We extend a recently proposed machine-learning-based iterative solver, i.e.
the hybrid iterative transferable solver (HINTS), to solve the scattering
problem described by the Helmholtz equation in an exterior domain with a
complex absorbing boundary condition. The HINTS method combines neural
operators (NOs) with standard iterative solvers, e.g. Jacobi and Gauss-Seidel
(GS), to achieve better performance by leveraging the spectral bias of neural
networks. In HINTS, some iterations of the conventional iterative method are
replaced by inferences of the pre-trained NO. In this work, we employ HINTS to
solve the scattering problem for both 2D and 3D problems, where the standard
iterative solver fails. We consider square and triangular scatterers of various
sizes in 2D, and a cube and a model submarine in 3D. We explore and illustrate
the extrapolation capability of HINTS in handling diverse geometries of the
scatterer, which is achieved by training the NO on non-scattering scenarios and
then deploying it in HINTS to solve scattering problems. The accurate results
demonstrate that the NO in HINTS method remains effective without retraining or
fine-tuning it whenever a new scatterer is given. Taken together, our results
highlight the adaptability and versatility of the extended HINTS methodology in
addressing diverse scattering problems.

我们扩展了最近提出的基于机器学习的迭代求解器，即混合迭代可转移求解器（HINTS），以解决具有复吸收边界条件的外部域中由 Helmholtz 方程描述的散射问题。

基于神经算子的快速求解器进行大规模散射

Large scale scattering using fast solvers based on neural operators

Large Language Models (LLMs), such as ChatGPT, are quickly advancing AI to
the frontiers of practical consumer use and leading industries to re-evaluate
how they allocate resources for content production. Authoring of open
educational resources and hint content within adaptive tutoring systems is
labor intensive. Should LLMs like ChatGPT produce educational content on par
with human-authored content, the implications would be significant for further
scaling of computer tutoring system approaches. In this paper, we conduct the
first learning gain evaluation of ChatGPT by comparing the efficacy of its
hints with hints authored by human tutors with 77 participants across two
algebra topic areas, Elementary Algebra and Intermediate Algebra. We find that
70% of hints produced by ChatGPT passed our manual quality checks and that both
human and ChatGPT conditions produced positive learning gains. However, gains
were only statistically significant for human tutor created hints. Learning
gains from human-created hints were substantially and statistically
significantly higher than ChatGPT hints in both topic areas, though ChatGPT
participants in the Intermediate Algebra experiment were near ceiling and not
even with the control at pre-test. We discuss the limitations of our study and
suggest several future directions for the field. Problem and hint content used
in the experiment is provided for replicability.

本文评估了 ChatGPT 的学习增益，比较其提示的功效，发现 ChatGPT 提示的质量虽比人工创作的提示部分高，但人工创作的提示所得的学习增益明显高于 ChatGPT，暗示其未来在教育领域中的应用潜力与局限性。

ChatGPT 与人类导师生成的代数提示之间的学习增益差异

Learning gain differences between ChatGPT and human tutor generated algebra hints

Confidence estimation aims to quantify the confidence of the model
prediction, providing an expectation of success. A well-calibrated confidence
estimate enables accurate failure prediction and proper risk measurement when
given noisy samples and out-of-distribution data in real-world settings.
However, this task remains a severe challenge for neural machine translation
(NMT), where probabilities from softmax distribution fail to describe when the
model is probably mistaken. To address this problem, we propose an unsupervised
confidence estimate learning jointly with the training of the NMT model. We
explain confidence as how many hints the NMT model needs to make a correct
prediction, and more hints indicate low confidence. Specifically, the NMT model
is given the option to ask for hints to improve translation accuracy at the
cost of some slight penalty. Then, we approximate their level of confidence by
counting the number of hints the model uses. We demonstrate that our learned
confidence estimate achieves high accuracy on extensive sentence/word-level
quality estimation tasks. Analytical results verify that our confidence
estimate can correctly assess underlying risk in two real-world scenarios: (1)
discovering noisy samples and (2) detecting out-of-domain data. We further
propose a novel confidence-based instance-specific label smoothing approach
based on our learned confidence estimate, which outperforms standard label
smoothing.

本论文介绍了一个在神经机器翻译中实现置信度估计的方法，通过让 NMT 模型向用户请求 “提示” 来确定置信度，进一步提出了利用置信度的标签平滑方法。实验结果表明，在真实世界的嘈杂和跨领域数据的情况下，该方法可以准确评估潜在的风险。