用神经网络学出更优的数值流限制器，自动优化且通用性强。
Learning second-order TVD flux limiters using differentiable solvers
- 用神经网络替代传统限制器，通过可微分求解器端到端训练。
- 在直线平流、布格尔斯方程等多类问题上精度超越经典方法。
- 训练后可直接接入现有计算流体代码，适合复杂流动模拟优化。
本文提出一种数据驱动框架，通过可微分模拟学习最优二阶总变差减小（TVD）通量限制器。在完全可微的有限体积求解器中，限制器函数被替换为神经网络。通过将限制器表示为Minmod与Superbee限制器的逐点凸组合，训练全程保持二阶精度与TVD约束。该方法利用自动微分实现误差从数值解到限制器参数的直接反向传播。我们在多种双曲守恒律方程上验证了有效性，包括线性平流方程、布格尔斯方程和一维欧拉方程。值得注意的是，仅在直线平流问题上训练的限制器，在包含激波与间断的各类问题中表现出强泛化能力，精度优于多数经典通量限制器。所学通量限制器可轻松集成至现有计算流体动力学代码中，该方法也为系统开发和优化复杂流动问题的限制器提供了灵活路径。
原文摘要 · Abstract (English)
This paper presents a data-driven framework for learning optimal second-order total variation diminishing (TVD) flux limiters via differentiable simulations. In our fully differentiable finite volume solvers, the limiter functions are replaced by neural networks. By representing the limiter as a pointwise convex linear combination of the Minmod and Superbee limiters, we enforce both second-order accuracy and TVD constraints at all stages of training. Our approach leverages gradient-based optimization through automatic differentiation, allowing a direct backpropagation of errors from numerical solutions to the limiter parameters. We demonstrate the effectiveness of this method on various hyperbolic conservation laws, including the linear advection equation, the Burgers' equation, and the one-dimensional Euler equations. Remarkably, a limiter trained solely on linear advection exhibits strong generalizability, surpassing the accuracy of most classical flux limiters across a range of problems with shocks and discontinuities. The learned flux limiters can be readily integrated into existing computational fluid dynamics codes, and the proposed methodology also offers a flexible pathway to systematically develop and optimize flux limiters for complex flow problems.
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