Automatically Learning Hybrid Digital Twins of Dynamical Systems

We appreciate the reviewer’s thorough evaluation and positive feedback.

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[P1] Clarifying computational budget

Thank you for raising this question. To clarify:

1. All neural baselines and each HDTwinGen model use identical training settings (2000 epochs, 1000 batch size), consuming the same number of function evaluations per model.
2. HDTwinGen evolves one new model every generation for 20 generations, using a higher total computational budget.
3. Exact evaluation counts are provided in the Questions response below.

Empirical comparisons with comparable budgets: We have included additional results comparing performance against DyNODE and SINDY given equivalent budgets. This additional budget is used towards HPT, where each baseline underwent 15 iterations of HPT, matching HDTwinGen's 15 evolutionary iterations. Results provided in [A3] demonstrate HDTwinGen's superior performance, highlighting the performance benefits of evolving model specification beyond HPT.

Conceptual differences: HDTwinGen automates specification and parameter design of hybrid digital twins, unlike neural baselines that only optimize expert-specified models' parameters. This automation accelerates and scales model development, improving cost- and time- efficiency. The increased computational complexity trades off with higher automation levels and the potential for better-performing models.

Actions taken: We now include the above in App H.

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[P2] Code release

Upon acceptance of the paper, we will release the code, accompanied by extensive reproducibility instructions in App E, to ensure reproducibility.

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[P3] Understanding reliance on underlying LLM

We appreciate this comment and agree that the performance of HDTwinGen depends on the capabilities of the underlying model. To understand how the algorithm scales with different models, we reported the performance of HDTwinGen using LLMs of varying capabilities (App H6). We observed that performance correlated with the capabilities of the underlying LLMs, supporting the hypothesis that HDTwinGen's performance will improve as the underlying models advance.

Actions taken: We now include this in S8 of the paper.

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Questions

• Composition operator: We have revised L55-56 to mention that we focus on additive composition.
• Bilevel objective: Your understanding is correct. The upper-level objective concerns model specification (evaluated on the validation set), while the lower-level objective concerns parameterization performance (evaluated on the training set). This has been clarified in L137-139.
• Additional details on benchmark: The three variants of the Lung Cancer dataset are synthetically generated from PKPD models, while the COVID-19 dataset is generated using an agent-based COVASIM simulator. The Plankton Microcosm (three species) and Hare-Lynx (two species) datasets are real-world ecological datasets. The Plankton dataset is measured in laboratory replication experiments, whereas the Hare-Lynx dataset is measured outdoors.
• Parameter count: On Lung Cancer:

• $\omega(\theta)$ in $\mathcal{P}^{(g)}$: Your understanding is correct, $\mathcal{P}^{(g)}$ is the set of optimized models and includes the optimized parameters $\omega(\theta)^*$. This has been revised in L188-L190.
• Clarification on modeling workflow: In our framework, the modeling process is divided into two subtasks: model generation (the modeling agent) and model evaluation (the evaluation agent). Each subtask has distinct instructions and performs different roles. Although there are two steps, both are implemented using a single LLM with different prompts and memory for each task. To clarify this, we will describe our system as a 'multi-step' agent rather than a multi-agent workflow. The main distinction is that multi-agent frameworks (e.g., AutoGPT, MetaGPT) involve dynamic interactions and task allocation, whereas our workflow is sequential and fixed. Thank you for this suggestion, which has improved the clarity and presentation of our work.
• Evolution of mechanistic component: We direct you to App H4, where we included the models returned by HDTwinGen in each generation and annotated it with some interpretations of the evolution (including mechanistic and neural components).

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Limitations

• Complexity and performance tradeoff: We addressed this concern in detail in [P1]. As a novel automated hybrid design strategy, HDTwinGen's computational complexity is indeed higher than manual model building, similar to other automated approaches (eg AutoML, NAS). However, this complexity is justified by: improved model performance, and enhanced scalability in model development, potentially reducing overall cost and time in modeling. This is further supported by our new results ([A3]), highlighting that given comparable budgets, HDTwinGen discovers better-performing models.
• Standardized comparison: Please see our response above clarifying synthetic vs real-world datasets. Regarding your comment about the performance benefits of HDTwinGen vs HPT of neural baselines, we addressed this with additional empirical results with comparable computational budgets, which we discussed in [P1].
• Sharpening claims: We have refined our claim in L79 to "a new approach to automatically design digital twins given human modeling priors."
• Explainability: While we have provided some preliminary interpretations of discovered models in App H4 and App H10, we agree that this is an important future direction. We have highlighted this by revising Section 8.

