Data-Driven Discovery of Dynamical Systems in Pharmacology using Large Language Models

We thank the reviewer for their constructive feedback and are glad the reviewer finds our paper interesting and appreciates our innovative application of LLMs to generate interpretable models of pharmacokinetic dynamics. We are also pleased that the reviewer acknowledges the clear writing and comprehensive experiments, and that they have learned from our work.

It is likely that OpenAI trained GPT-4 on relevant literature, leading to a significant knowledge leakage issue.

We appreciate this insightful concern regarding potential knowledge leakage. To address this, we conducted additional experiments using five semi-synthetic datasets specifically designed to ensure that the LLM had no prior exposure to their underlying equations. We ran D3-Hybrid on these datasets and compared the results with some of the baselines. The results, which are included in the supplementary material, demonstrate that D3-Hybrid consistently performs well even when the LLM has never encountered such models before. This empirical evidence strongly mitigates the concern of knowledge leakage and reinforces the robustness of D3-Hybrid in discovering well-fitting models from novel data.

It seems that this paper does not actually need to rely on simulated data and could use real high-dimensional time-series observational data for experiments.

Thank you for this suggestion. We agree with the importance of using real datasets and highlight that the Warfarin dataset used in our experiments is indeed a real-world dataset. Additionally, we conducted experiments involving up to 22 features, as shown in Figure 2 of Appendix G.2. The core focus of the D3 framework is to identify the most relevant features for modeling while disregarding irrelevant ones. This approach is particularly beneficial in real-world scenarios where distinguishing significant features from a large pool is crucial for accurate modeling.

The Table 3 did not report the performance of baseline models.

We acknowledge the oversight and appreciate the reviewer pointing this out. We have updated Table 3 in the manuscript to include the performance of baseline models for a comprehensive comparison. This additional information highlights the competitive edge of our proposed D3 framework over traditional baseline models, reinforcing the novelty and effectiveness of our approach.

Baseline methods like D-CODE [ICLR'22] and PGM are not compared.

We appreciate the suggestion to compare our work with D-CODE and other symbolic regression methods. However, our paper aims to introduce a framework that not only discovers interpretable models but also incorporates textual priors (context) and enables the acquisition of new features and samples on demand. While symbolic regression methods like D-CODE are powerful, they do not address the integrated feature acquisition and context utilization capabilities of D3. For comparison purposes, we included SINDy, a well-regarded symbolic regression method, and provided a detailed evaluation against it. We believe that a direct comparison with D-CODE, which focuses solely on symbolic regression, would not fully capture the broader capabilities and contributions of the D3 framework. Furthermore, the referred to PGM paper is out of scope as well, as it focuses on probabilistic graphical modeling of dynamical systems, whereas we focus on a deterministic best-fitting model for the dataset at hand.

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We hope that most of the reviewer’s concerns have been addressed and, if so, they would consider updating their score. We’d be happy to engage in further discussions.

We thank the reviewer for their constructive feedback and are glad the reviewer finds our work introducing the Data-Driven Discovery (D3) framework novel, innovative, and significant in its application to pharmacokinetic modeling.

Cost function considerations: Adding a few sentences or a small subsection to discuss the aspects of the cost function and its potential impact and tradeoffs will provide a more comprehensive understanding.

We agree with the reviewer's suggestion and appreciate the opportunity to clarify this point. The current description of the cost function $l(h_i)$ captures the complexity and computational resources required to acquire new features. However, it may not fully account for the practical challenges of acquiring specific biomarkers or data types that are costly or invasive to obtain. To address this, we have added a subsection discussing the potential impact and tradeoffs of different cost function considerations, emphasizing the importance of balancing accuracy improvements with practical feasibility. This addition provides a more comprehensive understanding of the cost-benefit analysis inherent in our feature acquisition process.

Certain biomarkers can be measured only in a small sample of patients. It would be interesting to discuss potential performance and limitations in light of missing data.

We agree, and this is indeed an assumption of the current presented method. Allow us to kindly re-iterate that the focus of the paper is to propose the Data-Driven Discovery (D3) framework, a novel approach leveraging Large Language Models (LLMs) to iteratively discover and refine interpretable dynamics models, advancing pharmacokinetic modeling. While we acknowledge that handling partially missing data is a critical aspect of real-world applications, we believe such a detailed analysis falls beyond the scope of this paper. However, we plan to explore this interesting topic in future work, investigating robust methods to address missing data scenarios within the D3 framework.

See specific numbers around time to completion wrt the numbers of features.

