We present a low-rank Koopman operator formulation for accelerating deformable subspace simulation. Using a Dynamic Mode Decomposition (DMD) parameterization of the Koopman operator, our method learns the temporal evolution of deformable dynamics and predicts future states through efficient matrix evaluations instead of sequential time integration. This yields log-linear scaling in the number of time steps and allows large portions of the trajectory to be skipped while retaining accuracy. The resulting temporal efficiency is especially advantageous for optimization tasks such as control and initial-state estimation, where the objective often depends largely on the final configuration. To broaden the scope of Koopman-based reduced-order models in graphics, we introduce a discretization-agnostic extension that learns shared dynamic behavior across multiple shapes and mesh resolutions. Prior DMD-based approaches have been restricted to a single shape and discretization, which limits their usefulness for tasks involving geometry variation. Our formulation generalizes across both shape and discretization, which enables fast shape optimization that was previously impractical for DMD models. This expanded capability highlights the potential of Koopman operator learning as a practical tool for efficient deformable simulation and design.
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We present a family of mixed Material Point Methods well suited for the CFL-rate simulation of stiff elastoviscoplastic materials, up to the incompressible limit. Our work builds upon the mixed discretization from Daviet and Bertails-Descoubes [2016a] and extends it to handle finite-strain vis-coelasticity and more general flow rules, allowing the simulation of a much wider range of materials. Our implicit integration scheme leads to a well-posed, symmetric optimization problem with compact stencils for which we propose an efficient GPU solver. We demonstrate our method on a variety of examples ranging from granular materials and snow to elastic solids, including two-way coupling with rigid-body solvers.
Discrete X-ray tomography reconstructs the internal structure of an object from X-ray projections, assuming that the volume is composed of a discrete set of known materials (e.g., steel, aluminum, and air). This is generally straightforward when many projections are available but becomes increasingly ill-posed as their number decreases. Discrete tomography has been extensively studied over the past five decades, resulting in a range of mature reconstruction algorithms. In this work, we introduce a new reconstruction method that draws inspiration from both classical computed tomography and recent advances in inverse rendering, demonstrating that a remarkably simple gradient-based inversion can significantly surpass the reconstruction quality of standard methods such as SIRT, DART, and TVR-DART. Our method represents each 3D location as a probability distribution over the set of known materials and minimizes a volumetric loss that encourages consistency with the measured projections. It supports nonlinear effects such as volumetric scattering and is simple to optimize and parallelize on compute accelerators. We evaluate our method on challenging 2D and 3D benchmarks, demonstrating superior performance particularly in sparse and limited-angle scenarios, where traditional techniques struggle with ambiguity.
We introduce a spatially discrete formulation of the Schrödinger bridge problem on meshes and grids that enables structure-preserving and scalable interpolation between probability distributions. Our approach builds on the duality between entropy-regularized optimal transport and the log-heat equation, deriving a discrete theory that is compatible with mesh-based finite element discretizations. The resulting Sinkhorn algorithm alternates application of the heat kernel with multiplicative updates to enforce marginal constraints. Compared to interpolation via Wasserstein barycenters, our formulation produces sharper interpolants for a given level of regularization and enforces exact endpoint marginals, in addition to enjoying faster computation. It also scales to high-resolution meshes and finer temporal discretizations, avoiding the prohibitive cost of directly discretizing dynamical transport. We demonstrate our approach across mesh- and grid-based applications, including displacement interpolation, shape interpolation, and color histogram manipulation, highlighting its ability to achieve geometric fidelity with computational efficiency.
We present an efficient B-spline finite element method (FEM) for cloth simulation. While higher-order FEM has long promised higher accuracy, its adoption in cloth simulators has been limited by its larger computational costs while generating results with similar visual quality. Our contribution is a full algorithmic pipeline that makes cloth simulation using quadratic B-spline surfaces faster than standard linear FEM in practice while consistently improving accuracy and visual fidelity. Using quadratic B-spline basis functions, we obtain a globally C1-continuous displacement field that supports consistent discretization of both membrane and bending energies, effectively reducing locking artifacts and mesh dependence common to linear elements. To close the performance gap, we introduce a reduced integration scheme that separately optimizes quadrature rules for membrane and bending energies, an accelerated Hessian assembly procedure tailored to the spline structure, and an optimized linear solver based on partial factorization. Together, these optimizations make high-order, smooth cloth simulation competitive at scale, yielding an average 2× speedup over linear FEM in our tests. Extensive experiments demonstrate improved accuracy, wrinkle detail, and robustness, including contact-rich scenarios, relative to linear FEM and recent higher-order approaches. Our method enables realistic wrinkling dynamics across a wide range of material parameters and supports practical garment animation, providing a new promising spatial discretization for high-quality cloth simulation.
