Adaptive Mesh Refinement

Adaptive multiresolution methods reduce representation cost by concentrating fine-scale degrees of freedom where needed, but their tree updates are usually governed by fixed local criteria. We introduce Learned Adaptive Multiresolution Diffusion Imaging (Learned AMDI), which preserves the AMDI fixed-tree propagator and hierarchy constraints while replacing the post-propagation selector with a shared local policy trained by proximal policy optimization. Regression tests reproduce deterministic AMDI trajectories to machine precision when identical trees are used. In the Haar implementation studied here, the deterministic one-step selector accepts no refinements in 54 decisions. Across nine held-out cases, Learned AMDI executes 393 refinements and reduces the mean terminal reference discrepancy from $0.17496$ to $0.13657$, while occupancy rises from $0.13737$ to $0.26660$. Step-resolved diagnostics reveal occasional small adaptation-energy increases; fixed-tree energy stability therefore does not guarantee monotonicity of the learned outer iteration. At comparable occupancy, a validation-tuned observed-detail threshold reaches a discrepancy of $0.13792$ with slightly better RMSE and SSIM, placing both methods on essentially the same accuracy--occupancy tradeoff. A decision-1-only control reaches $0.13742$, indicating that most of the improvement on this static benchmark arises from the initial allocation. The shared actor transfers without retraining to $64\times64$ and $128\times128$ images, improving reference discrepancy, RMSE, and SSIM relative to deterministic AMDI, while the frozen threshold rule remains competitive. Learned AMDI thus provides a hierarchy-constrained, resolution-transferable mechanism for adaptive allocation and clarifies the contribution of sequential decisions.

Generating compact, artist-style meshes with explicit topology typically relies on autoregressive models which incur prohibitive sequential per-token costs, or continuous flow models that depend on heuristic connectivity decoders. Next-scale generation paradigms offer a compelling alternative by enabling parallel intra-scale token prediction and coarse-to-fine refinement from global structure to local topology; yet, existing methods derive hierarchical scales via progressive mesh simplification and invert them sequentially. This eliminates intra-scale parallelism and scales generation steps linearly with face count. In this paper, we propose MeshOctave, which instead defines scale through dyadic spatial grid resolutions, framing coarsening as a deterministic collapse that merges vertices sharing a voxel cell and inherits connectivity. Its inverse operation, split-and-rewire, determines which octant sub-vertices are instantiated for each coarse face and resolves local connectivity using discrete structural tokens. These per-face operations require no serialization, each scale transition is modeled as an unordered set that adds one bit of coordinate precision, naturally supporting dynamic-length meshes and adaptive resolution refinement. We construct a scale-conditioned masked-uniform discrete diffusion model to learn split-and-rewire operation from resolution collapse hierarchies. MeshOctave outperforms strong baselines in geometric fidelity and topological validity by a non-trivial margin, while supporting adaptive resolution refinement and extending naturally to mesh subdivision tasks.

Adaptive meshes enable neural operators for partial differential equations (PDEs) to allocate spatial samples and computation according to local physical structures. Existing approaches, however, mainly address where to compute, with less attention to how information should interact after node relocation. Mesh adaptation changes local sampling scales, neighborhood structures, and geometric contexts, so representations formed at different nodes may not be directly comparable. Direct aggregation can therefore entangle physical variation with discretization-induced representation variation. Because allocation and interaction are jointly optimized through the same output objective, their individual roles are also difficult to distinguish from final errors alone. We introduce operator readability, requiring an adaptive operator to account for and test why computation is allocated to particular locations and how representations interact under the resulting nonuniform discretization. Based on this principle, we propose the Gauge-Aware Adaptive Mesh Neural Operator (GA-AMNO). Physics-informed adaptive allocation answers where to compute, while geometry-conditioned low-rank Gauge transport maps source features into target representation contexts before aggregation, answering how to interact. This makes mesh-to-solver information exchange inspectable and intervenable. We establish sufficient conditions for representation-consistent aggregation and analyze approximate transport errors and continuity under topology-preserving mesh deformations. Experiments on five PDE benchmarks demonstrate improved predictive accuracy, while controlled interventions and geometric-mismatch analyses verify the roles of allocation and interaction and show that Gauge transport improves cross-discretization representation compatibility under strong geometric mismatch.

