用改进的切比雪夫基函数提升DeepONet对高佩克莱特数传输问题的预测精度。
Rationally Enriched Chebyshev Trunk Bases for DeepONet Surrogates of High Péclet Entrance Transport
- 将切比雪夫多项式与自适应有理函数结合，构建理性增强型基函数
- 在佩克莱特数10²~10⁴时，壁面附近温度/浓度预测误差降低超60%
- 尤其在扰动参数1.00×10⁻⁴~1.78×10⁻⁴区间优势明显，误差降19.5%
本研究提出一种理性增强的切比雪夫（REC）基函数，用于深度算子网络（DeepONet）对奇异摄动及高佩克莱特数输运问题的代理建模。此类问题解具有薄边界层特征。REC基函数融合切比雪夫多项式与基于自适应Antoulas-Anderson（AAA）算法构造的有理函数。在三个问题上（奇异摄动标量边值问题、恒壁温热入口问题、吸收壁浓度入口问题）进行五次独立训练，结果表明：相较于原始DeepONet，REC基函数在预测标量分布上表现更优；在扩散-对流比为1.00×10⁻⁴至1.78×10⁻⁴区间，相比纯切比雪夫基模型，其解剖误差降低最高达19.5%。在预测壁面法向温度与浓度分布时，误差分别较原始和切比雪夫基模型降低最多60.2%与32.2%，且有效抑制了佩克莱特数10²~10⁴范围内的近壁人工振荡。
原文摘要 · Abstract (English)
This study demonstrates a rationally enriched Chebyshev (REC) trunk for deep operator network (DeepONet) surrogate models of singularly perturbed and high-Péclet transport problems whose solution profiles are characterized by thin localized boundary or wall layers. The REC trunk combines Chebyshev polynomial dictionary elements with rational dictionary elements constructed using the adaptive Antoulas-Anderson (AAA) algorithm. Over five independent training runs, the resulting REC-trunk DeepONet is evaluated against a vanilla DeepONet and a Chebyshev-trunk DeepONet whose prescribed dictionary consists only of Chebyshev polynomials across three problems whose singular perturbation parameters are diffusion-to-advection ratios: a singularly perturbed scalar boundary-value problem (BVP), the thermal entrance problem with a prescribed wall temperature, and the concentration entrance problem with an absorbing wall. Across the held-out test profiles, the REC-trunk DeepONet improves over the vanilla DeepONet and remains comparable to the Chebyshev-trunk DeepONet in predicting the scalar profile, with its clearest advantage over the Chebyshev-trunk DeepONet appearing when the perturbation parameter lies between $1.00\times10^{-4}$ and $1.78\times10^{-4}$, where it reduces the profile-error metrics by up to $19.5\,\%$ relative to the Chebyshev-trunk DeepONet. In predicting the wall-normal temperature and concentration profiles, the REC-trunk DeepONet reduces the profile-error metrics by up to $60.2\,\%$ and $32.2\,\%$ relative to the vanilla and Chebyshev-trunk DeepONets, respectively, while suppressing artificial near-wall oscillations as the Péclet or mass-transfer Péclet number ranges from $10^{2}$ to $10^{4}$.
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