Fluid simulation 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 HelmSim toward an accurate and interpretable simulator for fluid. Inspired by the Helmholtz theorem, we design a HelmDynamic block to learn the 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 HelmDynamic block into a Multiscale Integration Network, HelmSim can integrate learned Helmholtz dynamics along temporal dimension in multiple spatial scales to yield future fluid. Comparing with previous velocity estimating methods, HelmSim is faithfully derived from Helmholtz theorem and ravels out complex fluid dynamics with physically interpretable evidence. Experimentally, our proposed HelmSim achieves the consistent state-of-the-art in both numerical simulated and real-world observed benchmarks, even for scenarios with complex boundaries.

翻译：流体模拟因内在的高维非线性动力学而长期面临挑战。传统方法通常利用深度模型的非线性建模能力直接估计速度场以进行未来预测。然而，跳过固有物理属性而直接学习表层速度场，会使模型难以生成精确或物理可靠的结果。本文提出HelmSim，旨在构建精确且可解释的流体模拟器。受亥姆霍兹定理启发，我们设计了HelmDynamic模块来学习亥姆霍兹动力学，该模块将流体动力学分解为更易求解的无旋部分和无散部分，在物理上分别对应流体的势函数和流函数。通过将HelmDynamic模块嵌入多尺度积分网络，HelmSim能够在多个空间尺度上沿时间维度整合所学习的亥姆霍兹动力学，从而生成未来流体状态。相较于传统的速度估计方法，HelmSim严格基于亥姆霍兹定理推导，并以物理可解释的证据揭示复杂流体动力学。实验表明，我们提出的HelmSim在数值模拟和真实观测基准测试中均取得一致的最优性能，即使对于具有复杂边界的场景也不例外。