For simplicity, we omit the schedule coefficients in (5).

The standard definition of PAC learning (Valiant 1984) requires learners to succeed under all distributions -- even ones that are intractable to sample from. This stands in contrast to samplable PAC learning (Blum, Furst, Kearns, and Lipton 1993), where learners only have to succeed under samplable distributions. We study this distinction and show that samplable PAC substantially expands the power of efficient learners. We first construct a concept class that requires exponential sample complexity in standard PAC but is learnable with polynomial sample complexity in samplable PAC. We then lift this statistical separation to the computational setting and obtain a separation relative to a random oracle. Our proofs center around a new complexity primitive, explicit evasive sets, that we introduce and study. These are sets for which membership is easy to determine but are extremely hard to sample from. Our results extend to the online setting to similarly show how its landscape changes when the adversary is assumed to be efficient instead of computationally unbounded.

Synthesizing realistic microstructure images conditioned on processing parameters is crucial for understanding process-structure relationships in materials design. However, this task remains challenging due to limited training micrographs and the continuous nature of processing variables. To overcome these challenges, we present a novel process-aware generative modeling approach based on Stable Diffusion 3.5 Large (SD3.5-Large), a state-of-the-art text-to-image diffusion model adapted for microstructure generation. Our method introduces numeric-aware embeddings that encode continuous variables (annealing temperature, time, and magnification) directly into the model's conditioning, enabling controlled image generation under specified process conditions and capturing process-driven microstructural variations. To address data scarcity and computational constraints, we fine-tune only a small fraction of the model's weights via DreamBooth and Low-Rank Adaptation (LoRA), efficiently transferring the pre-trained model to the materials domain. We validate realism using a semantic segmentation model based on a fine-tuned U-Net with a VGG16 encoder on 24 labeled micrographs. It achieves 97.1% accuracy and 85.7% mean IoU, outperforming previous methods. Quantitative analyses using physical descriptors and spatial statistics show strong agreement between synthetic and real microstructures. Specifically, two-point correlation and lineal-path errors remain below 2.1% and 0.6%, respectively. Our method represents the first adaptation of SD3.5-Large for process-aware microstructure generation, offering a scalable approach for data-driven materials design.

For simplicity, we omit the schedule coefficients in (5).

In this work, we propose a novel methodology for robustly estimating particle size distributions from optical scattering measurements using constrained Gaussian process regression. The estimation of particle size distributions is commonly formulated as a Fredholm integral equation of the first kind, an ill-posed inverse problem characterized by instability due to measurement noise and limited data. To address this, we use a Gaussian process prior to regularize the solution and integrate a normalization constraint into the Gaussian process via two approaches: by constraining the Gaussian process using a pseudo-measurement and by using Lagrange multipliers in the equivalent optimization problem. To improve computational efficiency, we employ a spectral expansion of the covariance kernel using eigenfunctions of the Laplace operator, resulting in a computationally tractable low-rank representation without sacrificing accuracy. Additionally, we investigate two complementary strategies for hyperparameter estimation: a data-driven approach based on maximizing the unconstrained log marginal likelihood, and an alternative approach where the physical constraints are taken into account. Numerical experiments demonstrate that the proposed constrained Gaussian process regression framework accurately reconstructs particle size distributions, producing numerically stable, smooth, and physically interpretable results. This methodology provides a principled and efficient solution for addressing inverse scattering problems and related ill-posed integral equations.

In particular, we approximate percolation on a high-dimensional lattice using the Bethe lattice, a special system that corresponds to an infinite-dimensional lattice and admits exact solutions. For more details, the reader is referred to Stauffer and Aharony (1994), Bunde and Havlin (2012), and references therein. The Bethe lattice is an infinite connected undirected graph with no cycles (a tree) in which all nodes have equal degree z . The lattice centers on an arbitrary origin site, which connects to z neighbor sites. Each neighbor further connects to z 1 new sites, with each of those sites in turn connecting to z 1 new sites, such that the branching continues indefinitely 6 .

Nanomaterials exhibit distinctive properties governed by parameters such as size, shape, and surface characteristics, which critically influence their applications and interactions across technological, biological, and environmental contexts. Accurate quantification and understanding of these materials are essential for advancing research and innovation. In this regard, deep learning segmentation networks have emerged as powerful tools that enable automated insights and replace subjective methods with precise quantitative analysis. However, their efficacy depends on representative annotated datasets, which are challenging to obtain due to the costly imaging of nanoparticles and the labor-intensive nature of manual annotations. To overcome these limitations, we introduce DiffRenderGAN, a novel generative model designed to produce annotated synthetic data. By integrating a differentiable renderer into a Generative Adversarial Network (GAN) framework, DiffRenderGAN optimizes textural rendering parameters to generate realistic, annotated nanoparticle images from non-annotated real microscopy images. This approach reduces the need for manual intervention and enhances segmentation performance compared to existing synthetic data methods by generating diverse and realistic data. Tested on multiple ion and electron microscopy cases, including titanium dioxide (TiO$_2$), silicon dioxide (SiO$_2$)), and silver nanowires (AgNW), DiffRenderGAN bridges the gap between synthetic and real data, advancing the quantification and understanding of complex nanomaterial systems.

Accurate and efficient characterization of nanoparticles (NPs), particularly regarding particle size distribution, is essential for advancing our understanding of their structure-property relationships and facilitating their design for various applications. In this study, we introduce a novel two-stage artificial intelligence (AI)-driven workflow for NP analysis that leverages prompt engineering techniques from state-of-the-art single-stage object detection and large-scale vision transformer (ViT) architectures. This methodology was applied to transmission electron microscopy (TEM) and scanning TEM (STEM) images of heterogeneous catalysts, enabling high-resolution, high-throughput analysis of particle size distributions for supported metal catalysts. The model's performance in detecting and segmenting NPs was validated across diverse heterogeneous catalyst systems, including various metals (Cu, Ru, Pt, and PtCo), supports (silica ($\text{SiO}_2$), $\gamma$-alumina ($\gamma$-$\text{Al}_2\text{O}_3$), and carbon black), and particle diameter size distributions with means and standard deviations of 2.9 $\pm$ 1.1 nm, 1.6 $\pm$ 0.2 nm, 9.7 $\pm$ 4.6 nm, and 4 $\pm$ 1.0 nm. Additionally, the proposed machine learning (ML) approach successfully detects and segments overlapping NPs anchored on non-uniform catalytic support materials, providing critical insights into their spatial arrangements and interactions. Our AI-assisted NP analysis workflow demonstrates robust generalization across diverse datasets and can be readily applied to similar NP segmentation tasks without requiring costly model retraining.