Can Transformers Do Enumerative Geometry?

Authors: Baran Hashemi, Roderic Guigo Corominas, Alessandro Giacchetto
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: AI for Mathematics, Algebraic Geometry, Theorem Discovery, Transformers, Recursive functions, Interpretability Analysis and world model.

Abstract:
We introduce a Transformer-based approach to computational enumerative geometry, specifically targeting the computation of $\psi$-class intersection numbers on the moduli space of curves. Traditional methods for calculating these numbers suffer from factorial computational complexity, making them impractical to use. By reformulating the problem as a continuous optimization task, we compute intersection numbers across a wide value range from $10^{-45}$ to $10^{45}$. To capture the recursive nature inherent in these intersection numbers, we propose the Dynamic Range Activator (DRA), a new activation function that enhances the Transformer's ability to model recursive patterns and handle severe heteroscedasticity. Given precision requirements for computing the intersections, we quantify the uncertainty of the predictions using Conformal Prediction with a dynamic sliding window adaptive to the partitions of equivalent number of marked points. To the best of our knowledge, there has been no prior work on modeling recursive functions with such a high-variance and factorial growth. Beyond simply computing intersection numbers, we explore the enumerative "world-model" of Transformers. Our interpretability analysis reveals that the network is implicitly modeling the Virasoro constraints in a purely data-driven manner. Moreover, through abductive hypothesis testing, probing, and causal inference, we uncover evidence of an emergent internal representation of the the large-genus asymptotic of $\psi$-class intersection numbers. These findings suggest that the network internalizes the parameters of the asymptotic closed-form and the polynomiality phenomenon of $\psi$-class intersection numbers in a non-linear manner.

Decision: accept (poster)
Reviewer scores: 8, 5, 3, 3

Reviews and discussion:
<thread>
creation date:
1730192419422

thread_type:
official_review

<thread_object>
creator: reviewer_qddj

<content>
summary:
Unfortunately, I have no expertise at all in computational enumerative geometry. My review will thus be quite superficial.

*Summary*

This paper proposes to use transfomer models to tackle what I understood is a central problem in enumerative geometry: computing the phi-class intersection numbers on the moduli space of curves. From my pretty crude rudimentary and pragmatical ML perspective, the authors reduce this problem to learning a multi modal function mapping input tuples of the form (quantum Airy structure datum [a tensor / sequence of tensors], genus [integer], number of marked points [integer], partitions [permutation-invariant set]) to output intersection numbers [sequence of integers (?)]. The model is trained on solutions computed using brute-force methods up to some genus, and evaluated on its ability to extrapolate to find solutions for higher genus (geni?). 

The main technical contribution of the papers are methodological and consist in 

(i) designing a specific multi-modal transformer architecture suited to the problem at hand (combining mostly existing models / techniques)
(ii) introducing a novel activation function specifically suited to model recursive functions, which are crucial to solve the problem. 

Experiments on synthetic data are provided demonstrating that the model seems to be able to extrapolate to higher geni than the ones seen in the training data. The authors also provide some more qualitative analysis to investigate to which extent the internal representations learned by the model encode mathematical structures that are known to be relevant to solve the problem.

strengths:
(S1) Investigating to which extent the recent successes of transformer models can transfer to other tasks, such as the one of solving fundamental problems in mathematics, is worthwhile and relevant.

weaknesses:
(W1) The relevance and technical aspects cannot be well understood / evaluated unless the reader has some non-trivial background knowledge of enumerative geometry.

(W2) The writing and exposition of the material can be improved.

questions:
*Recommendation*

I recommend to reject the paper mainly because I believe ICLR is not a suited venue, both for referring this paper (this paper needs to be reviewed by at least one expert in enumerative geometry, I don't know if there are such reviewers at ICLR) and for disseminating it (a journal in the field of computational enumerative geometry may be more suited). Furthermore, in my opinion the presentation of the material can be improved in several aspects before publication.

