Title: Reasoning is Periodicity? Improving Large Language Models Through Effective Periodicity Modeling
Venue: NeurIPS 2025
Accepted: yes
Keywords: Large Language Model

Periodicity Modeling
Abstract: Periodicity, as one of the most important basic characteristics, lays the foundation for facilitating structured knowledge acquisition and systematic cognitive processes within human learning paradigms. However, the potential flaws of periodicity modeling in Transformer affect the learning efficiency and establishment of underlying principles from data for large language models (LLMs) built upon it. In this paper, we demonstrate that integrating effective periodicity modeling can improve the learning efficiency and performance of LLMs. We introduce FANformer, which adapts Fourier Analysis Network (FAN) into attention mechanism to achieve efficient periodicity modeling, by modifying the feature projection process of attention mechanism. Extensive experimental results on language modeling show that FANformer consistently outperforms Transformer when scaling up model size and training tokens, underscoring its superior learning efficiency. Our pretrained FANformer-1B exhibits marked improvements on downstream tasks compared to open-source LLMs with similar model parameters or training tokens. Moreover, we reveal that FANformer exhibits superior ability to learn and apply rules for reasoning compared to Transformer. The results position FANformer as an effective and promising architecture for advancing LLMs.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a periodic component into transformer attention layer. The change is inspired by recent Fourier based network development as well as backed by the argument of periodicity importance (motivation section 2). After describing the network changes, the authors test the architecture on LM benchmarks against the natural baselines with controlled variations (4.4.1) as well as language models with similar size and capacities. The model achieves clear superior performance compared to baselines on common sense reasoning, language modelling as well as controlled math tasks and logical reasoning. \n\nThe paper tries to find elements that can explain the improvements: the paper shows the advantage of learning curve as well as the scaling properties. Finally, the paper also shows the model expressiveness advantage measured by Lipschitz measure.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: Despite that the change is not straightforward, the paper is easy to follow where authors motivate the approach and explain clearly the changes that are brought to the existing architecture. The paper not only mentions the relevant work in the fourier network field but also connects with other attention improvement where the paper discusses the complementary roles of the current approach. The paper has extensive evaluations well organised. In terms of performance, the proposed model is evaluated against its natural baselines as well as compared with state-of-the-art models of similar size and show convincing improvements. Beyond that, the paper also makes a step to try to investigate where comes the improvement. Although the exact mechanism is still very difficult to find, the paper investigates into training curve, controlled task, scalability as well as representational power, laying a solid ground for future research to be built on top of the existing work. The work makes significant improvement on the seemingly well matured transformer architecture, so the contribution I judge to be significant to the field as well. \n\nWeakness: There are some natural questions that I have when reading the paper. The paper can be even more extensive although it is definitely good on the thoroughness for the experiments.\n\nQUESTIONS:\n1. Do authors have tested different p value for the architecture. when p = 0 it goes to a linear layer and one can also only have fourier layers so it seems interesting to see what this parameter balances in terms of downstream performance as well as characteristics.\n\n2. Do authors test on other benchmarks? As authors have mentioned, the improving mechanism is unknown so it is possible that other domains also simply improve and degradations are very useful to get noticed.\n\n3. It is surprised to see that the model only gets pretrained but the performance exceeds some models that are fine-tuned or distilled. Do authors have comments on this please?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a new method that integrates the Fourier basis into the attention module, primarily motivated by the periodicity in human data (as they claim). They demonstrated that transformers with this periodicity component outperform alternative transformers that lack this architecture. They did ablation studies to show that the benefit primarily stems from the periodicity component. They conducted extensive experiments comparing their method to existing transformers on a range of datasets, including those that incorporate reuse. They showed that FANformer scales up, performs okay, and takes fewer parameters.