Real-time path tracing increasingly operates under extremely low sampling budgets, often below one sample per pixel, as rendering complexity, resolution, and frame-rate requirements continue to rise. Superresolution is widely used in production because it reduces path-tracing cost by tracing rays on a coarser image grid and reconstructing missing details. This creates a uniform tradeoff between cost and spatial detail: every image region receives the same reduced ray budget, although path-tracing noise, reconstruction difficulty, and perceptual importance vary strongly across the image. Adaptive sampling offers a compelling alternative, but existing end-to-end approaches rely on approximations that break down in sparse regimes. We introduce an end-to-end adaptive sampling and denoising pipeline explicitly designed for the sub-1-spp regime. Our method uses a stochastic formulation of sample placement that enables gradient estimation despite discrete sampling decisions, allowing stable training of a neural sampler at low sampling budgets. To better align optimization with human perception, we propose a tone-mapping-aware training pipeline that integrates differentiable filmic operators and a state-of-the-art perceptual loss, preventing oversampling of regions with low visual impact. In addition, we introduce a gather-based pyramidal denoising filter and a learnable generalization of albedo demodulation tailored to sparse sampling. Our results show consistent improvements over uniform sparse sampling, with notably better reconstruction of perceptually critical details such as specular highlights and shadow boundaries, and demonstrate that adaptive sampling remains effective in the sub-1-spp regime.
We describe a method for computing signed distance to point clouds that allows fast pointwise evaluation at arbitrary spatial resolution. As input, our method takes a point cloud with normals; as output, it provides an analytical parameterization that allows queries of signed distance to the approximate underlying surface at arbitrary points — simultaneously providing reconstruction and distance. Our key idea is to reconstruct shapes by locally fitting point clouds with tori, which have closed-form signed distance functions. Tori are fitted in a feed-forward manner, using a pre-trained network to output per-point curvature and shift parameters. Importantly, our method does not require costly global optimization or spatial discretization, and is easily parallelizable. Underlying our method is a new theory that unifies signed distance with the classic reconstruction methods of winding numbers and Poisson surface reconstruction. We use our method to compute signed distance to point clouds arising from photogrammetry, meshes, 3D Gaussians, and neural implicits. Our method allows point clouds to be used directly in applications, without explicit surface reconstruction: as examples, we take offsets of point clouds, apply morphological and Boolean operations, and directly visualize offset surfaces using sphere tracing.
The explosion of generative 3D assets has created a massive demand for animation, yet current motion capture methods remain brittle, restricted to species-specific templates (e.g., SMPL) or requiring labor-intensive manual rigging. We introduce TopoCap, the first unified framework capable of extracting motion from monocular video and retargeting it onto characters with arbitrary, unseen skeletal topologies, i.e., from bipeds to hexapods and inanimate objects, without test-time optimization. Our key insight is that while skeletal structures are combinatorial and discrete, the underlying physics of motion occupy a continuous, low-dimensional manifold. We materialize this insight via a two-stage generative pipeline. First, we learn a Universal Motion Manifold using a Graph CVAE that compresses heterogeneous kinematic chains into a shared, fixed-length latent code. By explicitly conditioning the decoder on a structural embedding of the target rig, we disentangle motion dynamics from skeletal topology. Second, we treat video-to-animation as a conditional flow matching problem, predicting these topology-agnostic codes from visual features. To learn this generalized prior, we introduce Mobjaverse, a massive-scale dataset curated from Objaverse-XL. Comprising over 5,000 unique skeletal topologies and 2 million frames, it exceeds the structural diversity of existing datasets by two orders of magnitude. Extensive experiments demonstrate that TopoCapoutperforms specialist models on human and quadruped benchmarks while enabling zero-shot retargeting for the long tail of 3D creatures. Dataset is publicly available at https://huggingface.co/datasets/duckduckplz/Mobjaverse
We present Diceplay, a modular physical display for abstract visual composition built from a grid of identical dice. Each die has six faces with distinct geometric primitives, and images emerge through the placement and orientation of the dice. While this medium enables reusable and reconfigurable physical imagery, it poses a challenging design problem: images must be expressed through discrete, extremely low-resolution abstractions, making manual authoring difficult. To address this challenge, we introduce a computational design system that automatically generates Diceplay configurations from text prompts. Our key technical contribution is a grammar-based formulation that relaxes this discrete design space into a smooth optimization landscape, enabling gradient-based optimization using score distillation sampling. We show that our approach consistently produces meaningful abstractions for this medium, whereas state-of-the-art smoothing techniques fail in this extremely challenging regime. We demonstrate our method across a range of prompts and fabricated examples, showing how computationally generated abstractions can be realized as physical visual artifacts.