 
*Comments and questions*

- To which extent incorporating the conformal prediction framework in your analysis necessary? I am afraid this adds an additional layer of complexity that further hinders the communication of your findings. Maybe this discussion should be deferred to the appendix, keeping only what is strictly necessary to understand the main conclusion of your experiments in the main paper. 

- I don't understand the paragraph on top of p. 7, and I don't think this is due to my lack of expertise in enumerative geometry. In particular, what does "the neural network embedding p_g,n ... is a vector space"  means ? How can a function be a vector space? What does "go to the inner product space" means? These are (to me) very loose nonsensical mathematical statements.

*Minor comments & typos*

- p.3 the acronym COO has not been introduced

- Figure 5 should be included in the main part of the paper. In general, avoid forward references to far away, especially in the appendix without mentioning that it is in the appendix. 

- Use capitalization when reference tables, figures, sections, equations, etc. in the text (no capitalization needed when referring to figures or tables in general). E.g. Figure lines 168, 197, 334, Section lines 204, Table lines 274,298, Equation lines 319, 389 ... ... 

- line 196 we -> We

- line 421: the sentence "The interesting thing is that this is the performance of the non-linear probe." could be rephrased to better suit a formal publication.
</content>
</thread_object>

</thread>
<thread>
creation date:
1730390446969

thread_type:
official_review

<thread_object>
creator: reviewer_yn4i

<content>
summary:
The authors investigate the ability of transformers to compute the psi-intersection numbers in geometry and found that they perform (unsurprisingly) very well in distribution and quite well outside of distribution, and performed a series of analyses to understand which structures are being learned this way. They find in particular that the model learns the Dilaton equation and some information about the exponential growth of these psi-intersection numbers.

Overall, this paper is written very carefully, with excellent explanations of what is being done, although the importance of some details is unclear (like: what do we need to know about psi-intersection numbers? most of the sophisticated formulae are not really used in a meaningful way). However I am not sure the work is very interesting from a geometric point of view (the interesting thing is to gain theoretical insight into what psi-intersection numbers, not get somehow numerically accurate estimates of them) or from a machine learning point of view (it is not clear what is more interesting about these numbers than about any sequence in the OEIS, say). The experimental results are not particularly surprising given what is known about transformers, or at least I don't see it. 

While this work can be viewed as a first step towards making progress in applying machine learning to enumerative geometry, and the carefulness of the writing and experiments should be commended, I don't think it brings a lot of interesting new informations about machine learning or enumerative geometry.

strengths:
The care and clarity of the writing, the fact that some extensive research has been done, the general trust in the results that this paper inspires.

weaknesses:
What do we learn about machine learning or enumerative geometry? We seem to learn something that could be expected, a particular case of a general phenomenon.

questions:
It would be good if the authors could at least mention questions that would bring something interesting to the enumerative geometry (something feels interesting when they perform an analysis of the internal representation of the network, but it stops just before it gets interesting...).
</content>
</thread_object>

</thread>
<thread>
creation date:
1730612268614

thread_type:
official_review

<thread_object>
creator: reviewer_dajv

<content>
summary:
This paper introduced DynamicFormer to learn and predict the $\psi$-class intersection numbers. Experiments include both in-distribution results and out-of-distribution results. The author also presented some experiments to illustrate how transformers perform enumerative geometry. Meanwhile, the authors also investigated whether the proposed method could perform abductive reasoning and hypothesis testing to estimate the parameters of asymptotic form for intersection numbers.

strengths:
+ The idea of using transformers to do enumerative geometry is new. 
     + Meanwhile, the authors proposed a new activation function, DRA, which found to be useful to improve the prediction performance.
     + The authors compared DRA with other popular activations functions in Figure 1 and Table 2.

+ Experiments show some evidence of transformers can learn to predict the $\psi$-class intersection numbers.
     + Meanwhile, the authors also presented a discussion on how transformers being able to achieve that by inspecting internal vector space of the model.

+ The author also investigated how inputs affect the model’s understanding of $\psi$-class intersection numbers and the parameters for large genus.

weaknesses:
I am not an expert in "enumerative geometry". However, I think the paper lacks many important clarifications and discussions.
 
