2026年6月，中国工业与应用数学学会期刊《Journal of Machine Learning》（JML）上线发行2026年第二期。

JML期刊由鄂维南院士与鲁剑锋教授联合主编，致力于为全球机器学习学者搭建起高水平、可持续的学术交流平台，汇聚了来自全球机器学习及交叉领域的权威学者。

JML期刊是机器学习领域一本全新的期刊，现由中国工业与应用数学学会(CSIAM)、北京大学国际机器学习研究中心、北京科学智能研究院联合主办，环球科学出版社（Global Science Press）出版，并已成功被美国数学学会的在线数学评论和书目数据库MathSciNet收录。

期刊2026年第二期共3篇文章，所有发表文章均实行开放获取，论文目录、摘要及作者信息如下：

01

Zhongwang Zhang, Zhiwei Wang, Junjie Yao, Zhangchen Zhou, Xiaolong Li, Weinan E, Zhi-Qin John Xu

Anchor Function: A Type of Benchmark Functions for Studying Language Models

Abstract: Understanding transformer-based language models is becoming increasingly crucial, particularly as they play pivotal roles in advancing towards artificial general intelligence. However, language model research faces significant challenges, especially for academic research groups with constrained resources. These challenges include complex data structures, unknown target functions, high computational costs and memory requirements, and a lack of interpretability in the inference process, etc. Drawing a parallel to the use of simple models in scientific research, we propose the concept of an anchor function. This is a type of benchmark function designed for studying language models in learning tasks that follow an “anchor-key” pattern. By utilizing the concept of an anchor function, we can construct a series of functions to simulate various language tasks. The anchor function plays a role analogous to that of mice in diabetes research, particularly suitable for academic research. We demonstrate the utility of the anchor function with an example, revealing two basic operations by attention structures in language models: shifting tokens and broadcasting one token from one position to many positions. These operations are also commonly observed in large language models. The anchor function framework, therefore, opens up a series of valuable and accessible research questions for further exploration, especially for theoretical study.

02

Jin Liang, Jinshu Huang, Mingfei Sun, Chunlin Wu

A Deep Layer Limit Analysis of Transformer

Abstract: The Transformer architecture has a profound and wide impact on modern deep learning, enabling remarkable successes in natural language processing (NLP), computer vision, time series prediction, biological sequence modeling, and a variety of other fields. In this paper, we are interested in the dynamical system modeling and related analysis of a Transformer architecture without normalization steps. Different from existing works, we model both the encoder and decoder modules of the Transformer architecture as a system of difference equations in discrete-time setting and a system of differential equations in continuous-time setting, where the state variables of the encoder and decoder parts are coupled together in a special way. Our main result is establishing the $�$-convergence of the objective functionals of the learning problems from the discrete-time setting to the continuous-time setting. This leads to the convergence of the optimal value and subsequence convergence of the solution from the discrete-time learning problem to the continuous-time learning problem. Our results demonstrate in some sense the consistency and stability of the Transformer architecture from a dynamical system perspective.

03

Hongkang Yang, Zhi-Qin John Xu, Feiyu Xiong, Weinan E 

A First-Principles Theory of Slow Thinking and Active Perception

Abstract: As part of a series on first-principles modeling of cognitive functions, this paper attempts to provide a mathematical formulation of thinking and perception. It formally derives slow thinking or more generally, active perception, and encompasses the design, training and inference of slow thinking large language models. Our starting point is the lifting and projection of probability distributions on the observable and latent spaces,with the objective of representing complex data distributions by simple function families such as neural networks. A theory called “active lifting“ is proposed, based on the sampling of latent sequences and an intrinsic drive to reduce uncertainty with maximum rate. It derives a large design space, containing the slow thinking models in a subspace that we call the static theory. These models are positioned on the representation hierarchy and sampler hierarchy induced by the static theory, and can be upgraded by climbing the two hierarchies. Active lifting further derives an inference process with an internal time axis, and a training objective that resembles minimum-length coding as well as the invention of languages. Thus, it characterizes the agency of perception, including the emergence of the slow thinking formats. Technical by-products of this theory include a three-stage pathway for improving slow thinking models, a unified approach to constructing encoders and generative models for all data modalities, a priori formation of human-like visual representations, and a possible solution to policy collapse.

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期刊主页网站：https://jml.pub/

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