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“This Is Not How Mathematicians Are Trained”
数学家不是这样训练的
1, "Solving the forward problem (positive time) and the inverse problem (negative time) together has always been a desire of mathematicians, but they’ve never known where to begin. Neural networks, however, tackle this problem naturally" (将正时间的正问题与负时间的反问题同时求解,一直是数学家的梦想,但他们始终不知道从何入手。而神经网络却能自然地处理这个问题)
2. “Variables, coefficients, even coordinates, are changing. Everything is in flux. This is not how mathematicians are trained. It would be impossible for mathematicians to come up with such a design” (变量、系数,甚至坐标都在变化,一切都处于变动之中。数学家不是这样的训练。这样的设计,不可能出自数学家之手)
3. “Mathematicians tread carefully around composite functions with more than two layers, wary of the many pitfalls, yet neural networks solve them effortlessly, almost nonchalantly.” (数学家在处理超过两层的复合函数时格外谨慎,警惕其中诸多陷阱, 而神经网络却几乎漫不经心地轻松应对)
4. “Neural networks have stacked covers, mathematically speaking, whereas the Numerical Manifold Method typically uses only 3 to 4. In contrast, neural networks stack hundreds or even thousands of such covers. I never imagined anyone would take it that far”. (神经网络在数学上拥有堆叠的覆盖层,而数值流形通常只有三到四层覆盖。相比之下,神经网络则堆叠了上百乃至上千层。我从未想过有人会将其推进到这种程度).
That was what Gen-Hua Shi (石根华) told me after returning from a two-week vacation in early June 2024..
see rest of the story, click the link
open.substack.com/pub/deepma…
搭建 Multi-Agent 架构其实到现在也没有一个可以就是作为一个核心统一的那种扩展性强,泛化好的核心。
很多的研究集中在这个多 agent 架构的各个 agent 之间的博弈上。但是忽略了其他的比如共享状态、上下文上。
这篇论文 Mining the Design Space of Multi-Agent Workflows,总结了一下目前多 Agent 发展的这个忽视的地方和接下来的 agent 的开发原则、规范。
简单来说的话,过去像 AFlow 这类用 MCTS自动化寻找最优 Agent 工作流的方法,最终只会保留单条最优路径,成千上万次探索生成的中间代码与执行轨迹都被直接丢弃了。
作者基于 6 种可组合算子和 26 个强逻辑验证任务,完整保留了 MCTS 全量探索痕迹,构建了包含 28,568 个可执行 Multi-Agent 工作流的庞大数据集。
随后引入沙盒化 AutoResearch 挖掘循环,在稀疏化惩罚的目标函数下进行特征回归与跨任务泛化检验,最后呢是得到了几个结论:
很多工程架构习惯一上来就让多个 Agent 独立思考后做单步汇总聚合,但在高复杂度逻辑任务中,这不仅不会带来增益,反而引入了严重的早期噪声干扰。
第二个就是关于共享状态管理的,目前很多开源框架推崇多智能体互相之间的博弈容易发展为不是我们预期的事件。
但实验证明维护一份持久化的集中式共享文档/黑板记忆,比互相通信辩论的效果高出足足 13.5%。对长链推理来说,同步确定性的上下文状态远比一个博弈论要高效。
最后就是图结构,大家往往热衷于把调用图画得花里胡哨,但深入挖掘发现,算子内部的 prompt specificity 以及精准的占位符上下文路由,带来的收益提升高达 11.8%,影响力完全不亚于重写整套 Agent 拓扑。
其实这篇文章给日常做 Agent 开发、研究都有一些意义,其实你说这个论文对于 agent 发展的论点其实不大,甚至是心知肚明。
但是我们很好写出来做实验证明出来,我们在做编排、设计的时候往往都过于注重博弈、图结构的设计。但是忽视了上下文和状态管理。
arxiv:arxiv.org/abs/2610.31709
Where the Data End and the Mechanism Begins: Doubly Robust Causal Digital Twins for Treatment Strategies
arxiv.org/abs/2610.31706
へー。事後学習のデータについて、価値観ごとのデータ量にばらつきがあると、そのばらつきが増幅されながら学習されちゃうこともあるよ的な、そういう論文らしいっす👀
[2610.33371] How Does Post-Training Shape LLM Value Preferences? arxiv.org/abs/2610.33371
Recently, I've been missing the papers that made me fall in love with the field during my PhD, the time you could sit with a paper's ideas for days. So I've decided to revisit a few of them in a "paper nostalgia" series of mine.