On average, a complete run of D3 with a feature size of 3 features takes approximately 1 hour. This duration includes the iterative processes of model generation, evaluation, and feature acquisition, ensuring a thorough exploration and optimization of the pharmacokinetic models.

How plausibility of the model is evaluated.

The plausibility of the model is evaluated by the LLM through reflective analysis. Specifically, the LLM assesses the generated model based on domain knowledge, prior literature, and logical consistency. This reflective process involves checking the alignment of the model's predictions with known pharmacological principles and empirical observations. Additionally, the evaluation includes examining the model's ability to generalize across different datasets and its interpretability by human experts, ensuring both accuracy and clinical relevance.

mixing scientific and regular notation

Thank you for highlighting this inconsistency. We will revise the tables to ensure uniform notation, using scientific notation consistently throughout the manuscript. The mixed notation resulted from the automatic generation of tables from raw results using Pandas dataframes, and we appreciate your attention to detail in this matter.

The choice of LLM is quite significant and it is stated for the first time in Limitations on page 9. This could be brought earlier.

We agree and have now included the choice of the LLM in the introduction. Clearly stating this earlier provides context for the framework's capabilities and limitations, helping readers understand the significance of the LLM in driving the iterative discovery and refinement processes.

Questions

Why was MSE used as the metric for accuracy?

Mean Squared Error (MSE) is a standard metric for assessing model accuracy in pharmacokinetic modeling, as demonstrated in the cited PKPD model papers. It quantifies the average squared difference between observed and predicted values, providing a clear measure of model performance. We have defined and explained the use of MSE in the Appendix, ensuring transparency in our evaluation criteria.

Do you have ideas as to how to combat the limitations mentioned such as hallucinations?

To combat limitations such as hallucinations, we could employ Retrieval-Augmented Generation (RAG) techniques. RAG combines the retrieval of relevant documents or data with generative models, grounding the LLM's output in factual information. This approach reduces the likelihood of hallucinations by ensuring the generated content is based on verified sources, enhancing the reliability and accuracy of the models produced by D3.

Have you considered the inverse bias problem.

We acknowledge the inverse bias problem, where models like GPT-4 are restricted from using biased features that could be informative in the pharmaceutical context, such as ethnicity. While we have not addressed this issue in the current paper, we recognize its importance and plan to explore it in future work. Investigating strategies to balance the ethical considerations of bias with the need for accurate and individualized pharmacokinetic models is a promising area for further research.

I'm still not fully clear how the LLM acquires the unstructured context/metadata about each feature.

The LLM is provided with the feature's name, of which it is empirically observed that is enough information to understand the relevance and value of that feature given the current context and progress seen during operation.

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We hope that most of the reviewer’s concerns have been addressed and, if so, they would consider updating their score. We’d be happy to engage in further discussions.

Dear Reviewer QDKe, I am a NeurIPS 2024 Area Chair of the paper that you reviewed.

This is a reminder that authors left rebuttals for your review. We need your follow up answers on that. Please leave comment for any un-answered questions you had, or how you think about the author's rebuttal. The author-reviewer discussion is closed on Aug 13 11:59pm AoE.

Best regards, AC

We thank the reviewer for their constructive feedback and are glad the reviewer finds the D3 framework’s ability to iteratively improve models and acquire new features enhances its performance and robustness. We also appreciate that the reviewer acknowledges the clear writing, well-structured methodology, effective figures and tables, and the interesting problem setup involving LLM agents for clinically relevant tasks.

Limited Novelty: automated science labs (self-driving labs)

Thank you for raising additional related works. Allow us to kindly re-iterate that the focus of the paper is to propose a method that can both discover an interpretable model, incorporate textual prior (context), and have the possibility to acquire new features and samples. Existing automated science labs (e.g., The Automatic Statistician) do not leverage LLMs for iterative optimization and generally do not integrate feature acquisition in a dynamic manner. Our approach uniquely combines the strengths of LLMs with an iterative discovery process that can adjust to new data and contexts, providing a more flexible and scalable solution for pharmacological modeling.

Lack of Guardrails for Hallucinations

We agree with your suggestion and highlight that any LLM-based method or paper should have its outputs always checked by a human expert before producing the final result. We highlight that this is a common problem across any LLM-based method paper. To mitigate this risk, we have implemented a human-in-the-loop framework where clinician feedback is integrated into the iterative process. Additionally, future work could incorporate retrieval-augmented generation (RAG) techniques to further reduce hallucinations by grounding the LLM’s outputs in factual data.

