Solving diverse partial differential equations (PDEs) is fundamental in science and engineering. Large language models (LLMs) have demonstrated strong capabilities in code generation, symbolic reasoning, and tool use, but reliably solving PDEs across heterogeneous settings remains challenging. Prior work on LLM-based code generation and transformer-based foundation models for PDE learning has shown promising advances. However, a persistent trade-off between execution success rate and numerical accuracy arises, particularly when generalization to unseen parameters and boundary conditions is required. In this work, we propose OpInf-LLM, an LLM parametric PDE solving framework via operator inference. The proposed framework leverages small amounts of solution data to enable accurate prediction of diverse PDE instances, including unseen parameters and configurations, and provides seamless integration with LLMs for natural language task specification and physics-based reasoning of proper feature parameterization. Its low computational demands and unified solution pipeline further enable a high execution success rate across heterogeneous settings, opening new possibilities for generalizable reduced-order modeling in LLM-based PDE solving.
We present OctWorld, a video diffusion framework with persistent 3D memory for generating explorable, world-consistent, and high-fidelity visual scenes. Given a single image, OctWorld performs stable autoregressive world generation along user-specified camera trajectories. We focus on long-range generation, characterized by extended camera paths and wide viewpoint coverage, where preserving spatial consistency is particularly challenging when previously generated regions are revisited. To address this problem, we introduce OctMap, an extensible and spatially adaptive 3D memory that progressively fuses generated visual observations and their corresponding depth maps into a global representation. OctMap employs TSDF fusion within a dynamic sparse octree whose spatial resolution adapts to image evidence. This design preserves geometric and appearance details across diverse scene scales while maintaining low memory overhead. Experiments demonstrate that OctWorld generates long-range, spatially consistent videos and outperforms prior methods on both existing benchmarks and challenging long-range generation settings. OctMap also provides clear advantages over point-based caches and fixed-resolution TSDF volumes. Project page: https://github.com/microsoft/OctWorld