The first one is "Your Classifier is Secretly an Energy Based Model" by Will Grathwohl and others from ICLR 2020.
1/13
#DeepManifoldInterpretation
Personalization illustrates a fundamental property of neural networks: a model's fixed points are stochastic, not deterministic.
This means fixed points exist at the cohort level, not as unique fixed points for individuals.
From Deep Manifold, personalization changes the boundary conditions of inference, shifting the distribution of stochastic fixed-point classes within the learned Neural Fixed-Point Field.
Personalization does not create an individual fixed point. It changes which stochastic fixed-point cohort the inference is likely to reach.
Replying to @AlexWan40524978 @emnlpmeeting
Arxiv Link: arxiv.org/abs/2608.28833
I would like to thank my amazing collaborators and advisors at @uiuc_nlp and HKUST! @wuyuchen21 @qiancheng1231 @hyeonjeong_ai @shujin_wu @ambiyuan @JiayuJeff @hengjinlp @GeTiffanyWang
#DeepManifoldInterpretation
Cross-domain mechanism transfer is not magic. In Deep Manifold terms, it becomes possible because NN representations are propertyless, so they are not locked to one domain. What gets transferred is not the object itself, but the relationship structure, this is why category theory matters.
@Jiarui_Liu_, @chuanyang_jin, @leczhang, @_yongjinny, @yinghui_he_
IdeaScientist splits research ideation into three agents: a gap finder, an innovator that borrows mechanisms from other fields, and a report writer.
After RL post-training on Qwen3.6-27B, it scores 74.6 overall, higher than Claude Code with Opus 4.8 (68.7) and Codex with GPT-5.4 (69.5), all rated with Qwen3.6-27B as the judge. Plugging Opus 4.8 into the harness gives 77.7.
One of the caveats I find when reading the paper is that, the lead over CC/Codex comes entirely from their in-domain metric, which is a yes/no check for cross-domain transfer. Both CC+Opus 4.8 and Codex + GPT-5.4 score higher on the other novelty metrics and average quality. Moreover, their RL on Qwen3.6-27B adds only 1.1 points over the untrained harness with Qwen3.6-27B (73.5 → 74.6).
Another thing that I'm unsure about is that, in the paper, the two human experts agree at only r = 0.56, so I don't know how stable these scores are (both human judge and model judge's score), or whether a gap of a few points between systems means much. Overall, it is a good attempt to shifting the focus from solving an existing problem (do the experiments and finish a task) to proposing good ideas, which is the hard to quantify and evaluate on both models and humans.
After reinforcement learning, a model got more answers right on its first try. But given 256 attempts per question, its earlier version could solve a larger share of the test set.
A NeurIPS 2025 paper studied this difference across maths, coding and visual reasoning tasks.
The researchers examined reinforcement learning with verifiable rewards, where a model receives feedback based on checks such as whether its answer is correct or its code passes tests.
One experiment used Qwen2.5-7B on the Omni-MATH test set. After 450 steps of GRPO training:
◦ Single-attempt accuracy rose from 10.2% to 28.3%.
◦ The share of problems solved at least once across 256 attempts fell from 69.1% to 63.9%.
So training improved the chance of getting a correct answer quickly. At the same time, some problems the starting model could occasionally solve became harder to solve within the same number of attempts.
The authors' analysis suggests that much of the improvement came from making existing successful solution paths more likely to appear. A better single-attempt score therefore didn't necessarily mean the model could solve a wider range of problems.
