Combining Denoised Neural Network and Genetic Symbolic Regression for Memory Behavior Modeling via Dynamic Asynchronous Optimization

Dear Reviewer CdzH,

Thank you for your recognition of our work and for your valuable suggestions, particularly regarding the methodological innovation and empirical evaluation quality. We are delighted that you share an interest in the integration of neural networks with symbolic regression and acknowledge its potential in the field of memory modeling. As you noted, this research could inspire advancements not only in psychology, cognitive science, and neuroscience but also in other domains. Based on your suggestions, we have devised an improvement plan to enhance the clarity and readability of the manuscript, ensuring it is accessible to a broader academic audience. We plan to upload the revised version within 2-3 days and hope you will review it again and share your invaluable feedback.

Improvement Plans for Identified Weaknesses:

1. Simplified formulas and illustrations
To enhance readability, we will simplify lengthy formulas (e.g., the definition of mean squared error), retaining only essential content. Descriptions of Equations (8) and (9) will be condensed to reduce space usage. Additionally, the equations in Figure 2 will be moved to the main text to better highlight the core concepts of the alignment algorithm.

2. Highlighting table content
The current discussion of results from Table 2, particularly those generated by symbolic regression, is insufficient. In the revised version, we will provide a more detailed analysis of these results, focusing on the potential implications of symbolic regression formulas for psychological and neuroscience research.

3. Additional details for model reproducibility
We will include more detailed information about the model in the revised manuscript, such as neural network hyperparameters, training schedules, and symbolic regression initialization settings. This will ensure other researchers can accurately reproduce our experiments.

4. Background and main text optimization
We recognize some overlap between the background and introduction sections. In the revised version, we will merge overlapping content, retain key background knowledge, and use the saved space to expand discussions on model details and experimental results.

5. Clarification of terms and abbreviations
We will clearly define all abbreviations upon their first appearance in the revised manuscript. Additionally, abbreviations not used repeatedly will be removed to avoid distracting readers.

6. Significance testing
For the experimental results in Table 3, we will add statistical significance testing data (e.g., standard errors or error ranges) to enhance the credibility of the results.

7. Enhanced referencing of Table 2
In the revised version, we will explicitly reference the equations and related discussions in Table 2, elaborating on their significance in memory modeling.

Dear Reviewer tRhP,

Thank you for your recognition of our self-evolving psychological embedding neural network model and for taking the time to review our submission. We apologize for the shortcomings in the paper’s presentation. We have carefully considered your suggestions and made comprehensive revisions to improve its clarity and structure. We plan to upload the revised version within 2-3 days and hope you can review it again to provide further valuable feedback. Your comments have been invaluable in helping us refine our work, and we are deeply grateful.

Explanations and Responses to Weaknesses and Questions:

1. Incomplete description of the real-world task
We acknowledge that the original paper lacks clarity in describing the real-world task. In the revised version, we have clarified the problem setup with more detailed explanations and visual aids. Specifically, our study focuses on modeling learners’ memory states in a word memorization scenario. During the memory process, learners complete quiz questions through memory software. If a learner answers correctly, the word is marked as “mastered” (labeled as “correct”); if not, it is marked as “not mastered” (labeled as “incorrect”). Quiz formats include multiple-choice, fill-in-the-blank, listening, and matching questions. We have provided a more intuitive description of this task in the revised manuscript.

2. What is the model’s final output? Is it derived from the generated equations or the neural network?
The final output of the model is the prediction results from the neural network. All comparative and ablation experiments are evaluated based on the neural network’s output. The equations generated by symbolic regression are considered by-products of the model. In the equation comparison experiments, these equations are extracted and evaluated separately for their predictive performance. We have clarified the distinction between the two types of outputs in the revised manuscript to avoid confusion.

3. Confusing notation
We understand that the complexity of the notation may hinder readers' comprehension. To address this, we have thoroughly optimized the notation system in the revised manuscript. For example, we use $𝑚$ to represent a single time step and $1:𝑚$ to represent multiple time steps, and we have adjusted the mapping space of words to make the notation more intuitive and concise.

4. Undefined abbreviations
Thank you for pointing out the issue with undefined abbreviations. In the revised manuscript, we have resolved this by providing clear definitions for all abbreviations.

