FluxNet: 学习容量受限的局部传输算子用于保守且有界的PDE代理模型
FluxNet: Learning Capacity-Constrained Local Transport Operators for Conservative and Bounded PDE Surrogates
AI总结：
提出FluxNet，通过学习累积传输量实现精确离散守恒，并利用模块化容量受限传输头强制物理边界，解决自回归PDE代理中的守恒和边界违反问题。
摘要：
时间步进算子的自回归学习为数据驱动的偏微分方程（PDE）模拟提供了有效方法，但对于守恒律，它们面临一个基本挑战：学习到的更新可能在长时间滚动中违反全局守恒。对于质量守恒型方程的重要子类，固有的物理边界（例如，非负性或[0,1]中的浓度）使问题更加复杂，违反这些边界会进一步破坏预测的稳定性。我们引入FluxNet，它学习累积传输量，表示在完整代理时间间隔内每个单元与可配置邻域之间重新分配的总守恒量。保守更新通过构造保证精确的离散守恒；模块化容量受限传输头（L、U和D）通过架构设计强制下界、上界或接近零的双边界违反。与需要时间积分并因此继承CFL约束的通量率代理不同，FluxNet不涉及此类积分；可配置的传输邻域允许在全空间分辨率下进行大时间步预测。虚拟单元将框架扩展到非周期边界。在四个基准测试（一维对流扩散、二维浅水、一维交通流、二维Cahn-Hilliard）上的实验证明了精确守恒、结构边界保持、架构模块化以及在大的时间步长下优于通量率代理的稳定性。代码公开于：https://github.com/Lan-zs/FluxNet。
英文摘要：
Autoregressive learning of time-stepping operators provides an effective approach to data-driven partial differential equation (PDE) simulation, yet for conservation laws, they face a fundamental challenge: learned updates may violate global conservation over long rollouts. For the important subclass of mass-conservation-type equations, the problem is compounded by inherent physical bounds (e.g., nonnegativity or concentrations in [0,1]) whose violation further destabilizes predictions. We introduce FluxNet, which learns cumulative transport amounts representing the total conserved quantity redistributed between each cell and a configurable neighborhood over the full surrogate interval. A conservative update guarantees exact discrete conservation by construction; modular capacity-constrained transport heads (L, U, and D) enforce lower bounds, upper bounds, or near-zero dual-bound violations through architectural design. Unlike flux-rate surrogates that require temporal integration and thus inherit CFL constraints, FluxNet involves no such integration; configurable transport neighborhoods enable large-timestep prediction at full spatial resolution. Ghost cells extend the framework to non-periodic boundaries. Experiments on four benchmarks (1D convection--diffusion, 2D shallow water, 1D traffic flow, 2D Cahn--Hilliard) demonstrate exact conservation, structural bound preservation, architecture modularity, and superior stability over flux-rate surrogates at large temporal strides. The code is publicly available at: https://github.com/Lan-zs/FluxNet.
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