Prompt Dependency

Allow us to highlight that any LLM-based paper depends on the prompts that are input into it. However, we emphasize that our method’s strength lies in the combination of model feedback and LLM iteration. By iteratively refining prompts based on model performance and expert feedback, we mitigate the risks associated with prompt dependency and enhance the reliability and consistency of the generated models.

Scalability and Computational Efficiency

We find the method to still be scalable and the inner loop to train a small parameter hybrid model, where the parameter count on average is 245 parameters, to be feasible and a scalable approach. Compared to large black-box models like transformers with millions of parameters, our framework is computationally efficient and practical for larger datasets with reasonable computational resources.

Better performance of other baselines

Thank you for raising this. The focus of the paper is to propose a method that can both discover an interpretable model, incorporate textual prior (context), and have the possibility to acquire new features and samples. Existing black-box methods such as RNNs or transformers can fit datasets well but are unable to perform the feature acquisition and interpretability functions of D3, making them not directly comparable. Our method offers a unique blend of interpretability, adaptability, and performance, making it suitable for clinical applications where understanding the model is as crucial as its predictive accuracy.

Questions

Incorporate RAG?

We agree that incorporating RAG-based techniques could improve the method; however, we mark it as out of scope for our initial paper. We acknowledge the potential benefits and will include a discussion of RAG techniques in the final paper to highlight future directions for enhancing robustness against hallucinations.

Did you experiment with prompts multiple times to arrive at a reasonable framework?

Yes, we experimented with multiple prompts and iteratively refined them based on model performance and expert feedback. This process allowed us to develop the final proposed framework in the paper, ensuring that the prompts are well-suited to the task and yield reliable results.

Can you provide more information on the scalability and computational efficiency of your framework?

This is a great question. We find our hybrid models on average contain 245 parameters, making them more scalable than existing large black-box methods such as transformers with over a million parameters. Our approach balances computational efficiency with the ability to iteratively refine models, making it practical for real-world applications with limited computational resources.

How might the performance be improved to black-box methods?

It is a tradeoff to get an interpretable, hybrid/white-box model compared to a purely data-driven black-box method. Sacrificing some overall MSE can be beneficial when the process is interpretable and understandable by humans. Interpretable models provide insights that are crucial for clinical applications, where understanding the model’s behavior and its underlying assumptions is as important as its predictive accuracy.

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We hope that most of the reviewer’s concerns have been addressed and, if so, they would consider updating their score. We’d be happy to engage in further discussions.

Dear Reviewer b77T, I am a NeurIPS 2024 Area Chair of the paper that you reviewed.

This is a reminder that authors left rebuttals for your review. We need your follow up answers on that. Please leave comment for any un-answered questions you had, or how you think about the author's rebuttal. The author-reviewer discussion is closed on Aug 13 11:59pm AoE.

Best regards, AC

I don't think the gains in MSE are marginal when comparing the performance of the transformer versus the D3-white box model in Table 2. The MSE of the D3-white box is roughly 5-10 times higher than that of the transformer across almost all datasets. The point was, if a model is already a poor fit, is its interpretability reliable?

I have also updated my score in response to the comments on lack of guardrails and hallucinations as the authors pointed out they did not observe any significant deviation from expert consensus

We thank the reviewer for their constructive feedback and are glad the reviewer finds our paper's presentation of the innovative general strategy for searching the space of pharmacokinetic models through our D3 framework to be accessible and acknowledges the superior performance of our D3-Hybrid approach on the datasets.

What happens when the Modeling Agent produces an invalid model?

Thank you for raising this important point. When an invalid model is generated, typically due to a coding error, we have implemented a robust mechanism to address this. Specifically, if the generated code contains syntax errors or logical inconsistencies that prevent it from running correctly, the system automatically re-queries the LLM up to 10 times to regenerate the model. In practice, this iterative querying resolves most coding errors and ensures that the model is valid and executable. We have now included a detailed description of this procedure in the experimental setup section in Appendix F to enhance the transparency and reproducibility of our approach. This method effectively mitigates the issue of invalid models, ensuring continuous and reliable model generation.

What models are possible?

The models generated by our framework are represented as PyTorch models. Therefore, any model that can be expressed using PyTorch's capabilities is possible within our framework. This includes a wide range of mathematical white-box models that can incorporate various operations such as logarithms, maxima, minima, and trigonometric functions. Additionally, our framework supports complex neural architectures, including multi-layer perceptrons with various regularization techniques (e.g., dropout), parameter initialization schemes, and activation functions. In practice, we observe that the generated models often consist of a combination of interpretable mathematical components and neural network elements, optimized for both performance and interpretability. This flexibility allows our framework to adaptively explore a diverse model space and discover well-fitting models tailored to the specific dataset and task at hand.