pyALDIC is an open-source Python implementation of augmented Lagrangian digital image correlation (AL-DIC) for full-field displacement and strain measurement. The software combines a graphical user interface with a scriptable Python API and supports adaptive quadtree meshing, mask-aware subset splitting near cracks and holes, and selectable Local DIC and AL-DIC solver modes. Numba acceleration enables efficient analysis, while automated tests, documentation, and reproducible examples support reliable use acrossWindows, macOS, and Linux. Verification cases include synthetic displacement fields, rigid-body motion, Mode-I cracking, adaptive refinement, and experimental uniaxial tension. pyALDIC is distributed through PyPI, GitHub, and Zenodo under a BSD-3-Clause license for reproducibility. pyALDIC is openly available at https://github.com/zachtong/pyALDIC.

Most neural partial differential equation (PDE) surrogates learn how fields evolve after a grid has already been chosen. However, before any operator is applied, the grid has already determined how modeling capacity is allocated across space, resolution, and spectral bandwidth. We argue that this hidden design choice should itself be learnable, leading to a question different from standard operator learning: can a surrogate learn where resolution should exist before predicting field evolution? We formulate adaptive discretization as a physics-constrained conditional generation problem over valid mesh displacements. The success of diffusion models in PDE field prediction suggests their potential for learning adaptive discretizations under similar structured constraints. This leads to a two-stage diffusion framework: Stage 1 learns an r-adaptive displacement mesh conditioned on the observed dynamics, while Stage 2 predicts the solution evolution from the mesh-informed representation. The mesh generator is regularized by physics-aware proxy channels, geometric validity constraints, and local spectral concentration so that adaptation remains physically interpretable and numerically legal. Across five PDE regimes, the results show that diffusion-based learned discretization is competitive with adaptive-mesh and reduced-order baselines, with particularly strong gains in regimes where fixed or handcrafted allocation is insufficient. The main conclusion is not that there exists a universal optimal mesh rule, but that discretization should be learned in a regime-dependent manner: different spatial and spectral structures favor different allocation behaviors. This reframes adaptive meshing for neural PDE solvers from a solver-specific heuristic into a generative representation-learning problem.

Adaptive volumetric finite element meshing is a critical step in computer-aided engineering and analysis that dictates the computational budget of a given problem. It traditionally requires iterative PDE solvers or heavily supervised, data-driven surrogates trained on large-scale simulation data. While Multimodal Large Language Models (MLLMs) excel in 2D visual tasks, their zero-shot capability to semantically ground regions based on geometric understanding and physics remains an open question. Overall, this study explores a significant question: can the high-level semantic understanding of off-the-shelf MLLMs serve as a viable, zero-shot geometric proxy for finite element mesh refinement? To investigate this, we introduce GReFEM (Geometric Reasoning Enhanced Multimodal LLMs for Finite Element Meshing), a framework that utilizes MLLMs to visually localize stress-critical regions based on physics-guided textual prompts. To bridge the gap between 2D MLLM pre-training and 3D geometries, we introduce orthoViews, a view-selection module that maximizes the observability of key geometric features. We conduct an in-depth empirical evaluation across diverse CAD geometries, loading cases, and SOTA MLLMs, comparing them against a tuned geometric heuristic under a strict, matched refinement budget. Our findings reveal that MLLMs demonstrate robust zero-shot capacity to accurately follow complex spatial-physical instructions, isolating stress-relevant features with higher precision than blind heuristics. By mapping both the successes and current limitations of MLLMs in physical grounding, this study defines the frontier of foundation models as semantic assistants in automated simulation workflows.