We present a global, discrete optimization framework for quad-dominant mesh reduction that formulates simplification as an edge selection problem under coupled topological and geometric constraints. Unlike greedy edge-collapse heuristics, which make irreversible local decisions, and rigid poly-chord removal schemes, which over-constrain global structure, our method casts quad-dominant reduction as an Integer Linear Programming (ILP) problem that jointly optimizes edge flow continuity and geometric fidelity in a single global optimization. To make this formulation both tractable and adaptive, we segment poly-chords at geometrically meaningful breakpoints to relax otherwise rigid topological constraints, and introduce geometry-aware soft constraints that steer the optimization toward preserving high-curvature and deformation-critical regions. Extensive experiments demonstrate that our approach consistently outperforms state-of-the-art methods in preserving quad structure and edge flow, while achieving competitive geometric accuracy and significantly benefiting downstream editing applications.
The low-poly style is a popular genre of vector graphics that depicts objects and scenes as flat-shaded meshes of low polygon count, often with a limited palette. In this paper, we propose a method to generate 2D low-poly meshes to abstract images. While it has been possible to achieve this look with general-purpose image generators and vector-based diffusion models, the resulting images are not guaranteed to be valid polygonal meshes. A key problem is that polygons overlap or intersect and, in the case of pixel-based image generators, the shapes are often not polygons. Moreover, the colors are not guaranteed to be constrained to a fixed palette. Aside from aesthetic considerations, this has practical consequences: it complicates editing and fabricating the results. We solve this problem by representing an image as a 2D polygonal mesh and optimizing the topology, geometry and coloring of the mesh using score distillation sampling, while enforcing geometric constraints, such as manifoldness and bijectivity. This presents unique challenges due to the discrete nature of the topology, which we handle using a fine-to-coarse strategy based on mesh simplification. By also constraining the colors to a fixed palette, we are able to produce various fabrications such as mosaics, embroidery, crocheting, patchwork and stencils from the resulting vector images. Code is at https://github.com/rubenwiersma/alps.
Developing controllers capable of completing a wide range of tasks in a natural and life-like manner is a key challenge in enabling practical applications of physics-based character animation. In this work, we introduce Generative Pretrained Controllers (GPC), which leverage tokenization and next-token modeling to create general-purpose, reusable generative controllers from large-scale motion datasets. Our framework utilizes end-to-end reinforcement learning to jointly optimize a "motion vocabulary", modeled via Finite Scalar Quantization (FSQ), along with a corresponding control policy that can map the discrete codes to physics-based controls. After the "codebook" has been learned, the underlying structure of this large vocabulary is modeled by training a GPT-style autoregressive transformer, leading to a powerful generative controller that generates controls for a physically simulated character by performing next-token prediction. Once the generative controller has been trained, we propose a suite of adaptation techniques for finetuning the controller for new downstream tasks. Our proposed framework greatly simplifies the training process compared to previous tokenized methods, and achieves a 99.98% success rate in reproducing a vast corpus of motion clips. The generative controller exhibits a variety of natural emergent behaviors, such as responsive behaviors to perturbations and recovery behaviors after falling. This results in highly robust general purpose controllers for a variety of downstream applications.