There is a practical distinction here. Getting a correct answer somewhere among 256 responses costs much more than answering once. It also doesn't mean the model knows which response is correct. The improvement in first-attempt accuracy still matters.
These results apply to the models and training methods tested. They show why reasoning evaluations need to measure both how reliably a model succeeds and how many different problems it can solve.
#DeepManifoldInterpretation
The paper reveals how persistent abstractions can reproduce errors across agent tasks.
We extend this insight into a Deep Manifold Waypoint proposal: build an adaptive waypoint network over the Neural Fixed-Point Field (NFPF).
Instead of accumulating potentially misleading instructions, agents retain verified intermediate states and trajectories. A pack-level harness coordinates exploration, verification, and recovery.
The NFPF remains unchanged; the navigation network evolves through experience.
@DongruoZ, @iu_ruhan
#DeepManifoldInterpretation
The harder the problem, the sparser the AI's internal representations become. A fascinating finding
From Deep Manifold, this connects directly to neural plasticity. Training progressively develops interconnected local manifold covers, forming the Neural Fixed-Point Field (NFPF).
Hidden-state activation is the local deformation of this field induced by input boundary conditions. Harder problems may produce more localized deformation and sparser activation.
The deeper question is how neural plasticity shapes the geometry of intelligence.
What Happens Inside an AI Model When a Problem Becomes Too Difficult?
New research reveals a fascinating pattern: the harder the problem, the sparser the AI's internal representations become.
We know that large language models perform remarkably well on familiar questions but often struggle with complex reasoning, conflicting information, and unfamiliar problems.
But what actually changes inside a model when the difficulty increases?
A new research paper, Farther the Shift, Sparser the Representation: Analyzing OOD Mechanisms in LLMs, investigates exactly this question.
And the findings could change how we understand AI reasoning, model robustness, and the way we design more intelligent inference systems.
1. The surprising discovery: harder problems produce sparser representations
Researchers investigated how the internal activations of LLMs change when they encounter increasingly challenging inputs.
They examined models from the Qwen and Llama families across different reasoning tasks and model sizes.
The central finding was remarkably consistent.
As task difficulty increases, the model's final hidden-state representation becomes more sparse.
In simpler terms, more of the representation's activation energy becomes concentrated in a smaller subset of dimensions.
The researchers observed this pattern across four different types of difficulty:
Complex reasoning: Harder mathematical problems produce sparser representations.
More answer choices: Adding plausible incorrect answers increases representational sparsity.
Conflicting information: When provided context contradicts the model's learned knowledge, activations become more concentrated.
Longer contexts: Increasing the amount of information the model must process produces similar effects.
Importantly, this is a statistical pattern across groups of examples, not a guaranteed rule for every individual input.
The researchers summarize their discovery with a simple principle:
The farther the distribution shift, the sparser the representation.
2. Why does this happen?
One of the most interesting parts of the research is its explanation of how these patterns develop during training.
The researchers studied a smaller Transformer trained from scratch in a controlled environment.
They discovered two distinct phases of representation learning.
Phase 1: Feature selection
During the early stages of learning, the model increasingly concentrates its activations in a limited set of dimensions.
It appears to emphasize features that are useful for learning the task.
Representations become more concentrated while the model is still learning the underlying structure.
Phase 2: Feature consolidation
As training progresses and the model becomes more familiar with the data, the opposite pattern emerges.
Representations become more distributed across multiple dimensions.
The model develops broader internal representations for patterns it has successfully learned.
This leads to an intriguing interpretation.
Familiar problems can activate rich, distributed representations, while unfamiliar problems may rely on narrower activation patterns.
In other words, representation density may reflect how well a model has internalized particular patterns during training.
This is especially interesting because the model was not explicitly trained to produce the observed sparsity–difficulty relationship.
The pattern emerged through learning.
Final thoughts
The most interesting implication of this research is not simply that difficult questions produce sparser activations.
