HelmFluid: Learning Helmholtz Dynamics for Interpretable Fluid Prediction

TL;DR

HelmFluid tackles the challenge of predicting high-dimensional fluid dynamics from partial observations by grounding learning in physical principles. It introduces HelmDynamics blocks that learn Helmholtz dynamics via a potential function $\Phi$ and a stream function $\mathbf{A}$, yielding the curl-free and divergence-free velocity components $\nabla\Phi$ and $\nabla\times\mathbf{A}$, respectively. These components are integrated through a Multiscale Multihead Temporal Integral Architecture using Runge–Kutta-based time integration with BFECC, enabling accurate predictions across scales and complex boundaries while preserving interpretability. Empirically, HelmFluid achieves state-of-the-art performance across synthetic and real-world benchmarks, including scenarios with unknown boundaries, and demonstrates robust generalization and potential for extension to 3D fluid prediction.

Abstract

Fluid prediction is a long-standing challenge due to the intrinsic high-dimensional non-linear dynamics. Previous methods usually utilize the non-linear modeling capability of deep models to directly estimate velocity fields for future prediction. However, skipping over inherent physical properties but directly learning superficial velocity fields will overwhelm the model from generating precise or physics-reliable results. In this paper, we propose the HelmFluid toward an accurate and interpretable predictor for fluid. Inspired by the Helmholtz theorem, we design a HelmDynamics block to learn Helmholtz dynamics, which decomposes fluid dynamics into more solvable curl-free and divergence-free parts, physically corresponding to potential and stream functions of fluid. By embedding the HelmDynamics block into a Multiscale Multihead Integral Architecture, HelmFluid can integrate learned Helmholtz dynamics along temporal dimension in multiple spatial scales to yield future fluid. Compared with previous velocity estimating methods, HelmFluid is faithfully derived from Helmholtz theorem and ravels out complex fluid dynamics with physically interpretable evidence. Experimentally, HelmFluid achieves consistent state-of-the-art in both numerical simulated and real-world observed benchmarks, even for scenarios with complex boundaries.

Figures (27)

• Figure 1: Comparison on dynamics and fluid modeling. Different from the numerical method ruzanski2011casa and optical-flow-based deep model sun2018pwc, HelmFluid infers the dynamics from the inherent physics quantities: potential and stream functions.
• Figure 2: HelmDynamics block, which learns spatiotemporal correlations $\mathbf{c}(\mathbf{r})$ both in the domain and on the boundary to estimate potential and stream functions of fluid from past observations for composing the Helmholtz dynamics.
• Figure 3: Transform potential and stream functions to velocity.
• Figure 4: HelmFluid architecture (left part), which employs Runge-Kutta with BFECC kim2005flowfixer as a TempoIntegral Block to integrate the learned Helmholtz dynamics along the temporal dimension (right part) at multiple scales with multiheads to generate future fluid field. Especially, a residual connection across different scales is utilized to ensure the consistency of learned multiscale dynamics.
• Figure 5: Summary of five experiment benchmarks, including (a) simulated and (b) real-world data.