Simulating free-surface flows requires capturing the effects of incompressibility, viscosity, and surface tension. Existing particle-based methods often rely on operator splitting, which introduces coupling artifacts and limits stability. We propose a unified nonlinear optimization framework that achieves a strong coupling of these three effects within a single solver. By leveraging peridynamics, we formulate the discretization of distinct fluid mechanisms under a consistent variational principle. Specifically, we recast fluid motion as a nonlinear variational optimization problem over particle positions, which is solved via the semi-implicit successive substitution method. Moreover, the framework incorporates separate treatments for bulk and shear viscosity, allowing for more refined control of different viscous fluid behaviors. To the best of our knowledge, this is the first particle-based unified solver capable of fully resolving the interdependence of incompressibility, viscosity, and surface tension, thereby significantly enhancing stability in complex simulations of free-surface flows. The source code for the paper is publicly available at https://github.com/peridyno/peridyno.
Boundary Representation (B-rep) is the most commonly used data format in Computer-Aided Design (CAD) due to its analytical precision and direct support for parametric editing. However, its heterogeneous data structure— continuous parametric geometry with discrete topological graphs—poses fundamental challenges for deep learning models. Existing methods often directly predict the heterogeneous B-rep graph, relying on fixed-size padding or sequential tokenization to handle the varying cardinality of the geometric primitives. These approaches struggle with the combinatorial complexity of CAD models. The discrete, non-differentiable nature of the graph data structure prevents end-to-end optimization of the geometry and watertightness. In this work we introduce DualBrep, a novel continuous representation that unifies B-rep geometry and topology within a fully structured Euclidean domain. DualBrep encodes a CAD model using dual scalar fields: a Signed Distance Function (SDF) to represent the global shape geometry, and an Unsigned Distance Field (UDF) that implicitly encodes the topological structure via a Voronoi partitioning of the surface elements. Rather than processing these fields independently, we compress them into a single latent space. While the dual-field formulation alone already gives reconstruction a flexible, primitive-free segmentation signal that adapts to arbitrary face counts and surface types, the shared latent also makes generation tractable: a Flow Matching model can sample geometry and topology jointly from a single code, avoiding the error accumulation that plagues sequential or autoregressive B-rep predictors. Finally, we use a neural rebuilder to extract explicit B-rep models—comprising both prismatic and free-form primitives—directly from our continuous dual scalar fields. We demonstrate that DualBrep serves as a robust, unified backbone for CAD B-rep learning, achieving strong performance in both reverse engineering from raw point clouds and generative modeling via latent flow matching. Code is available at https://github.com/AutodeskAILab/DualBrep.
As Large Language Models (LLMs) increasingly shape online content, how to remove targeted information from well-trained LLMs (also known as LLM unlearning) has become increasingly critical for web governance. A key challenge in LLM unlearning lies in the sample-wise imbalance within the forget set: different samples exhibit widely varying unlearning difficulty, leading to asynchronous forgetting speeds where some knowledge remains insufficiently erased while others become over-forgotten. To address this challenge, we propose BalDRO, a novel and efficient framework for balanced LLM unlearning. BalDRO formulates unlearning as a min–sup process, where the inner process identifies a worst-case data distribution that adaptively emphasizes hard-to-unlearn samples, while the outer process updates model parameters based on the worst-case data distribution. We instantiate this formulation through two efficient variants: BalDRO-G, a discrete GroupDRO-based approximation that focuses on high-loss subsets, and BalDRO-DV, a continuous Donsker–Varadhan dual method that enables smooth, adaptive weighting within standard LLM training pipelines. Extensive experiments on the TOFU and MUSE benchmarks demonstrate the effectiveness of our proposed BalDRO, yielding significant improvements in both forgetting quality and model utility over existing methods. For reproducibility, we have released the code for BalDRO. https://github.com/nxZhai/BalDRO.
User interest is not static but rather a continuously evolving process shaped by diverse interactions across multimodal web platforms. Traditional recommendation systems often model user preferences as discrete snapshots or oversimplified trajectories, failing to capture the nuanced dynamics of interest evolution. In this work, we propose Drifting with Intent, a novel framework that formalizes user interest as a continuous-time stochastic process governed by spatio-temporal coupled stochastic differential equations (SDEs). Our approach fundamentally departs from diffusion-based methods by modeling interest evolution as a directed drift toward meaningful content rather than random diffusion. The core innovation lies in a multi-granularity hypergraph encoder that captures cross-scale user-item interactions, coupled with a unified SDE solver that generates personalized interest trajectories across temporal and relational dimensions. Unlike prior work employing redundant twin networks, we introduce a generative-discriminative co-optimization framework that efficiently balances content generation and recommendation precision within a single parameter space. Extensive experiments across eight diverse datasets demonstrate that our framework not only outperforms state-of-the-art methods in recommendation quality but also provides interpretable interest trajectories that reveal how user preferences evolve across different modalities and time scales. Our work bridges the gap between generative modeling and practical recommendation systems by treating user interest as a purposeful drift rather than a random walk.