+  The paper lacked discussion of the reasons/motivations of using transformers.  At the moment, the paper seemed only a combination of  a popular neural network architecture and a new mathematical problem.
+ The "Related Work" section is quite weak at the moment: the authors spent only one paragraph to discuss related works and then summarized their contributions.
+  From my perspective, the proposed DRA is not the only way to capture the periodic behavior in data. This lacks sufficient discussion in the paper.
+ From experiments in Figure 1, the authors did not apply DRA to other neural network architectures (eg MLP), and provided readers with more discussions on that.
+ Lack of theoretical discussion on the proposed method.
+ Code is not available.

questions:
+ I wonder why the authors choose transformers as the regression function? 
+ In Figure 1, have you tried to apply DRA to MLP or other potential neural networks?
+ Will the code and the datasets be available?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730667263622

thread_type:
official_review

<thread_object>
creator: reviewer_ncfx

<content>
summary:
The paper proposes and tests the usage of transformers in the field of enumerative geometry, specifically regarding topological recursions and $\psi$-class intersection numbers. To accomplish this, the paper proposes a new class of activation functions called Dynamic Range Activators (DRAs), and presents evidence of their performance in predicting a simple recursive function as part of a fully connected neural network, and then their ability to predict $\psi$-class intersection numbers as part of their DynamicFormer architecture. The paper then attempts to investigate the trained DynamicFormer to see if it can predict other concepts in enumerative geometry, including the Dilation equation that stems from Virasoro constraints, as well as the asymptotic behavior of $\psi$-class intersection numbers using abductive reasoning, verified using counter-factual intervention.

strengths:
* The new DRA functions, motivated by the evidence presented in the paper, are a significant contribution that may interest machine learning scientists.
* Training a DynamicFormer to predict $\psi$-class intersection numbers, which then allows one to investigate a system's deeper geometry, is a significant, novel contribution that will interest mathematicians investigating enumerative geometry.
* The use of Conformal Prediction to estimate uncertainty provides a concrete measure of confidence in the experimental results, contributing to the paper's soundness.
* The figures are clear and high-quality, with informative captions.
* The writing is clear and mostly organized, including the mathematical background, methodology, and results.

weaknesses:
**Notice: These weaknesses have been adressed during the discussion phase, and apply only to the initial version of the manuscript. However, these will remain unedited for posterity.**

### Section 2
* The equations in this section use $\hbar$ without defining it in the text. It may be worth explicitly calling it the reduced Planck constant in the text.
* The last paragraph mentions excluding the tensor $C$ due to a decreased impact on the computed $\psi$-class intersection numbers, observed during experimentation. Appendix C justifies this exclusion, yet is not referenced in the text, making for seemingly unsound reasoning for excluding $C$. The authors should consider referencing appendix C here to further justify the exclusion of $C$, and expanding on this point within appendix C proper.

### Section 3
* In the first two paragraphs, the paper presents the DynamicFormer for the first time and references a figure placed within an unrelated appendix, resulting in a disjointed reading experience. The authors may consider moving some parts of section 3 (such as its first two paragraphs) into a new appendix showcasing the DynamicFormer in detail and including the figure close by.
* In the same paragraphs, the authors use the initials COO without previously defining them. These initials seem to appear nowhere else in the main text, and only in appendix B are they defined as Coordinate List. Besides hurting the paper's readability, this seems to be an implementation detail that does not need to appear in the main text.
* The last paragraph mentions the [DYN] registry tokens, but fails to reference appendix B1. It may be appropriate to reference it here.