It is that the geometry of a model's internal representations may carry meaningful information about how familiar or challenging an input is.
That changes how we might investigate AI reasoning.
Rather than studying only the final answers generated by language models, we can also examine the internal structures that emerge while they process information.
Understanding these structures could help researchers build systems that adapt more intelligently to unfamiliar problems.
Perhaps one of the next breakthroughs in AI will come not only from making models larger, but from understanding their internal representations well enough to guide how they reason.
The future of AI may depend as much on understanding how models represent difficulty as on improving how they answer questions.
📄 Research paper: Farther the Shift, Sparser the Representation: Analyzing OOD Mechanisms in LLMs
arxiv.org/abs/2603.03415
💻 Code: github.com/MingyuJ666/sparsi…
#ArtificialIntelligence #LLM #AIResearch #MachineLearning #DeepLearning
#DeepManifoldInterpretation
This paper findings show that Transformers and SSMs can have very different global activation geometries while preserving similar local relationships, including in their late-layer representations.
This is expected in Deep Manifold Theory: different architectures can develop different Neural Fixed-Point Fields (NFPFs), yet produce similar local activation geometry with global context when they learn similar data relationships and achieve comparable performance. Activations are physical covers, the outputs of the NFPF, not the NFPF itself.
1/ 🎉 Excited to present at #EMNLP2026 🎉
Transformers and Mamba are fundamentally different. Do they also represent things differently?
Globally, they differ. Locally, they are similar. 🧵
#DeepManifoldInterpretation
Linear probes observe local manifestations of the neural fixed-point field, while circuit tracing follows selected computational relationships. Neither alone reconstructs the underlying manifold.
That is not a criticism of CTA; it identifies the next level of mathematical interpretation.
💥Paper at #EMNLP2026: Probes tell us whether a model has linearly encoded a concept; they don't tell us which features or circuits produce that encoding.
We introduce Concept-Targeted Attribution (CTA), attribution graphs built around a probe's concept direction. 🧵
Can LLMs infer human actions and motives the way humans do?
In our ACL Findings paper led by @ShresthaKaleen and co-authored by Abhinav Gupta, Harish Dukkipati, and Maja Mataric, we examine this question using a validated game theory framework from cognitive science. We find that current LLMs struggle both to recognize statistical patterns in human behavior and to identify humans’ underlying motives.
These findings highlight important limitations in LLMs’ social reasoning capabilities, raising concerns about their ability to accurately simulate human behavior.
Our paper can be found here: aclanthology.org/2026.findin…, and feel free to reach out to Kaleen for further questions!
#DeepManifoldInterpretation
Neural CDE models the observed flow of agent trajectories, but the underlying computational landscape is the Neural Fixed-Point Field (NFPF), developed during training and deformed during inference.
Agent reasoning follows boundary-conditioned, sampled, iterated integral paths through this field. A Neural CDE approximates those paths; it does not recover the field itself.
Trajectory equivalence is not geometric or topological equivalence.
When an LLM agent executes a long task, the hardest question is usually not whether it failed but where. Which step actually broke the run? 🧵
That’s the failure attribution problem, and it’s becoming central to debugging agentic systems. But current approaches have a real cost problem:
1. Prompting-based pipelines, where an LLM reads the whole trajectory and judges each step, are slow and expensive to run.
2. Post-trained attribution models need failure trajectories labeled at the step level, which are painful to collect and hard to scale.
We think a practical failure attribution model should be lightweight and shouldn’t need step-level labels on failures at all.
So in our #neurips2026 paper, “Tracing Agentic Failure from the Flow of Success,” we ask: can you find where an agent went wrong by learning only what going right looks like?
Meet OAT 🌾
The idea: treat failure attribution as one-class learning. We train only on successful trajectories, using neural controlled differential equations to model how successful runs evolve in latent space. Think of it as learning the “flow” of a healthy agent trajectory.
At inference, we take a failed trajectory and score every step by how far it drifts from those learned success dynamics. Steps that deviate most get flagged as the likely error steps. The algorithm doesn't rely on any step-level annotation.
