We propose a unified, few-step generative modeling framework based on cumulative flow maps for long-range transport in probability space, inspired by flow-map techniques for physical transport and dynamics. At its core is a cumulative-flow abstraction that connects local, instantaneous updates with finite-time transport, enabling generative models to reason about global state transitions. This perspective yields a unified few-step framework built on cumulative transport and cumulative parameterization that applies broadly to existing diffusion- and flow-based models without being tied to a specific prediction instantiation. Our formulation supports few-step and even one-step generation while preserving synthesis quality, requiring only minimal changes to time embeddings and training objectives, and no increase in model capacity. We demonstrate its effectiveness across diverse tasks, including image generation, geometric distribution modeling, joint prediction, and SDF generation, with reduced inference cost.

We introduce a novel approach to simulate the interaction between fluids and thin elastic solids without any penetration. Our approach is centered around an optimization system augmented with barriers, which aims to find a configuration that ensures the absence of penetration while enforcing incompressibility for the fluids and minimizing elastic potentials for the solids. Unlike previous methods that primarily focus on velocity coherence at the fluid-solid interfaces, we demonstrate the effectiveness and flexibility of explicitly resolving positional constraints, including both explicit representation of solid positions and the implicit representation of fluid level-set interface. To preserve the volume of the fluid, we propose a simple yet efficient approach that adjusts the associated level-set values. Additionally, we develop a distance metric capable of measuring the separation between an implicitly represented surface and a Lagrangian object of arbitrary codimension. By integrating the inertia, solid elastic potential, damping, barrier potential, and fluid incompressibility within a unified system, we are able to robustly simulate a wide range of processes involving fluid interactions with lower-dimensional objects such as shells and rods. These processes include topology changes, bouncing, splashing, sliding, rolling, floating, and more.

We propose Leapfrog Flow Maps (LFM) to simulate incompressible fluids with rich vortical flows in real time. Our key idea is to use a hybrid velocityimpulse scheme enhanced with leapfrog method to reduce the computational workload of impulse-based flow map methods, while possessing strong ability to preserve vortical structures and fluid details. In order to accelerate the impulse-to-velocity projection, we develop a fast matrix-free Algebraic Multigrid Preconditioned Conjugate Gradient (AMGPCG) solver with customized GPU optimization, which makes projection comparable with impulse evolution in terms of time cost. We demonstrate the performance of our method and its efficacy in a wide range of examples and experiments, such as real-time simulated burning fire ball and delta wingtip vortices.

We propose the V ortex P article F low M ap (VPFM) method to simulate incompressible flow with complex vortical evolution in the presence of dynamic solid boundaries. The core insight of our approach is that vorticity is an ideal quantity for evolution on particle flow maps, enabling significantly longer flow map distances compared to other fluid quantities like velocity or impulse. To achieve this goal, we developed a hybrid Eulerian-Lagrangian representation that evolves vorticity and flow map quantities on vortex particles, while reconstructing velocity on a background grid. The method integrates three key components: (1) a vorticity-based particle flow map framework, (2) an accurate Hessian evolution scheme on particles, and (3) a solid boundary treatment for no-through and no-slip conditions in VPFM. These components collectively allow a substantially longer flow map length ( 3–12 times longer) than the state-of-the-art, enhancing vorticity preservation over extended spatiotemporal domains. We validated the performance of VPFM through diverse simulations, demonstrating its effectiveness in capturing complex vortex dynamics and turbulence phenomena.

We present NeuralFluid, a novel framework to explore neural control and design of complex fluidic systems with dynamic solid boundaries. Our system features a fast differentiable Navier-Stokes solver with solid-fluid interface handling, a low-dimensional differentiable parametric geometry representation, a control-shape co-design algorithm, and gym-like simulation environments to facilitate various fluidic control design applications. Additionally, we present a benchmark of design, control, and learning tasks on high-fidelity, high-resolution dynamic fluid environments that pose challenges for existing differentiable fluid simulators. These tasks include designing the control of artificial hearts, identifying robotic end-effector shapes, and controlling a fluid gate. By seamlessly incorporating our differentiable fluid simulator into a learning framework, we demonstrate successful design, control, and learning results that surpass gradient-free solutions in these benchmark tasks.

This paper introduces a two-phase interfacial fluid model based on the impulse variable to capture complex vorticity-interface interactions. Our key idea is to leverage bidirectional flow map theory to enhance the transport accuracy of both vorticity and interfaces simultaneously and address their coupling within a unified Eulerian framework. At the heart of our framework is an impulse ghost fluid method to solve the two-phase incompressible fluid characterized by its interfacial dynamics. To deal with the history-dependent jump of gauge variables across a dynamic interface, we develop a novel path integral formula empowered by spatiotemporal buffers to convert the history-dependent jump condition into a geometry-dependent jump condition when projecting impulse to velocity. We demonstrate the efficacy of our approach in simulating and visualizing several interface-vorticity interaction problems with cross-phase vortical evolution, including interfacial whirlpool, vortex ring reflection, and leapfrogging bubble rings.

We propose a novel Particle Flow Map (PFM) method to enable accurate long-range advection for incompressible fluid simulation. The foundation of our method is the observation that a particle trajectory generated in a forward simulation naturally embodies a perfect flow map. Centered on this concept, we have developed an Eulerian-Lagrangian framework comprising four essential components: Lagrangian particles for a natural and precise representation of bidirectional flow maps; a dual-scale map representation to accommodate the mapping of various flow quantities; a particle-to-grid interpolation scheme for accurate quantity transfer from particles to grid nodes; and a hybrid impulse-based solver to enforce incompressibility on the grid. The efficacy of PFM has been demonstrated through various simulation scenarios, highlighting the evolution of complex vortical structures and the details of turbulent flows. Notably, compared to NFM, PFM reduces computing time by up to 49 times and memory consumption by up to 41%, while enhancing vorticity preservation as evidenced in various tests like leapfrog, vortex tube, and turbulent flow.

We propose a novel numerical scheme to simulate interactions between a magnetic field and nonlinearly magnetized objects immersed in it. Under our nonlinear magnetization framework, the strength of magnetic forces is effectively saturated to produce stable simulations without requiring any parameter tuning. The mathematical model of our approach is based upon Langevin's nonlinear theory of paramagnetism, which bridges microscopic structures and macroscopic equations after a statistical derivation. We devise a hybrid Eulerian-Lagrangian numerical approach to simulating this strongly nonlinear process by leveraging the discrete material points to transfer both material properties and the number density of magnetic micro-particles in the simulation domain. The magnetic equations can then be built and solved efficiently on a background Cartesian grid, followed by a finite difference method to incorporate magnetic forces. The multi-scale coupling can be processed naturally by employing the established particle-grid interpolation schemes in a conventional MLS-MPM framework. We demonstrate the efficacy of our approach with a host of simulation examples governed by magnetic-mechanical coupling effects, ranging from magnetic deformable bodies to magnetic viscous fluids with nonlinear elastic constitutive laws.