物理信息分组数据处理方法：函数表示的自适应构建及其在Navier-Stokes方程中的应用
Physics-Informed Method of Group Data Handling: Adaptive Construction of Functional Representations with an Application to the Navier-Stokes Equations
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中文总结 AI 辅助
提出物理信息分组数据处理（PI-GMDH）方法，在求解中自适应构建函数表示，通过首次变分和块坐标阻尼Gauss-Newton优化，在Taylor-Green基准上实现高精度，优于固定结构方法。
中文摘要 AI 辅助
物理信息计算方法通常在一个预先固定结构的函数表示中优化参数。本工作提出了一种物理信息分组数据处理方法（PI-GMDH），在该方法中，耦合物理场的表示在求解过程中逐步构建。候选函数方向通过完整物理与观测目标的首次变分进行评估，该目标以包（packages）形式引入，随后进行块坐标阻尼Gauss-Newton系数优化。该框架通过不可压缩Navier-Stokes方程上的张量积Chebyshev函数进行演示，并使用二维时间依赖的Taylor-Green基准测试。在测试配置下，自适应PI-GMDH达到了验证损失和保留测试损失分别为5.299e-19和5.296e-19，对于u、v和p分别使用了204、201和175个活跃函数。完整的逐度（degree-by-degree）和全项（all-terms）PI-GMDH变体，以及选定的PINN和KAN参考配置，用于检验结构构建策略的影响。结果表明，对于这个受控的合成基准，选择性渐进构建可以提供准确性、表示大小和墙钟时间的有利组合。该比较是说明性的，而非声称相对于其他物理信息方法的普遍优越性。
英文摘要
Physics-informed computational methods usually optimize parameters within a functional representation whose structure is fixed in advance. This work proposes a Physics-Informed Method of Group Data Handling (PI-GMDH), in which representations of coupled physical fields are progressively constructed during solution. Candidate functional directions are evaluated through the first variation of the complete physical and observational objective, introduced in packages, and followed by block-coordinate damped Gauss-Newton coefficient optimization. The framework is demonstrated with tensor-product Chebyshev functions on the incompressible Navier-Stokes equations using a two-dimensional time-dependent Taylor-Green benchmark. Under the tested configuration, adaptive PI-GMDH reached validation and held-out test losses of 5.299e-19 and 5.296e-19 with 204, 201, and 175 active functions for u, v, and p. Complete degree-by-degree and all-terms PI-GMDH variants, together with selected PINN and KAN reference configurations, are used to examine the effect of structural construction policy. The results show that, for this controlled synthetic benchmark, selective progressive construction can provide a favorable combination of accuracy, representation size, and wall-clock time. The comparison is illustrative rather than a claim of universal superiority over alternative physics-informed approaches.
发表机构
- JSC KIEP（基辅经济学院股份公司）
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