面向输运主导问题的概率型盖根鲍尔重构的物理信息学习
Physics-Informed Learning of Probabilistic Gegenbauer Reconstruction for Transport-Dominated Problems
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中文总结 AI 辅助
针对输运主导问题数据驱动方法的数值振荡难题，提出物理信息机器学习框架，预测盖根鲍尔参数对的概率分布，经实验可降低数值误差1至2个数量级，实现更优的精度-成本权衡。
中文摘要 AI 辅助
输运主导问题对数据驱动方法而言仍具挑战性，这类方法常因全局支撑基函数或过于平滑的假设空间，在激波或陡峭梯度附近出现严重的数值振荡。盖根鲍尔重构已展现出缓解此类振荡的潜力，但其有效性关键取决于重构参数，尤其是权重参数λ和截断阶数m。对于数据驱动模型，控制问题、训练数据及模型架构的变化，使得系统的参数选择极具挑战性。为解决该问题，我们提出一种物理信息机器学习框架，该框架可预测候选盖根鲍尔参数对的概率分布，从而在考虑参数不确定性的同时实现概率加权重构。我们采用两阶段策略：首先预训练一个通用预测器，随后针对目标问题进行微调，以平衡精度与计算成本。该框架针对降阶模型和神经算子模型（分别以POD-Galerkin和DeepONet为代表）进行了评估。对一维和二维输运主导问题的数值实验表明，该框架可学习到有效的空间自适应参数分布。与传统重构策略相比，其可将数值误差降低1至2个数量级，且比针对特定问题的模型重训练实现了更优的精度-成本权衡。
英文摘要
Transport-dominated problems remain challenging for data-driven methods, which often exhibit severe numerical oscillations near shocks or steep gradients due to globally supported basis functions or overly smooth hypothesis spaces. Gegenbauer reconstruction has shown promise in mitigating such oscillations, but its effectiveness critically depends on the reconstruction parameters, particularly the weight parameter $λ$ and truncation order $m$. For data-driven models, variations in governing problems, training data, and model architectures make systematic parameter selection particularly challenging. To address this issue, we propose a physics-informed machine-learning framework that predicts probability distributions over candidate Gegenbauer parameter pairs, enabling probabilistically weighted reconstruction while accounting for parameter uncertainty. A two-stage strategy is adopted, in which a general predictor is first pre-trained and then fine-tuned for target problems to balance accuracy and computational cost. The framework is evaluated for reduced-order and neural operator models, represented by POD-Galerkin and DeepONet, respectively. Numerical experiments on one- and two-dimensional transport-dominated problems show that the framework learns effective spatially adaptive parameter distributions. Compared with conventional reconstruction strategies, it reduces numerical errors by up to one to two orders of magnitude and achieves a more favorable accuracy--cost trade-off than problem-specific model retraining.