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We hope the reviewer’s concerns are addressed and they will consider updating their score. We welcome further discussion.

We thank the reviewer for their thoughtful and helpful review. We’re glad that the reviewer finds our approach to be both highly novel and of high potential impact, though we agree that the addition of existing hybrid models improves the paper significantly.

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[P1] Extending the literature review

We appreciate the references to related works, which consider hybrid models combining mechanistic equations with neural components. [R1] integrates prior ODE/PDE knowledge into a hybrid model, using specialized regularization to penalize the neural component's information content. [R2] investigates hybridization in the latent space, employing semantically grounded expert latent variables and neural latent variables that are regularized to reduce divergence from the expert component.

Our work differs significantly in both motivation and methodology:

• Motivational difference: Our method introduces an automated framework to jointly optimize hybrid model specification (evolving both mechanistic components and neural architecture) and its parameters. This contrasts with related works, where experts specify the hybrid model design and optimization is limited to parameters.
• Methodological novelty: Our approach uniquely integrates LLMs within an evolutionary framework for automated design, guided by expert-provided task context and data-driven feedback. This leverages LLMs' capabilities in symbolic manipulation, contextual understanding, and learning, enabling the exploration of a vast combinatorial space of hybrid models previously infeasible with standard techniques.

Actions taken: (1) Extended discussions of related works [R1,R2] in L269-280. (2) Introduced APHYNITY, an existing hybrid model, as an additional baseline in the global response [A1], demonstrating HDTwinGen's superior performance.

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[P2] Demonstrating the benefits of automated design of existing hybrid models

We agree and have added an experiment demonstrating HDTwinGen's ability to further optimize a human-specified APHYNITY model. Specifically, we seed the process with an expert-designed hybrid model (combining logistic tumor growth model [R7] with three-layer MLP). The evolution process is visualized in [A2] (in the PDF), showing HDTwinGen further evolving the model specification, resulting in improved performance over the initial model specification.

Our automated approach offers substantial benefits: it enables model development at scale, significantly reducing the time and cost compared to human-driven development, involving humans only at key moments (e.g., providing initial context or iterative feedback if desired). Importantly, our approach has the potential to uncover novel and effective hybrid model designs that might elude human designers.

Actions taken: (1) Added additional results demonstrating HDTwinGen's ability to further evolve an expert-specified APHYNITY model in [A2], (2) Included discussion on benefits of automated design in S8.

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[P3] Clarifications on the algorithm

Thank you. We highlight that low-level details are already contained within App E,F,G. Here we clarify each aspect:

• Search spaces: We do not impose predefined restrictions on mechanistic models (e.g., primitive/terminal sets in symbolic regression). We also do not specify any architecture primitives (e.g., in neural architecture search) for neural components. The search space is implicitly constrained by the symbolic language, as the evolved model must be valid/executable PyTorch python code.
• Hybrid model: Our hybrid model considers an additive composition of mechanistic and neural components of the form $dx(t)/dt = f(x(t))$, and $f = f_{neural} + f_{mechanistic}$. Evolution optimizes the specification of both mechanistic and neural components, and corresponding parameters.
• Training/evaluation and metrics: The hybrid model is trained jointly to minimize the training MSE loss (Eq 5, App F) using the Adam optimizer. Model evaluation is based on the val MSE and val MSE per component (corresponding to val loss per state dimension, Eq 6).

Actions taken: We have highlighted App E more prominently in the main paper.

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[P4] Understanding the variability of results

Variability. Thank you for your observation regarding performance variability in Table 1. As HDTwinGen optimizes both model specification and parameters, this is equivalent to searching in a much larger hypothesis space compared to baselines (that only search in the parameter space). The variability stemming from model specifications is also evident in the higher std dev of ZeroOptim results (optimized model based on zero-shot LLM generated specifications). Additionally, despite higher variability, HDTwinGen-discovered models exhibit several key strengths: superior OOD performance (Table 2), sample efficiency (Fig 2) and easier evolvability to changing dynamics (Fig 4).

Actions taken: Updated manuscript to discuss variability and the trade-offs of automation explicitly in S8.