### Section 5
* Equation 5.4 is presented without proof, with the authors claiming they used an approach described in Eynard et al. (2023). A sketch of the proof (perhaps in an appendix) will contribute to the work's soundness.
* Figure 3, and the relevant experiment, are based on the assumption that $A$ is rational. The authors should consider justifying the choice of testing only rational values of $A$, perhaps by connecting it back to equation 5.3, as proven by Aggarwal (2021).
* Figure 3 presents a significantly higher value of $R^2$ for $A=2/3$ compared to the values for $A=4/6$ and $A=6/9$, despite being identical numbers. This issue does not appear for other such sets of identical rational numbers, such as $A=3/4$ and $A=6/8$. Since the rest of the subsection on Abductive Reasoning relies on $A=2/3$ being the correct answer, **this error calls the entire subsection into question and significantly hurts the paper's soundness and overall rating**. The authors must justify how the $R^2$ of $A=2/3$ is different from the other two values, or replace the figure (and perhaps rewrite some of the supporting text). Based on the other values of the figure, it should be expected to see a maximal $R^2$ around $A=2/3$, but without such a significant jump.

### Typos
* Section 5.1 line 319: "The topological recursion formula equation 2.4 [...]". Consider removing either "formula" or "equation", or placing all of "equation 2.4" in parentheses.
* Section 5.1.1. has multiple citations included in sentences with their parentheses. The ICLR 2025 formatting instructions (section 4.1) require such references to not have parentheses except around the year. The references in question appear in lines 371, 378, and 381.
* Section 5.1.1 line 417: "As a result, We find an evidence [...]". "We" does not need to be capitalized, and "an" should be removed.
* Appendix C line 950: "Figure 6 shows (s) numerical [...]".
* The title of appendix D and the caption of figure 7 both mistakenly write Princip**le** Component Analysis instead of Princip**al** Component Analysis.

questions:
**Notice: These questions have been answered during the discussion phase, and remain unedited for posterity.**

* What is the significance of $\hbar$ in the quantum Airy structure? How is it relevant specifically to training the DynamicFormer?
* Figure 3 may be a discrete sampling of an underlying (continuous?) map that gives an $R^2$ for each $A$, with a maximum at $A=2/3$. Can the authors characterize this map?
* Figure 4 shows a significantly weaker causal impact of $B$ on the number of intersection points, compared to $n$ and $d$. Though the authors call this unexpected in section 5's last paragraph, is there any explanation regarding the weak causal impact of $B$?
</content>
</thread_object>

</thread>
<thread>
creation date:
1732548480680

thread_type:
official_comment

<thread_object>
creator: authors

<content>
comment:
Dear Reviewers, 

Thank you for your thoughtful feedback and for raising concerns regarding the suitability of our work for ICLR. Given that the primary area of this paper is applications to physical sciences (physics, chemistry, biology, etc.), we respectfully disagree and believe that our study offers several key contributions to both pure ML and AI for Science communities, extending beyond typical toy mathematical datasets such as arithmetic problems:

1. **Development of DynamicFormer:** We introduce DynamicFormer, a novel multi-modal Transformer-based model enhanced with the Dynamic Range Activator (DRA). This architecture is specifically designed to handle the complex, high-variance recursive data with heteroscedasticity inherent in enumerative geometry. By augmenting Transformer architectures, we demonstrate their capability to accurately model recursive functions exhibiting factorial growth. This addresses a long-standing challenge of modeling recursive function, representing a significant advancement beyond existing models, which often struggle with such intricate structures. **To the best of our knowledge, we are the first to explore and successfully model recursive with factorial growth functions.**

2. **Explainability and Uncertainty Quantification:** We have implemented a robust conformal uncertainty estimation and interpretability analyses into our model, ensuring that predictions are not only accurate but also reliable and interpretable. This is particularly crucial in AI for Mathematics contexts, where precision and rigor are paramount. To the best of our knowledge, **no prior work in AI for Mathematics has attempted to integrate uncertainty quantification and explainability methods specifically tailored for research-level mathematical problems.** 

3. **Discovery of Mathematical Insights:** Our analysis revealed that the network autonomously learned very deep identities and constraints in a pure data-driven manner, providing insights into the model’s mechanisms for out-of-distribution (OOD) generalization. Additionally, by investigating the asymptotic behavior of the intersection numbers, our interpretability analyses offered valuable insights into the underlying mathematical structures and network's "World Model". This is especially beneficial for theorem building, where the network's guidance or hints can suggest the existence of relations or identities. Furthermore, our approach facilitates the conjecturing of properties related to the asymptotic formulas of other intersection numbers. Collectively, these findings pave the way for data-driven human-machine collaboration in mathematical discovery.

