发表机构
ELTE Eötvös Loránd University; Central European University（罗兰大学; 中欧大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立传统数值方法与神经网络的联系，通过两个最小参数网络分别重现Godunov方法和模拟斜率限制器，合并后构建二阶重构格式，并串联成深度网络，以适当损失训练获得稳定且不增复杂度的改进方案。
AI 中文摘要
本文建立了求解一维标量守恒律的传统数值方法与神经网络近似之间的明确联系。重点在于凸通量函数情形下构造合适的通量项以改进经典格式。这里开发的第一个神经网络能够重新发现Godunov方法，而第二个神经网络则模拟二阶斜率限制器的行为。通过将两者合并，可以开发出基于二阶重构的格式。本文提出的网络使用最少的参数，与以往方法相比显著降低了复杂度。这些网络还可以连续连接，得到对应于多个时间步的深度网络。使用适当的损失函数训练它们可以得到稳定的格式，甚至在不增加复杂度的情况下改进经典方法。
英文摘要
A clear link is established between conventional numerical methods and neural network approximations for solving one-dimensional scalar conservation laws. The focus is on the construction of an appropriate flux term in the case of convex flux functions for improving the classical schemes. The first neural network developed here is able to rediscover Godunov's method, while the second one emulates the behavior of a second-order slope-limiter function. In this way, by merging them, second-order reconstruction-based schemes can be developed. The networks presented here employ a minimal number of parameters, significantly reducing the complexity compared to previous approaches. These networks can also be linked consecutively to get a deep one corresponding to multiple time steps. Training them with an appropriate loss leads to stable schemes, improving even the classical methods without increasing their complexity.
Journal refFekete, I., Izsák, F., Kupás, V.P. Neural network-assisted refinement of traditional schemes for one-dimensional scalar conservation laws. Adv Comput Math 52, 88 (2026)
DOI：10.1007/s10444-026-10357-w