Typically you want interpretable and performance pharmacokinetic models, so there is a tradeoff between validation performance (e.g., MSE) and simplicity.

We completely agree with this observation, and it aligns with one of the central goals of our research. Our D3 framework is specifically designed to navigate the tradeoff between model performance and interpretability. To achieve this, we employ a hybrid approach that combines the strengths of both white-box and black-box models. Our white-box models are inherently interpretable, providing clear insights into the underlying pharmacokinetic processes. At the same time, our hybrid models incorporate neural network components to capture complex dynamics that might be missed by purely white-box models.

In our experiments, we systematically evaluate both types of models to ensure that we achieve a balance between accuracy and simplicity. For example, in our case study on Warfarin pharmacokinetics, we discovered a new, interpretable model that outperformed existing literature models in terms of validation MSE while maintaining clinical plausibility. This demonstrates the framework's ability to produce models that are not only accurate but also easy to understand and interpret by domain experts. Detailed insights into how we balance these tradeoffs can be found in Sections 4 and 5 of our paper, where we discuss the model evaluation metrics and provide expert clinical commentary on the interpretability of the discovered models.

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We hope that most of the reviewer’s concerns have been addressed and, if so, they would consider updating their score. We’d be happy to engage in further discussions.

Dear Reviewer 2i8U,
I am a NeurIPS 2024 Area Chair of the paper that you reviewed.

This is a reminder that authors left rebuttals for your review. We need your follow up answers on that. Please leave comment for any un-answered questions you had, or how you think about the author's rebuttal. The author-reviewer discussion is closed on Aug 13 11:59pm AoE.

Best regards, AC

We thank the reviewer for their constructive feedback and are glad the reviewer finds the results promising for using LLMs for pharmacokinetic modeling and the paper well-organized.

symbolic regression methods like D-CODE [ICLR'22] are not compared.

Thank you for raising additional related works. Allow us to kindly re-iterate that the focus of the paper is to propose a method that can both discover an interpretable model, incorporate textual prior (context), and have the possibility to acquire new features and samples. Existing symbolic regression works such as D-CODE and others cannot acquire new features on demand, as outlined in the related works table in Table 1. We did compare against SINDy as a baseline, and believe D-CODE to be out of scope.

I would like to see several simulated datasets with different simulation equations for further investigation.

We agree with your invaluable suggestion and created five additional semi-synthetic datasets running D3-Hybrid across them, combined with some of the baselines, the results of which can be seen in the additional rebuttal PDF, as outlined in the global response. We observe that D3-Hybrid still performs well, especially when the LLM has never seen such a model. This provides empirical evidence to mitigate the concern of potential knowledge leakage issues due to pre-trained LLMs, as D3-Hybrid is still able to discover well-fitting models of the underlying synthetic equation it has never observed before.

no ablation study on model design or case study on different module choice

We can see how the existing ablations were overlooked. We did include ablations in most experimental results as ablations of D3, specifically with the additional baselines of a model zero-shot generated from D3 called ZeroShot and the same model with its parameters optimized called ZeroOptim. We have revised the baseline method description and tables to make these ablations more prominent to the reader now. Thank you for the suggestion. In terms of ablation results, these ablation results verify the components and complexity of D3 to achieve better performance.

can LLM agents provide explanations on their output results to make a more transparent solution

We agree that this is indeed possible and already achieved by the evaluator agent, see Figure 1.

Questions

how is the memory module implemented in this paper

The memory module is simply a buffer of the top-k performing programs represented as code.

Cost for training D3, API cost

The cost for training models is equivalent to training a standard 3-4 layer MLP with 128 hidden unit parameters. The API cost is around $0.075 per D3 run in total. Open-source LLMs could be used, however, D3's performance would correlate with the underlying LLM's ability.

RNN and Transformer perform differently on several datasets.

We agree some datasets are difficult to model due to the large variation of the features and complex underlying feature interactions, especially for the Lung Cancer model when analyzing the underlying model. This leads to pure parameter optimization techniques getting stuck in local minima, and hence, starting with random seeds of random weight initialization can produce different final models on these complex datasets.

Ethics review flag

Thank you for being cautious. However, we would like to clarify that we did not perform any research involving human subjects, only analyzing existing collected open-source medical data that is available in the public domain and providing links and descriptions to all datasets used within the paper.

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We hope that most of the reviewer’s concerns have been addressed and, if so, they would consider updating their score. We’d be happy to engage in further discussions.