\n\nSTRENGTHS AND WEAKNESSES:\n__Strengths__ \n\nThe paper's strength lies in its extensive experiments and evaluations on a wide range of datasets, as well as investigations into the influence of hyperparameters on the loss. The organization and structure of the paper is also very clear. \n\n__Weaknesses__ \n\nWhat is currently lacking is a comparison with alternative models that also employ periodicity in their architecture, which have also claimed that these models perform better than transformers, as it is not the first paper to find ways to cook periodicity into its architecture. For example, how does Fanformer compare to state space models (SSMs), such as mamba, with recurrence that naturally support periodic dynamics?\n\nI found the motivation for introducing periodicity rather unconvincing.\nThe paper claims that periodicity facilitates structured knowledge acquisition. However, this is conceptually vague and poorly grounded. Knowledge, particularly in domains like language, does not naturally exhibit periodic structure. While one might metaphorically describe some mental representations as \"wave-like,\" structured knowledge is not inherently represented as waveforms.\n\nLine 31:\nThe authors cite Lake et al. and Buzsáki (2006) as part of the motivation for periodicity. However, these references tangentially relate to the claim. Lake et al. emphasize human learning efficiency through intuitive theories and compositional representations, periodicity is never mentioned. Buzsáki’s work hypothesizes that rhythmic activity underlies some brain functions, but this remains speculative. Furthermore, most supporting evidence pertains to animal studies and does not directly suggest that periodicity is a principle governing human learning processes, especially not in high-level domains like language.\n\nMoreover, pattern recognition—which is a plausible explanation for human generalization—is not equivalent to with periodicity.\nLastly, it remains completely unclear how periodicity should manifest in language modeling. Mathematical functions like x mod 5 may be periodic, but this kind of structure is far removed from the kind of compositional and hierarchical patterns present in natural language.\n\nQUESTIONS:\nFrom Section 4.4.3, it appears that FAN introduces more holes in unwanted regions around 99/98 in the linear regression task (figure 5 right most plot). Does FANFormer introduce new problems that transformers do not have?\n\nThere are a small number of holes in the modular addition task, but why does the transformer have holes shaped like a square? \n\nEquation 1: what is p with an overline and p without an overline? \n\nHow is the tokenization done? Why does FANformer need a smaller number of training tokens? \n\nThe authors observe that effective periodicity modeling improves the learning efficiencies of LLMs. Does it induce loss in some other aspect, mainly when knowledge is not described in a periodic form? \n\nHow are the Lipschitz constants evaluated? \n\nLine 27:\n“Transformer-based models are also known for their immense demand for data and computational resources during training [Kaplan et al., 2020; Hoffmann et al., 2022; Chowdhery et al., 2023]. In comparison, humans can accomplish similar learning tasks with far fewer resources. This discrepancy suggests that existing LLM architectures still suffer from low learning efficiency.”\nThis is a valid observation, but it remains unclear whether the proposed FAN model addresses this gap in learning efficiency. The connection between periodicity and improved learning efficiency is not substantiated in the paper.\n\nLIMITATIONS:\nThe authors included limitations, but a discussion with other large models that take periodicity into account is currently lacking.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces FANformer, a novel large language model (LLM) architecture that improves learning efficiency and performance by enhancing the Transformer attention mechanism with periodicity modeling via Fourier Analysis Network (FAN). The authors argue that periodic patterns, ubiquitous in human reasoning, are poorly modeled by standard Transformers. FANformer modifies the attention mechanism to operate partially in the frequency domain using sine and cosine projections, resulting in better representation of periodic features. Extensive experiments show FANformer achieves comparable or superior performance to baseline Transformers with significantly fewer parameters and training tokens. It also demonstrates better rule-based reasoning, logical inference capabilities across several downstream tasks.\n\nSTRENGTHS AND WEAKNESSES:\nInnovation of Approach:\nThe use of a modified Fourier-based projection within the QKV structure of attention (the ATF module) is an original solution. By embedding periodic patterns directly into attention computation, the model diverges meaningfully from standard Transformer variants. While previous work explored Fourier Analysis Network, this paper proposes a principled and scalable adaptation.\n\nSignificance:\nThis work has high potential impact. Periodicity is underexplored in LLMs, and this approach opens a novel and elegant path for improving Transformer-based architectures, especially in domains where pattern repetition and rule-based reasoning are essential.\n\nClarity and Fluency:\nThe writing is generally clear, and the motivations are well articulated. However, a few sections (e.g., the mathematical formalism in Section 2) may be too dense for readers unfamiliar with group theory or frequency-domain analysis. Pseudocode and diagrams are helpful but could be further clarified.