks that the model has mastered during the pretraining phase. By comparing their individual error bounds, we provide novel theoretical insights into the statistical superiority of advanced prompt engineering techniques.

Effectively implementing quantum algorithms on noisy intermediate-scale quantum (NISQ) processors is a central task in modern quantum technology. NISQ processors feature tens to a few hundreds of noisy qubits with limited coherence times and gate operations with errors, so NISQ algorithms naturally require employing circuits of short lengths via quantum compilation. Here, we evaluate a reinforcement learning (RL)-based quantum compiler on a superconducting processor. Our experiments reveal that for two-qubit circuits, the RL-based compiler surpasses conventional methods, demonstrating its ability to discover hardware-amenable circuits with near-optimal lengths. However, for three-qubit circuits, the RL-based compiler does not achieve unity theoretical fidelity. To address this limitation, we integrate a variational strategy with the RL-based compiler, highlighting their complementary strengths. Systematic experiments show that this variational RL-based compiler consistently identifies near-optimal circuits, even under stringent hardware constraints, outperforming conventional techniques. Furthermore, we analyze the impact of decoherence and gate errors, providing critical insights into the practical performance of RL-based compilers on quantum hardware. These findings exemplify the codesign of the software with hardware for efficient quantum compilation, offering valuable insights for the advancement of RL-based compilers.