5. Source of $\beta_t$ and similarity to diffusion models
The noise scheduling parameter $\beta_t$ indeed shares similarities with noise generation equations in diffusion models. In the revised version, we have explicitly acknowledged this connection and added relevant citations. Additionally, we have included a detailed proof in the appendix to demonstrate the consistency between our noise generation method and the DDPM perturbation kernel.

6. Lack of clear descriptions of alternative methods in the ablation study
Our proposed method combines asynchronous training and dynamic optimization. Asynchronous training refers to the joint training of the neural network and symbolic regression model, while dynamic optimization adjusts the loss weights of the neural network dynamically. For instance, if asynchronous training is enabled without dynamic optimization, the model’s loss weights remain as fixed hyperparameters and do not change dynamically based on performance. It is important to note that dynamic optimization requires asynchronous training to be activated.

In the ablation study, the configurations are as follows:

• Baseline model (DKT-F): Does not include the denoising module or the symbolic regression module.
• Asy (asynchronous training): Combines the neural network and symbolic regression with a waiting strategy during training, but the loss weights are fixed hyperparameters.
• DyOp (dynamic optimization): Introduces dynamic loss weights that adjust based on model performance. If DyOp is not enabled, the loss weights remain fixed hyperparameters.

In the revised manuscript, we have provided detailed explanations and refined the analysis of the experimental results.

The revised version of our manuscript will be uploaded to OpenReview within three days. We sincerely hope you can review it then. If you have any additional suggestions, we would be delighted to incorporate them to further improve our work!

Dear Reviewer tRhP,

Thank you for pointing out the insufficient explanation of Deep Knowledge Tracing (DKT) in our manuscript. In the revised version, we have added a detailed description of DKT in the Background section to enhance the clarity of this concept. Additionally, we have carefully reviewed all abbreviations in the manuscript to ensure their clear and consistent usage. Regarding the citation formatting issues you mentioned, we have corrected them in the latest revision. We sincerely appreciate your thorough review and valuable suggestions, which have been instrumental in improving the quality of our paper.

Sincerely,

Authors of Paper 8572

Dear Reviewer qnLQ,

Thank you for recognizing the value of our work, particularly the innovative approach and potential of integrating neural networks with symbolic regression for memory modeling. Indeed, our research is greatly inspired by the PINN models from the field of physics, and we also believe our method has the potential for applications in other domains. We will mention this point in the revised manuscript. Additionally, we have included an analysis comparing the discovered memory equations with existing theories, highlighting our method’s potential in uncovering new psychological theories.

We sincerely apologize for the shortcomings in the presentation of the manuscript, as noted by you and other reviewers. We have carefully considered your suggestions and have revised the manuscript to improve its clarity and presentation. We plan to upload the revised manuscript within 2-3 days and hope you can review it again and provide us with further valuable feedback. We truly appreciate your insightful comments, which have been immensely helpful in refining our work.

Responses and Improvements to the Weaknesses:

1. Lengthy and redundant mathematical notations:
We acknowledge that the current version’s notations are overly complex, which may hinder the reader’s understanding. In the revised manuscript, we have simplified the notations and removed unnecessary redundancies. Furthermore, we have restructured the content to present mathematical symbols and formulas more clearly and concisely, reducing the cognitive load for readers.

2. Lack of clarity in baseline method descriptions:
Thank you for pointing out the inadequacies in the explanation of baseline methods. In the revised manuscript, we have enhanced the descriptions of baseline methods, including background details on the DKT-F and FIFKT models, and provided a thorough explanation of each variant in the ablation studies. We have also clarified the effects of combining different components, enabling readers to better understand the significance of various model configurations.

3. Enhancements to background methods and problem description:
Since the paper involves multiple approaches, including symbolic regression, deep learning, memory modeling, and PINNs, we have reorganized the manuscript’s logical structure and provided more detailed explanations of these background methods in the revised version. These changes will help readers gain a more comprehensive understanding of the research context and core contributions.

Responses to Specific Questions:

1. Combination of asynchronous training and dynamic optimization:
Our proposed method integrates asynchronous training and dynamic optimization. Asynchronous training refers to the joint training of the neural network and the symbolic regression model in an asynchronous manner, while dynamic optimization involves dynamically adjusting the neural network’s loss weights. For instance, when asynchronous training is enabled but dynamic optimization is disabled, the loss weights remain fixed hyperparameters and do not change dynamically with model performance. It should be clarified that dynamic optimization requires asynchronous training to be enabled. We will further clarify these experimental settings in the revised manuscript.