Although our study investigates concepts from enumerative geometry, we have made concerted efforts to present the material in a manner accessible to the ML community. We provide the necessary background and context to ensure that the paper is understandable without requiring specialized knowledge in enumerative geometry, as evidenced by Reviewer ncfx's positive assessment. **Our objective is indeed to bridge the gap between these fields, demonstrating how advanced ML techniques can tackle complex mathematical problems and, conversely, how these problems can inspire new developments in AI.** We also have to note the emerging body of work at major ML conferences that intersects with pure mathematics. For instance, the recent paper **"Machine Learning Detects Terminal Singularities" presented at NeurIPS 2023** applies ML methods to problems in algebraic geometry. This exemplifies an interest and recognition within the machine learning community for such interdisciplinary research, supporting the suitability of our work for ICLR.

We hope that these clarifications and enhancements address your concerns and demonstrate the meaningful impact and innovative nature of our research. Thank you again for your comments, which have been invaluable in improving the quality and clarity of our paper. If our arguments resolve your concerns we would appreciate if you would consider raising your score!

Best Regards,

The Authors
</content>
</thread_object>

</thread>
<thread>
creation date:
1733225065378

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Summary of Rebuttal and Discussion Period

comment:
Dear Reviewers, AC and SAC, 

We would like to express our gratitude to the reviewers and ACs for their time, effort and valuable feedback on our work. We have carefully considered all comments and have made significant revisions to the manuscript to address the concerns raised during the review and discussion phases.

**Our Main Contributions:**

1. **Development of DRA:**
   - We introduced the **Dynamic Range Activator (DRA)**, a new non-linear activation function, specifically designed to handle complex, high-variance recursive data with factorial growth. We address the long-standing challenge of modeling recursive functions with high variance, representing a significant advancement over existing models that struggle with such intricate structures. **To the best of our knowledge, we are the first to explore and successfully model these recursive functions.**

2. **Uncertainty Quantification:**
   - We incorporated and adapted a proper conformal uncertainty estimation method ensuring that our model's predictions are not only accurate but also reliable, which is important in mathematics where precision and rigor are paramount. **Our work is the first to apply uncertainty quantification in AI-based computational mathematics.**

3. **Discovery of Mathematical Insights:**
   - Our analysis revealed that the network autonomously learned **Virasoro constraints**, providing insights into how the model generalizes to out-of-distribution data.
   - By investigating the asymptotic behavior of $\psi$-class intersection numbers, our interpretability analyses offered valuable insights into underlying mathematical structures.
   - **This finding opens new avenues for data-driven human-machine collaboration in closed-form expression discovery, theorem formulation and conjecturing properties related to asymptotic formulas of other enumerative problems.**


**Responses to Reviewers:**

- **Reviewer ncfx:**
  - We really appreciate their detailed and constructive feedback.
  - After addressing all concerns, the reviewer acknowledged our efforts, stating: "This paper is solid and interesting, and no doubt will be of interest to the greater ICLR community."
  - The reviewer increased their score accordingly, reflecting their positive assessment.

- **Reviewer daJv:**
  - We thoroughly addressed the concerns raised, including expanding on the motivations for using Transformers, enhancing the Related 
     Work section, code provision, and improving the presentation of our findings.
  - Despite our detailed responses and acknowledging the novelty and strengths of our work, Reviewer daJv dismissed our responses 
    entirely and unexpectedly raised concerns about the paper's suitability for ICLR, even though similar works have been accepted at 
    previous ICLR and other major AI conferences. (provided in our last message to all reviewers and ACs.)

- **Reviewer Yn4i:**
  - They engaged minimally with our contributions. Despite our detailed clarifications on our cotribution both from ML and Computational 
    Algebraic Geometry perspective , their feedback remained vague and uninformative, ultimately expressing reservations about the 
    paper's excitement.