What we found:
⚡ With just 100 successful trajectories for training, OAT runs 200-5000× faster than prompting-based baselines (including GPT-5 judge).
📈 It also outperforms them on accuracy: about +20% F1 score.
So you don’t have to trade cost for quality here. You can get both.
Why I’m excited about this: successful runs are cheap and plentiful, while carefully annotated failures are not. Building diagnostics on top of the data we already have feels like the right direction as agent systems grow longer and more complex.
Huge thanks to my amazing student @Samuel861025 for leading the effort, and collaborators from Microsoft Research @ShaleenDeep and Yiwen Zhu.
📄 Paper: arxiv.org/abs/2607.12747
Goodfire’s Monitoring and Discovering Reward Hacking with Internal Representations during LLM Evaluations (Leon Bergen et al., arXiv:2609.19101) demonstrates that reward hacking produces detectable signatures in neural activations.
From the Deep Manifold perspective, reward hacking reflects a deeper structural source of misalignment. The propertyless, full-rank relational structure of neural networks enables powerful connections, including between semantically unrelated representations.
This flexibility can also enable unintended computational shortcuts that satisfy reward conditions while violating task objectives.
Such shortcuts can be interpreted as misaligned trajectories through the learned neural fixed-point field (NFPF).
Activation probes observe these trajectories through physical covers, but detection does not necessarily correct the underlying mathematical covers.
What makes AI powerful is also an inherited source of misalignment.
very good questions...
“Completely break our intuition”?
Yes, if we still view neural network geometry through the lens of Riemannian manifolds.
No, if we view it through the lens of stacked, piecewise manifolds.
xitter.cf/BetaTomorrow/status/20…
one of the strangest things about deep learning is that high-dimensional spaces behave in ways that completely break our intuition.
1. in high dimensions, almost all the volume of a sphere concentrates near its surface. weird, right?
2. randomly chosen vectors become almost perpendicular to each other as dimensionality increases.
3. in thousands of dimensions, two random vectors will typically have a cosine similarity close to zero.
4. neural networks can use these high-dimensional spaces to represent different features as directions in their activation space, without necessarily making them overlap too much.
5. but features can still interfere with each other and share dimensions. this is one reason understanding what a neural network has actually learned is so difficult.
6. gradient descent doesn’t just adjust weights. through training, it also shapes the geometry of the representations the network learns.
To channel my inner Schmidhuber, we found this softmax "bug" ~1.5 years ago and published in this paper arxiv.org/pdf/2603.11682
In reality, it's pretty hard to come up with a training scenario where this matters because gradient w.r.t. highly likely tokens is tiny anyway. Nonetheless I recommend fixing it -- one less thing to worry about! Check out @jon_barron 's work for an elegant fix!
#DeepManifoldInterpretation
LEAP learns stochastic action over complete trajectories, rather than local transitions.
Deep Manifold reading: Training develops a Neural Fixed-Point Field with stochastic fixed-point classes.
Stochasticity perturbs trajectories within or between these classes, while boundary conditions shape the realized path.
Inference is an iterated integral through the learned field.
@liaw_sarah, @YuanqiD, @du_yilun
We introduce action learning: learning a stochastic action that scores how plausible an entire trajectory is, directly from observed trajectories.
No known forces. No potential energy function. No simulator likelihood.
We think this is a new paradigm for reasoning about trajectory data.
Joint work with @du_yilun and @YuanqiD.
🔥 Recovers underlying physical dynamics with over 2,000× lower error than local regression in a double well system.
🪐 Identifies a hidden 0.01% perturbation in the Sun's mass from trajectories of a 31 body orbital system.
🧬 Reuses a single learned molecular action for trajectory completion, steering, constrained optimization and temperature transfer.
The core idea: One learned action → reuse for many possible downstream tasks!























































