In this section, we describe in detail the proposed architecture and its implementation. Our architecture is composed by an encoder and a decoder. The encoder receives as input a predefined number of learnable latent probes LP, together with the point coordinates of the target point cloud XT. Each layer of the encoder performs an operation of cross-attention between LP and XT followed by a self-attention on LP. Each attention is followed by a feed-forward layer.

Variational methods that rely on a recognition network to approximate the posterior of directed graphical models offer better inference and learning than previous methods. Recent advances that exploit the capacity and flexibility in this approach have expanded what kinds of models can be trained. However, as a proposal for the posterior, the capacity of the recognition network is limited, which can constrain the representational power of the generative model and increase the variance of Monte Carlo estimates. To address these issues, we introduce an iterative refinement procedure for improving the approximate posterior of the recognition network and show that training with the refined posterior is competitive with state-of-the-art methods. The advantages of refinement are further evident in an increased effective sample size, which implies a lower variance of gradient estimates.

Finally, PDE-Refiner's connection to diffusion models enables an accurate and efficient assessment of the model's predictive uncertainty, allowing us to estimate when the surrogate becomes inaccurate.

Diffusion Language Models (DLMs) offer a promising parallel generation paradigm but suffer from slow inference due to numerous refinement steps and the inability to use standard KV caching. We introduce CDLM (Consistency Diffusion Language Models), a training-based acceleration method that simultaneously tackles both bottlenecks. CDLM integrates consistency modeling to drastically reduce the number of required sampling steps by enabling multi-token finalization. Furthermore, we enforce a block-wise causal attention mask during fine-tuning, making the model fully compatible with KV caching. Experiments show CDLM achieves 3.6x-14.5x lower latency while maintaining competitive accuracy on math and coding tasks. The full training and evaluation code is available at https://github.com/SqueezeAILab/CDLM.

From a Bayesian perspective, score-based diffusion solves inverse problems through joint inference, embedding the likelihood with the prior to guide the sampling process. However, this formulation fails to explain its practical behavior: the prior offers limited guidance, while reconstruction is largely driven by the measurement-consistency term, leading to an inference process that is effectively decoupled from the diffusion dynamics. To clarify this structure, we reinterpret the role of diffusion in inverse problem solving as an initialization stage within an expectation--maximization (EM)--style framework, where the diffusion stage and the data-driven refinement are fully decoupled. We introduce \textbf{DAPS++}, which allows the likelihood term to guide inference more directly while maintaining numerical stability and providing insight into why unified diffusion trajectories remain effective in practice. By requiring fewer function evaluations (NFEs) and measurement-optimization steps, \textbf{DAPS++} achieves high computational efficiency and robust reconstruction performance across diverse image restoration tasks.

The generation of questions and answers (QA) from knowledge graphs (KG) plays a crucial role in the development and testing of educational platforms, dissemination tools, and large language models (LLM). However, existing approaches often struggle with scalability, linguistic quality, and factual consistency. This paper presents a scalable and deterministic pipeline for generating natural language QA from KGs, with an additional refinement step using LLMs to further enhance linguistic quality. The approach first clusters KG triplets based on their relations, creating reusable templates through natural language rules derived from the entity types of objects and relations. A module then leverages LLMs to refine these templates, improving clarity and coherence while preserving factual accuracy. Finally, the instantiation of answer options is achieved through a selection strategy that introduces distractors from the KG. Our experiments demonstrate that this hybrid approach efficiently generates high-quality QA pairs, combining scalability with fluency and linguistic precision.

Motion blur caused by camera shake, particularly under large or rotational movements, remains a major challenge in image restoration. We propose a deep learning framework that jointly estimates the latent sharp image and the underlying camera motion trajectory from a single blurry image. Our method leverages the Projective Motion Blur Model (PMBM), implemented efficiently using a differentiable blur creation module compatible with modern networks. A neural network predicts a full 3D rotation trajectory, which guides a model-based restoration network trained end-to-end. This modular architecture provides interpretability by revealing the camera motion that produced the blur. Moreover, this trajectory enables the reconstruction of the sequence of sharp images that generated the observed blurry image. To further refine results, we optimize the trajectory post-inference via a reblur loss, improving consistency between the blurry input and the restored output. Extensive experiments show that our method achieves state-of-the-art performance on both synthetic and real datasets, particularly in cases with severe or spatially variant blur, where end-to-end deblurring networks struggle. Code and trained models are available at https://github.com/GuillermoCarbajal/Blur2Seq/