- **Reviewer qDDj**
  - We improved the manuscript based on their feedback, including correcting errors and elaborating more on key concepts, providing examples on similar works that have been accepted at previous ICLR and other major AI conferences. However, the 
     reviewer did not update their evaluations or engage further in the discussion.


**Final Remarks:**

We believe that our work offers important contributions to the field of AI for Science. This work bridges the gap between ML and advanced mathematics, aligning with ICLR's interest in innovative and interdisciplinary research.

We respectfully request that the area chairs and senior area chairs consider our comprehensive responses and the positive evaluation from Reviewer ncfx when making their decision. We are confident that our work will be of interest and value to the ICLR community.

Thank you for your time and consideration. We remain available to provide any additional information or clarification as needed.

Best regards,

The Authors
</content>
</thread_object>

</thread>


Metareview:
metareview:
**Summary:**  
This paper introduces a Transformer-based model, DynamicFormer, with a novel Dynamic Range Activator (DRA) activation function tailored to model recursive functions with high variance and factorial growth. The study applies this methodology to computational enumerative geometry, specifically in computing \(\psi\)-class intersection numbers. Additionally, the paper incorporates conformal prediction for uncertainty quantification and provides interpretability analyses that uncover evidence of the model learning mathematical structures, such as the Virasoro constraints.

**Strengths:**  
1. **Novel Methodology:**  
   - The DRA activation function addresses challenges in modeling recursive functions with factorial growth, a significant contribution beyond standard applications of transformers.  
   - Incorporates uncertainty quantification in a rigorous and tailored manner for high-variance recursive data.  

2. **Interdisciplinary Contribution:**  
   - Bridges machine learning and enumerative geometry, showcasing how AI can assist in tackling complex mathematical problems.  
   - Reveals new insights into mathematical structures through model interpretability, contributing to the AI for Science community.  

3. **Solid Experimental Design:**  
   - Comprehensive evaluation of DRA against other activation functions on synthetic data.  
   - Demonstrates the ability of DynamicFormer to generalize to out-of-distribution cases.  
   - Insightful analysis of the network's internal representations and their alignment with known mathematical properties.  

**Weaknesses:**  
1. **Complexity and Accessibility:**  
   - The paper's interdisciplinary nature makes it challenging for readers without expertise in enumerative geometry or advanced mathematics to fully grasp the significance of the contributions.  
   - The initial presentation of material was fragmented, though significantly improved during the discussion phase.  

2. **Limited Exploration of Applications:**  
   - While the paper demonstrates the feasibility of using transformers for \(\psi\)-class intersection numbers, broader applications and deeper theoretical insights into the DRA activation function could strengthen the work further.  

3. **Venue Suitability Concerns:**  
   - Some reviewers questioned whether ICLR is the most appropriate venue for this work, given its mathematical focus. However, the paper aligns with the growing interest in interdisciplinary research at ML conferences.  

**Discussion:**  
The reviewers expressed mixed opinions. One reviewer rated the paper highly, highlighting its interdisciplinary novelty and methodological contributions. Other reviewers raised concerns about the presentation and accessibility but acknowledged the potential of the work. The authors addressed these concerns through substantial revisions, clarifications, and added context, improving the paper significantly. Despite its complexity, the paper offers novel contributions to both the ML and mathematical communities.

**Suggestions for Camera-Ready Submission:**  
1. Continue improving the clarity and accessibility of the manuscript, particularly for readers with limited background in enumerative geometry.  
2. Highlight broader implications and potential applications of the DRA activation function in modeling recursive functions beyond the specific mathematical context explored.  
3. Ensure all implementation details and code are easily accessible to facilitate reproducibility.  

**Conclusion:**  
This paper represents an original and significant contribution to the intersection of AI and advanced mathematics. While the application is domain-specific, the methodologies and insights have broader relevance to the AI for Science community. The constructive feedback from reviewers has been well-addressed, and I recommend the revised manuscript for acceptance.