2. Performance of symbolic regression when trained alone:
The primary objective of our study is to design a neural network model, with symbolic regression introduced to aid the neural network in better modeling learners’ memory states. In the original version, we did not provide the standalone performance results of the symbolic regression model. In the revised manuscript, we have added these results and analyses to provide a more comprehensive comparison.

3. Selection of operator sets in symbolic regression:
Thank you for raising this question. Due to an oversight, we did not explicitly specify the operator set used in symbolic regression. In this study, we adopted operators from traditional memory equations, and the $log$ function mentioned in the manuscript refers to the natural logarithm ($ln$). In the revised manuscript, we have corrected and standardized the symbolic descriptions to ensure consistent presentation and avoid confusion.

Thank you once again for your valuable feedback. We will upload the revised manuscript to OpenReview within three days, and we look forward to your further review and suggestions.

3. On referring to ANN architecture as TNN

In our paper, we use the term Temporal Neural Network (TNN) to refer collectively to neural network models designed for time-series data modeling (e.g., RNN, LSTM, GRU, Transformer). We acknowledge that this terminology, coined for convenience, may lack precision, and we deeply regret any confusion it may have caused.

There are existing analogous naming conventions, such as Spatio-temporal Neural Network[2] and Temporal Convolutional Neural Network[3]. The use of TNN in our paper aimed to generalize a series of neural network models for time-series data modeling, including RNN, LSTM, Transformer, etc. Specifically, in our method, we employ LSTM to capture dynamic memory states and further process these states with an MLP.

Once again, thank you for pointing out the potential confusion caused by this terminology!

References

[1] Makke N, Chawla S. Interpretable scientific discovery with symbolic regression: a review. Artificial Intelligence Review, 2024, 57(1): 2.

[2] Ye J, Sun L, Du B, et al. Co-prediction of multiple transportation demands based on deep spatio-temporal neural network. In Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining. 2019: 305-313.

[3] Pelletier C, Webb G I, Petitjean F. Temporal convolutional neural network for the classification of satellite image time series. Remote Sensing, 2019, 11(5): 523.

We hope the above responses address your concerns clearly. Should you have further questions, we would be delighted to discuss them further! Thank you again for your support and suggestions regarding our work!

Dear Reviewer gLam,

We sincerely appreciate your detailed review of our submission and your recognition of the method integrating neural networks with equation optimization. Your valuable feedback has provided crucial guidance for improving our manuscript. We apologize for the shortcomings in the presentation of the paper. We have carefully considered your suggestions and revised the manuscript to enhance its clarity and presentation. We plan to upload the revised version within 2-3 days and hope you can review it again and provide further insights. Your comments have been instrumental in improving our work, and we are deeply grateful.

Improvements and Explanations for Weaknesses:

1. Clarification of the paper’s objectives:
Our primary goal is to construct a knowledge-driven neural network model suitable for memory behavior modeling. To achieve this, we incorporate classical memory theory equations to constrain the neural network’s modeling process. However, existing memory theories are often debated and lack the explanatory precision seen in physical equations. Therefore, we initialize the neural network with classical memory theory equations and refine these equations using symbolic regression. This approach enables the neural network to absorb knowledge from classical memory theories while achieving collaborative optimization between the memory equations and the neural network model.

In this process, the memory equations derived from symbolic regression serve as proxy models to explain the neural network and offer potential new perspectives for psychological theories. Our primary focus is not on proposing a novel symbolic regression method but on leveraging existing symbolic regression algorithms to address key challenges in memory behavior modeling.

We chose genetic symbolic regression algorithms based on three considerations:

1. Genetic algorithms are a classic and well-researched family of symbolic regression methods, offering numerous resources and references.
2. They allow us to set initial populations, enabling classical memory theory equations to serve as initialization equations.
3. Genetic algorithms can strictly control the depth of symbolic trees, ensuring that the derived equations remain interpretable.

Our framework is versatile and compatible with various genetic symbolic regression algorithms. PySR was selected as the codebase for our experiments, and we will clarify this in the revised manuscript. Additionally, we have added experiments to demonstrate the generality of our approach. Theoretically, our method also has the potential to integrate more advanced symbolic regression algorithms, which we plan to explore in future research to further enhance its applicability.