\n\nWeaknesses:\n1. The theoretical advantages of periodic modeling were not explained in further detail.\n2. While promising, the reliance on sinusoidal expansions could introduce constraints in non-periodic linguistic domains. Furthermore, there is insufficient analysis of the model’s behavior in handling irregular or non-periodic sequences — a potential limitation for broad NLP applicability. While generalization is claimed, more rigorous checks (e.g., across languages and tasks) would have bolstered the argument.\n3.  While periodic tasks are ideal tests for FANformer, real-world linguistic periodicity is more nuanced; extrapolation from synthetic setups (e.g., modular arithmetic) to natural language warrants caution.\n4. Lack of Cross-Domain Validation– All tasks use English and a narrow band of benchmarks. Cross-linguistic or multimodal validation is absent.\n\nQUESTIONS:\n1. Explanation of Validity\nAuthors should provide more detailed explanations as to why periodic modeling can enhance the capability of reasoning and why it can improve the efficiency of learning, rather than merely illustrating through experiments. Intuitively, modeling in the frequency domain would decrease the model efficiency.\n\n2. Limited Scope of Evaluation Benchmarks\nThis paper primarily evaluates standard common-sense question-answering tasks and synthetic mathematical reasoning datasets, lacking systematic testing in more real-world task scenarios.\n\n3. Real-world Applicability:\n Have you benchmarked the inference speed and GPU memory usage of FANformer relative to Transformer in deployment settings? Would the frequency-domain calculations add significant latency?\n\n4. Generalization Beyond Commonsense Tasks:\nHave you evaluated FANformer in other structured or periodic domains (e.g., code generation) where periodicity may be even more prevalent?\n\n5. How to balance the model's capability in periodic modeling and its ability in non-periodic task modeling?\n\nLIMITATIONS:\nYES", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper's primary contribution is introducing periodicity as a mechanism to enhance language modeling and reasoning capabilities in large language models. The authors provide both theoretical foundations and practical implementations for this approach.\n\nSTRENGTHS AND WEAKNESSES:\nContribution and strengths:\n\n--Incorporating Fourier transforms into Transformer architectures—and potentially future LLMs—is a challenging problem. On the other hand, it is well known that such integration could be a breakthrough for handling data with inherent periodicity. This paper explores new model architectures to address this challenge and demonstrates strong empirical performance.\"\n\n--The paper includes a wide range of downstream tasks, which makes the proposed architecture particularly appealing.\n\n--The authors provide detailed descriptions of their model architectures, hyperparameters, and training procedures, ensuring reproducibility of their experimental results.\n\n--The paper also offers theoretical explanations for why periodicity facilitates language modeling and reasoning, supported by formal proofs of their methodological claims, and computational cost analysis.\n\n\n\nWeaknesses:\n--Computational Resource Constraints: Due to limits in computational resources, the researchers were only able to pretrain FANformer-1B model. The result would be more convincing if it is observed on larger models. \n\n--FANformer architecture is designed to be orthogonal to other existing approaches for revising the attention mechanism, meaning it can seamlessly incorporate them. However, in this work, only Flash Attention was incorporated for necessary acceleration. The exploration and integration of other attention variants, are left for future research.\n\nQUESTIONS:\n--How to detech if a setting suits to use periodicity modeling? Is there a preliminary step to determine whether to introducting a FANformer vs a regular transformer?\n\n----While the study demonstrates that enhancing a language model's ability to model periodic patterns improves performance, the underlying mechanisms responsible for this improvement remain mysterious. Why there is naturally a periodic structure in reasoning and generalization?\n\n--Limited testing scope - the approach has only been tested on a few datasets.\n\n--The scope of the claims may be limited by the experimental setup. It remains unclear how well this new method generalizes to other settings.\n\n--Results may depend on implicit assumptions that need to be articulated.\n\nLIMITATIONS:\n--Computational Resource Constraints: Due to limits in computational resources, the researchers were only able to pretrain FANformer-1B model. The result would be more convincing if it is observed on larger models. \n\n--FANformer architecture is designed to be orthogonal to other existing approaches for revising the attention mechanism, meaning it can seamlessly incorporate them. However, in this work, only Flash Attention was incorporated for necessary acceleration. The exploration and integration of other attention variants, are left for future research.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}