Digital maps are crucial for web-based location services. The continuous collection of vehicle trajectories has made trajectory data a vital source for map inference. State-of-the-art (SOTA) methods formulate trajectory-based map inference as an image processing task: they first rasterize trajectories into density-based images, then extract keypoints and infer their connections to construct the road map. Despite performing well on standard road structures, these methods still suffer from low topological accuracy in complex scenarios because i) The discrete rasterized representation struggles to capture road adjacency in multi-level road structures. ii) The limited contextual awareness of keypoint-based inferring strategy leads to connectivity misjudgment in dense road areas. To solve these limitations, we propose D2Map, a Dual-view Map Inference Framework via Primal-Dual Graphs Co-generation. To precisely encode road adjacency, we introduce the serialized trajectory view as a complement to the rasterized view to reflect traversable relationships between roads, and devise a strategy-adaptive fusion module that dynamically selects and executes the optimal fusion operator to integrate dual view representations, yielding map element embeddings. To eliminate connectivity errors, we extend road map modeling from a keypoint-centric primal graph to primal and dual graphs. In the dual graph, roads are explicitly modeled as nodes, enabling context-aware topology inference. A co-generation strategy is then employed to jointly infer both graphs while maintaining their geometric consistency. Extensive experiments on two real-world datasets demonstrate the superiority of D2Map, which outperforms SOTA baselines by 11.44% in the TOPO metric.
Source localization has attracted increasing attention in recent years due to its vital role in governing the harmful propagation. However, existing localization methods do not fully consider the temporal characteristics in propagation and struggle to leverage the continuous-time information of real-world propagation scenarios. Moreover, the aggregation ability of GNN based localization models is limited by the structural noise commonly present in complicated real-world topologies. To address these challenges, a Spectral Neural Ordinary Differential Equation (SNODE) is proposed to infer the source in real-world continuous-time scenarios. First, the forward propagation is formulated as a flow based ODE system, and the source localization problem is transformed into an inverse ODE modeling task. Second, a neural process based on a graph variational autoencoder is introduced to encode global latent propagation patterns as a conditioning variable for the ODE system. Third, a spectral graph optimization is performed to suppress topological noise by filtering out high-frequency components that degrade the quality of graph aggregation in the neural process. Comprehensive experiments demonstrate that SNODE not only outperforms the optimal baseline in real-world continuous-time propagation scenarios with an average performance improvement of 43.8%, but also achieves consistently superior performance in synthetic discrete-time datasets with an improvement of 4.5%, highlighting its strong generalization ability in different propagation settings. Our code is available at https://github.com/cgao-comp/SNODE.
The explosive growth of multimodal web data demands communication that transmits meaning rather than raw bits. Existing semantic-communication systems often fail under noise, missing modalities, and distribution shifts because they optimize surface features instead of modality-invariant knowledge. We present Grasp, a knowledge-centric framework for cross-modal communication. Grasp segments streams into semantic blocks and builds a graph over them; a lightweight Graph Neural Networks (GNN) produces schedulable, importance-weighted representations. At its core is knowledge purification : we minimize a conditional mutual information upper bound to perform a three-way disentanglement—strongly related, weakly related, and task-irrelevant components—so that only essential semantics are transmitted while non-essential factors are suppressed. To maintain synchrony, we introduce one-to-two temporal contrastive learning to achieve triple alignment of video, audio, and text despite sampling asynchrony. For efficient transmission, Grasp uses a cross-modal shared vector-quantization codebook—a discrete knowledge codebook —updated by multimodal attention. At the receiver, a soft-recovery mechanism leverages this shared knowledge to robustly reconstruct semantics under low signal-to-noise ratio (SNR) or missing modalities, yielding graceful degradation. Across web tasks—including cross-modal retrieval and missing-modality inference—Grasp improves knowledge consistency, semantic fidelity, and downstream performance over strong baselines while maintaining low latency. These results show that communication structured around purified knowledge is key to building robust, semantic-aware systems for the modern web.