Controlling systems governed by partial differential equations is an
inherently hard problem. Specifically, control of wave dynamics is challenging
due to additional physical constraints and intrinsic properties of wave
phenomena such as dissipation, attenuation, reflection, and scattering. In this
work, we introduce an environment designed for the study of the control of
acoustic waves by actuated metamaterial designs. We utilize this environment
for the development of a novel machine-learning method, based on deep neural
networks, for efficiently learning the dynamics of an acoustic PDE from
samples. Our model is fully interpretable and maps physical constraints and
intrinsic properties of the real acoustic environment into its latent
representation of information. Within our model we use a trainable perfectly
matched layer to explicitly learn the property of acoustic energy dissipation.
Our model can be used to predict and control scattered wave energy. The
capabilities of our model are demonstrated on an important problem in
acoustics, which is the minimization of total scattered energy. Furthermore, we
show that the prediction of scattered energy by our model generalizes in time
and can be extended to long time horizons. We make our code repository publicly
available.

介绍了一个用于研究有源变形材料设计控制声波的环境，并开发了一种基于深度神经网络的机器学习方法，可以有效地学习声波的动态。该方法可预测和控制散射波能量，在声学中解决了最小化总散射能量的重要问题。

驯服波动：物理可解释的机器学习框架用于可实现波动控制

Taming Waves: A Physically-Interpretable Machine Learning Framework for  Realizable Control of Wave Dynamics

Large Language Models (LLMs), such as the Generative Pretrained Transformer
(GPT), have achieved tremendous success in various language tasks, but their
emergent abilities have also raised many questions, concerns, and challenges
that need to be addressed. To gain a better understanding of the models' inner
mechanisms, we analyze the hidden state and channel wave dynamics in a small
GPT, focusing on the coherence of wave patterns in terms of cross-channel
correlation and individual auto-correlation. Our findings suggest that wave
dynamics offer consistent and repeatable intrinsic oscillation modes, along
with context-aware plasticity and expressiveness in language generation. By
analyzing wave patterns, coherence, and clustering, we provide a systematic way
to identify and interpret the functionality of the hidden state channels,
paving the way to understand and control higher-level language pattern
formation. In addition, we investigate the Poisson statistics of spelling
errors in text sequence generation across various levels of model training and
observe a phase-transition-like process. As coherence builds up, there is a
competition between the generation of correct and misspelled words. However,
once the model is adequately trained and significant coherence has emerged, the
coherent process becomes strong enough to effectively suppress spelling errors,
preventing the cascade amplification of defects. The distribution of correct
spellings transitions from Poissonian to Sub-Poissonian, while the distribution
of misspellings shows the opposite trend. By leveraging concepts and techniques
from quantum physics, we gain novel insights into the dynamics of the small
GPT. This approach can be extended to larger language models that exhibit more
complex coherent language patterns, opening up opportunities to interpret their
emergent capabilities and develop more specialized models.

研究了波动力学在小型 GPT 中的表现，发现波动动力学提供了一种内在的振荡模式，同时具有上下文感知的可塑性和表达能力，在分析波动模式、相干性和聚类的基础上，为人们提供了系统地识别和解释隐藏状态通道功能的方法。此外，研究了在文本序列生成中不同阶段的拼写错误的泊松统计，探讨了正确和错误拼写的比较，并得出了正确拼写从泊松分布向亚泊松分布转变的规律，为深入理解大型语言模型的能力和开发更专业的模型开辟了新机遇。