Multi-fidelity data are common in various scientific and engineering fields. High-fidelity data, often more accurate, come with greater expense, such as precision experimental testing or high-resolution simulation. Conversely, low-fidelity data are less accurate but more cost-effective. Multi-fidelity surrogate modeling, which integrates multi-fidelity data to build a model, is widely used for its ability to reduce costs while improving modeling accuracy. In this work, we introduce an innovative method named active learning-assisted multi-fidelity surrogate modeling based on geometric transformation (AL-MFSGT). The AL-MFSGT method comprises three essential components: the two-fidelity surrogate based on geometric transformation (TFSGT), an active learning (AL) strategy, and a multi-fidelity modeling framework. TFSGT utilizes geometric transformations to adjust the low-fidelity surrogate to align it more closely with the high-fidelity data. Then, the transformed low-fidelity surrogate and the high-fidelity surrogate are coupled using correlation functions. The AL strategy combines accelerating error convergence and enhancing sample set diversity to judiciously select high-fidelity incremental samples. Subsequently, the two-fidelity surrogate modeling is extended to a comprehensive multi-fidelity framework. To validate the efficacy of AL-MFSGT, we conduct extensive comparative experiments using ten numerical examples. The results demonstrate the superiority of AL-MFSGT over several compared multi-fidelity surrogate modeling methods. Furthermore, we apply AL-MFSGT to two practical engineering cases, demonstrating its effectiveness in real-world modeling scenarios.

中文翻译：

---

基于几何变换的主动学习辅助多保真代理建模

多保真数据在各个科学和工程领域都很常见。高保真数据通常更准确，但费用也更高，例如精密实验测试或高分辨率模拟。相反，低保真度数据的准确性较低，但更具成本效益。多保真代理建模通过集成多保真数据来构建模型，因其能够降低成本同时提高建模精度而被广泛应用。在这项工作中，我们引入了一种创新方法，称为基于几何变换的主动学习辅助多保真代理建模（AL-MFSGT）。 AL-MFSGT 方法包含三个基本组成部分：基于几何变换的双保真度代理 (TFSGT)、主动学习 (AL) 策略和多保真度建模框架。 TFSGT 利用几何变换来调整低保真度替代项，使其与高保真度数据更紧密地对齐。然后，使用相关函数将变换后的低保真代理和高保真代理耦合起来。 AL 策略结合了加速误差收敛和增强样本集多样性来明智地选择高保真增量样本。随后，双保真度代理建模被扩展到综合的多保真度框架。为了验证 AL-MFSGT 的功效，我们使用十个数值示例进行了广泛的比较实验。结果证明 AL-MFSGT 优于几种比较的多保真替代建模方法。此外，我们将 AL-MFSGT 应用到两个实际工程案例中，证明了其在现实建模场景中的有效性。