2. Lack of comparisons with state-of-the-art SR algorithms:
To address your suggestion, we have expanded the experimental section in the revised manuscript by including performance evaluations of advanced symbolic regression methods such as TPSR, DSR, and GP-SBP. These results have been incorporated into Table 2 to enrich the experimental comparisons.

3. Missing error bars in empirical results to assess significance:
Regarding the significance of empirical results, we have added error bars for all experimental results in the revised version and included statistical significance tests to enhance the credibility and completeness of the experimental findings.

4. Lack of details about the MLP architecture and adjustments:
We have provided detailed information about the architecture of SPsyINN in the revised manuscript, including the MLP architecture, hyperparameter settings, and training schedules, to ensure clarity and precision in describing the model.

Responses to Specific Questions:

1. How does SPsyINN perform on existing equation discovery benchmark datasets?
Our primary research focus is on designing a model capable of effectively modeling learners’ memory states, rather than creating a general-purpose symbolic regression algorithm. Consequently, SPsyINN has not been evaluated on general equation discovery benchmarks (e.g., SRBench or SRSD). Nevertheless, we believe our proposed method holds potential for applications in other scenarios, and future research will explore its performance in broader domains.

2. How do advanced SR algorithms perform?
In response to your suggestion, we have included the performance of advanced symbolic regression methods such as TPSR, DSR, and SBP-GP in the memory modeling scenario. These results have been updated in Table 2 of the revised manuscript, providing a comprehensive comparison with SPsyINN to demonstrate the applicability of symbolic regression algorithms in memory modeling tasks.

3. Addition of error bars in experiments:
We have addressed this issue by including error bars (e.g., standard deviations or interquartile ranges) in the revised manuscript to highlight significant differences in the results.

4. Citations related to Table 2:
For the results in Table 2, we have added detailed discussions and included references to the theoretical equations involved in the revised manuscript, enabling readers to better understand the significance of the results.

The revised version will be uploaded to OpenReview within three days. We earnestly request your review at that time and thank you once again for your thorough review and invaluable suggestions!

The authors' revision has greatly improved clarity, so I will be increasing the presentation score to 3. The authors' have also identified key SOTA SR methods for comparison, which improves the quality of the experiments.

However, I do have some new issues with the results. To verify the results of the paper, I am testing the obtained equations from Table 5 on the dataset given in the link. However, I am unable to reproduce the MAE scores of Table 5 (which is linked to Table 1). First, there seems to be an error for the results of SBP-GP (MaiMemo) in Table 5, where the average MAE is .3988 but .2660 in Table 1. Second, on the duolingo dataset, using the data extracted from the train_loader and test_loader (and also normalizing with the max and min code variables: "data = (data - min) / (max - min)", and changing nan values to 0: "data = torch.nan_to_num(data, 0)"), I get the following results:

DUOLINGO dataset Wickelgren Model, "0.89*(1+0.0003*X[:,1])**-0.0003", MAE(using the provided "masked_mae" function provided by the authors): 0.1238

PySR Model, "0.92 - X[:,0]*(X[:,4]+0.03)", MAE: 0.0987

DSR Model, "np.cos(X[:,0]-X[:,3]+np.exp(X[:,0]*X[:,1]X[:,2](-X[:,2]-X[:,3]-X[:,4])+X[:,4]))", MAE: 0.4667

SPsyINN-C-F Model, "0.56np.exp(-1.75X[:,3]*X[:,0]*X[:,2])", MAE: 0.4150

SPsyINN-W-F Model, "0.56*np.exp(X[:,0]X[:,3](X[:,2]-X[:,0]))", MAE: 0.4149

Could the authors', for the sake of easing the process for the reviewers to test reproducibility, extract a csv file with just the 6 features that are used for prediction (e.g., $\delta_1$, $\delta_2$, $\delta_3$, $\delta_4$, $\delta_5$, $\delta_6$), along with the output, R?

Also, I have noticed in the code that these seem to be the normalized values based on the train set, I think for greater clarity, the authors should mention this in both Table 5 and Appendix A.2 for reproducibility. If $\delta_1$, $\delta_2$, $\delta_3$, $\delta_4$, $\delta_5$, $\delta_6$ refer to the unnormalized version instead, please correct me.