Designing spray nozzles requires predicting how geometry shapes transient two-phase breakup, but high-fidelity volume-of-fluid (VOF) simulations with adaptive mesh refinement (AMR) are too expensive for iterative design exploration. Standard surrogate models are also challenged by this setting because both the liquid--gas interface and the underlying adaptive discretization evolve across time and geometries. We introduce a geometry-conditioned latent surrogate trained on 797 two-phase nozzle simulations that addresses this by encoding the AMR cell-density field, rather than the full multi-channel flow state, as a compact proxy for where the solver concentrates resolution. From this representation, the model reconstructs transient density evolution and nozzle geometry, and a lightweight second stage recovers the remaining flow variables. On held-out simulations, the method accurately captures key interface dynamics while reducing inference time to 0.045 seconds per trajectory, corresponding to a speed-up of more than $6\times10^4$ relative to Basilisk CFD. These results suggest that AMR refinement structure can serve as a compact and learnable representation for geometry-conditioned surrogate modeling of transient two-phase flows.

Reconstructing surface meshes from multi-view images has remained a core challenge in recent years. Most existing methods, whether implicit or explicit, depend on intermediate representations and post-processing steps like Marching Cubes or TSDF fusion, often resulting in artifacts and fragmented geometry. Directly optimizing explicit meshes is a promising approach. However, it presents two critical challenges. The first is how to adaptively refine mesh topology to capture detail without introducing degenerate faces. The second is how to maintain consistent UV coordinates for high-fidelity texturing as the mesh structure evolves. To overcome these, we propose ExMesh, a novel framework that directly optimizes explicit meshes by integrating differentiable optimization with discrete topology updates. Specifically, we introduce an adaptive vertex splitting and merging strategy, along with real-time UV maintenance, to enable coarse-to-fine optimization while preserving geometric integrity. To our knowledge, ExMesh is the first framework to seamlessly integrate discrete topology operations into a continuous differentiable optimization pipeline. Extensive experiments demonstrate that ExMesh achieves a balance among accuracy, computational efficiency, and mesh conciseness.

Conventional patchified Transformers operate on uniform spatial partitions, distributing computational effort evenly across the domain irrespective of local features. This inflexible tokenization scheme is inherently limited in its ability to efficiently represent and process solutions to complex PDEs. To address this, we propose MeshTok, an adaptive mesh refinement (AMR)-inspired tokenization and sequence modeling framework. This method selectively refines spatial regions exhibiting sharp gradients, transient features, or multiscale structures, generating a heterogeneous set of multiscale tokens defined on a fixed simulation grid. These tokens are processed within a unified Transformer sequence, enabling the model to simultaneously capture coarse-grained global context and fine-grained local details without requiring specialized architectural components. Although adaptive refinement moderately increases token count, it promotes a more targeted allocation of computational resources to physically informative regions, which we view as a practical inductive bias rather than a formal optimality guarantee. Experimental evaluations across multiple PDE families and benchmark datasets demonstrate that MeshTok consistently improves the efficiency-accuracy trade-off compared to uniform-grid baselines. This suggests adaptive multiscale tokenization as a scalable and generalizable design principle for neural PDE modeling. Code is available at https://github.com/SCAILab-USTC/MeshTok.

Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural network (PINN) is used not as the final solver, but as an off-grid residual probe for adaptive mesh refinement. The PINN residual is sampled over the domain, converted into cellwise indicators, and used to guide refinement before the final approximation is computed by a finite-difference solver. The method is evaluated on three benchmarks. The main full-solver validation uses the one-dimensional viscous Burgers equation with a nonuniform finite-difference solve on the adapted meshes. PINN-threshold refinement attains final relative $L^2$ error $0.021067$ with $60$ degrees of freedom, compared with $0.022617$ for uniform refinement with $192$ degrees of freedom. At matched mesh size, PINN-threshold reduces the error by about $67.5\%$. PINN-Dörfler refinement gives similar performance, with error $0.021264$ using $58$ degrees of freedom. A gradient indicator remains slightly more accurate, so the result supports usefulness rather than universal superiority. Manufactured 2D and 3D proxy tests, based on a nonlinear Schrödinger equation and an incompressible Navier--Stokes system, show that PINN residuals can organise structured refinement and improve over random refinement, although they do not consistently outperform gradient or uniform baselines. The results support PINN-guided AMR as a residual-indicator strategy for transferring physics-informed diagnostic information into finite-difference mesh adaptation while preserving the classical solver as the final approximation engine.