Fast Subspace Fluid Simulation with a Temporally-Aware Basis

TL;DR

The paper tackles the challenge of real-time, high-fidelity fluid animation by proposing a reduced-order approach built on Dynamic Mode Decomposition (DMD). By learning a temporally-aware reduced Koopman operator $\hat{K}=\Phi \Lambda \Phi^*$ that evolves the velocity field in a low-dimensional basis, the method enables direct querying of fluid states at arbitrary times via $\mathbf{u}(t+\Delta t)=\boldsymbol{\Phi} \boldsymbol{\Lambda} \mathbf{z}(t)$ and supports interactive editing, time-reversal, and upsampling. It integrates OptDMD for noisy data, randomized SVD for memory efficiency, and DMDc for incorporating external forces, while preserving boundary and divergence-free constraints. The approach demonstrates superior reconstruction with far fewer basis functions than traditional spatial ROMs across 2D and 3D scenes and remains solver-agnostic, enabling broad applicability in graphics pipelines and interactive animation.

Abstract

We present a novel reduced-order fluid simulation technique leveraging Dynamic Mode Decomposition (DMD) to achieve fast, memory-efficient, and user-controllable subspace simulation. We demonstrate that our approach combines the strengths of both spatial reduced order models (ROMs) as well as spectral decompositions. By optimizing for the operator that evolves a system state from one timestep to the next, rather than the system state itself, we gain both the compressive power of spatial ROMs as well as the intuitive physical dynamics of spectral methods. The latter property is of particular interest in graphics applications, where user control of fluid phenomena is of high demand. We demonstrate this in various applications including spatial and temporal modulation tools and fluid upscaling with added turbulence. We adapt DMD for graphics applications by reducing computational overhead, incorporating user-defined force inputs, and optimizing memory usage with randomized SVD. The integration of OptDMD and DMD with Control (DMDc) facilitates noise-robust reconstruction and real-time user interaction. We demonstrate the technique's robustness across diverse simulation scenarios, including artistic editing, time-reversal, and super-resolution. Through experimental validation on challenging scenarios, such as colliding vortex rings and boundary-interacting plumes, our method also exhibits superior performance and fidelity with significantly fewer basis functions compared to existing spatial ROMs. The inherent linearity of the DMD operator enables unique application modes, such as time-reversible fluid simulation. This work establishes another avenue for developing real-time, high-quality fluid simulations, enriching the space of fluid simulation techniques in interactive graphics and animation.

Figures (13)

• Figure 1: Long-time Single Step Integration. We demonstrate that our method can perform integration into arbitrary points in time via the exponential integration of a single matrix (Sec. \ref{['sec:arbitrary_time_step']}). Since the DMD operator is diagonal within the reduced basis, it is trivial to find the matrix that evolves the initial velocity field to the field at any point in time, significantly accelerating the integration as compared to traditional methods required by PCA. Although the DMD operator is trained over the velocity field $\bm{u}$, we show the corresponding density field at each time point for clearer visualization.
• Figure 2: Reconstruction of 3D Plume Simulations. From left to right: standalone plumes, plumes interacting with a sphere, and plumes interacting with a bunny. For each configuration, the last frame of the original MacCormack selle2008unconditionally fluid simulation is shown on the left, alongside the last frame of our subspace simulation with SVD rank ranging from $r = 2$ to $r = 61$. Remarkably, the fluid dynamics demonstrate strong resilience to low-rank bases, highlighting a key advantage of our proposed reduced-order pipeline. Additionally, these scenarios illustrate the robustness of our method in handling increasingly complex boundary conditions. The reconstruction quality improves as the number of basis functions increases, enabling more accurate capture of finer details around the boundaries. While reasonable results are achieved with a small basis (r=9), increasing the basis significantly enhances the fidelity of the simulations. More details on the temporal evolution of the flow and other basis configurations can be found in the Fig. \ref{['fig:appendix_plume']}, Fig. \ref{['fig:appendix_sphere']} and Fig. \ref{['fig:appendix_bunny']}.
• Figure 3: Editing Temporal Dynamics of the Plume with Bunny with the Koopman Operator Approximation. In this experiment, we demonstrate the impact of changing the moduli of low-frequency and high-frequency modes in a 4:1 (left) and 1:4 (right) ratio. As mentioned in Sec. \ref{['sec:editing']}, in real applications, users can modify and manipulate the dynamic of different scales of vorticity by adjusting the reduced-order parameters.
• Figure 4: Comparison of PCA and Our Method for Low-Rank Flow Reconstruction. The leftmost column shows the reference high-resolution simulation, while the right grid presents reconstructions at different rank truncations ($r$). Each pair in the grid compares Principal Component Analysis (PCA) (left) and our method (right). Lower ranks ($r=2,9$) fail to capture large-scale turbulence while increasing $r$ improves accuracy. Our method retains more detailed structures at lower ranks than PCA, demonstrating improved efficiency in capturing complex flow dynamics.
• Figure 5: Comparison of Relative Error and Vorticity Confinement Between PCA kim2013subspace and DMD (Ours). Left to right: (1) relative error over time for PCA (150 basis) (dashed) and Ours (50 basis) (solid), showing comparable or lower error with fewer basis functions. (2) reference high-resolution simulation with vorticity confinement $1.5$, with a zoomed-in region marked. (3) comparison of the zoomed region in (2) under novel unseen vorticity confinement force $1.51$ (magenta), $1.6$ (blue), $2.5$ (dark green).