Dear Reviewer gLam,

Thank you so much for taking the time to review our manuscript and providing such valuable feedback. Your comments have been immensely helpful in improving and refining our work.

We have made every effort to address the issues and suggestions you raised, and we hope the revised manuscript meets your expectations. On this basis, we would like to kindly inquire whether, if you find our improvements satisfactory, you might consider raising the overall score for our paper.

We greatly respect your professional judgment and would be happy to engage in further discussion if you have any additional questions or suggestions.

Once again, thank you for your support and invaluable input throughout this process!

Sincerely,

Authors of Paper 8572

As a supplementary note, when calculating directly using the CSV data, the performance metrics must be computed in the format specified in the code documentation. This is because the label data contains zeros, which cannot be used as a denominator. We still recommend using the simple performance testing document we provided, as it offers a more convenient way for you to conduct the tests. We have set a mask in the calculation of metrics because there are 0 labels in the label data, and these would be treated as denominators in the metric calculations, which can lead to errors. This approach is consistent with the calculation process in FIFKT and is a standard practice in modeling. All standard calculation methods in our work are consistent. We have provided examples of how to calculate the corresponding metrics in the CSV.

Dear Reviewer gLam,

1. Regarding the inconsistency between equation performance and annotations
   We believe there may have been some misunderstanding. We provided a simple implementation test for the performance of all equations (located in the code documentation under Test for functions/Function_test.py). We have carefully reviewed the composition of the CSV data and the performance of the equations, and we did not find any anomalies.

We recommend reviewing the settings for equations in Test for functions/Function_test.py. We have carefully validated its reproducibility. Below are examples of the performance for specific equations:

MaiMemo Dataset

• SPsyINN-C Function: 0.30229160710443679**((delta1*delta5)**(delta4 + 0.14383100468725032))
  • Test: MAE: tensor(0.2213) | MAPE: tensor(22.1332)
• SPsyINN-W Function: 0.49258729183071737**((delta1 + 0.008248828395980482)**(delta4**0.6295170754361378))
  • Test: MAE: tensor(0.2157) | MAPE: tensor(21.5667)
• SPsyINN-I Function: 0.4733634187963817**((delta1**1.109281583119398)**(delta4**0.7567579728325409))
  • Test: MAE: tensor(0.2251) | MAPE: tensor(22.5074)

Duolingo Dataset

• SPsyINN-C Function: -(delta6 + delta2)*(delta1 - delta2) + 0.9135621262263904
  • Test: MAE: tensor(0.1052) | MAPE: tensor(12.6823)
• SPsyINN-W Function: ((delta5 + 0.0190478124994636)**(delta1-delta2))*0.9257066642765628**e**delta6
  • Test: MAE: tensor(0.1041) | MAPE: tensor(12.5264)
• SPsyINN-I Function: 0.92171514151 ** (0.2101281541 * delta1+ exp(delta6))
  • Test: MAE: tensor(0.1017) | MAPE: tensor(12.3638)

2. Regarding the inconsistency between the initial, revised, and final versions of equations
   Our GSR module uses the PySR method, which provides a candidate set of equations during the symbolic regression process. We retained data files from each training session and selected the final equations from these files. When updating the equations, we rigorously confirmed their reproducibility. We have supplemented the code documentation with information on the equation sets for SPsyINN-C, SPsyINN-W, and SPsyINN-I for your review.

3. Regarding the confusion about regression labels being 1
   In the Duolingo dataset, this issue may arise from the dataset itself. Duolingo’s testing sessions allow learners only one incorrect response per word. Specifically, for a given word, testing stops only if the number of incorrect responses is less than one. This is explicitly reflected in the original dataset, and we observed this behavior. During model performance testing, we avoided equations and models that predict all values as 1. We have added examples of the original data to the code documentation to clarify this issue.

4. Regarding the validity of the R² score
   We have also considered this issue. While the identified equations perform well on MAE and MAPE metrics, their performance on the R² metric is poor. We believe this is likely due to the nature of the dataset. As you mentioned, most label values are 1, making it challenging for the equations to achieve a good R² score.

We hope the above responses address your concerns. Once again, thank you for pointing out these issues.

Sincerely,
Authors of Paper 8572