Empirical comparisons with comparable budgets: We have included additional results comparing performance against DyNODE and SINDY given equivalent budgets. This additional budget is used towards HPT, where each baseline underwent 15 iterations of HPT, matching HDTwinGen's 15 evolutionary iterations. Results provided in [A3] (in additional PDF) demonstrate HDTwinGen's superior performance, highlighting the performance benefits of evolving model specification beyond HPT.

Actions taken: Included empirical comparisons given comparable computational budget in App H.

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[P5] Clarifying spatial dimensions

The spatial dimensions of considered benchmarks are Cancer PKPD (4), COVID-19 (4), Plankton-Microcosm (3), Hare-Lynx (2).

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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 thorough review. We are pleased that the reviewer finds our approach interesting, addressing an important research topic, with careful experimental analysis.

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[P1] Incorporating additional baselines

Thank you for this suggestion. While we included SINDy for discovering closed-form governing equations, it doesn't represent an evolutionary search method. To address this, we've added genetic programming (GP) for symbolic regression [R5] as a baseline to discover symbolic equations on $\mathcal{D}\_{train}$, using implementation and tuned hyperparameters from [R6].

We have included the additional results in global response [A1], observing that our method discovered superior models. This can be attributed to two key factors:

1. Hybrid model design: HDTwinGen simultaneously optimizes both the specification (functional form) and parameters of mechanistic and neural components. This hybrid approach enables more flexible and powerful modeling compared to symbolic discovery methods, which are limited to purely symbolic equations.
2. Efficient search: By leveraging LLMs, HDTwinGen utilizes domain knowledge, contextual understanding, and learning to enhance search efficiency, leading to more accurate solutions.

Actions taken: We have incorporated GP-based SR as an additional baseline in our empirical analysis.

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[P2] Hyperparameter optimization search

To broaden our comparison with non-LLM methods, we conducted an additional experiment using Bayesian hyperparameter tuning (HPT) search for SINDy and DyNode baselines. For a fair comparison, we matched the number of HPT searches to HDTwinGen's evolutionary search steps (15 precisely). Results presented in [A3] demonstrate HDTwinGen's superior performance against these HPT-optimized baselines, highlighting the performance benefits of evolving model specification beyond HPT.

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[P3] Extending the literature review

We appreciate the references to related works, which consider hybrid models combining mechanistic equations with neural components. [R1] integrates prior ODE/PDE knowledge into a hybrid model, using specialized regularization to penalize the neural component's information content. An alternative approach investigates performing hybridization in the latent space. [R2, R3] consider settings where an expert equation is known, but equation variables are unobservable. Correspondingly, they employ two sets of latent variables: one governed by expert equations and another linked to neural components. [R2] introduced specialized regularization to reduce the divergence of the overall model from the mechanistic component, and semantically grounded expert latent variables. [R4] leverages the assumptions that expert models remain valid OOD to sample augmented training distributions from expert latent variables.

Our work differs significantly in both motivation and methodology:

• Motivational difference: Our method introduces an automated approach to jointly optimize hybrid model specification (evolving both mechanistic component and neural architecture) and its parameters. This contrasts with related works, where experts specify the hybrid model design and optimization is limited to only its parameters. The benefits of our automated approach are substantial: it enables model development at scale, significantly reducing the time and cost associated with human-driven development. Our method involves humans only at key moments (e.g. providing initial context or iterative feedback, if desired), minimizing efforts required from experts. Importantly, our approach has the potential to uncover novel and effective hybrid model designs that might elude human designers.
• Methodological novelty: Our approach uniquely integrates LLMs within an evolutionary framework to discover optimal models, guided by human-provided task context and data-driven feedback. By leveraging LLMs' advanced capabilities in symbolic manipulation, contextual understanding, and learning, we enable the exploration of a vast combinatorial space of hybrid models. This exploration was previously infeasible with standard evolutionary techniques, marking a significant advancement in automated model discovery and optimization.

Actions taken: In response to your suggestion, we have (1) extended discussions of related works [R1-R4] in L269-280. (2) We introduced APHYNITY as an additional baseline in global response [A1], where we observed HDTwinGen outperforming APHYNITY. (3) We additionally demonstrated HDTwinGen's ability to further evolve a human-specified APHYNITY model in [A2], highlighting the flexibility of our framework in accommodating various degrees of expert involvement through automated optimization.

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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. We’re glad the reviewer finds the approach general and of high potential.

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[P1] Clarifying "Multi-agent" terminology

We appreciate your comment on our system's classification as "multi-agent". Upon reflection, we agree that our framework doesn't fully embody all characteristics typically associated with LLM multi-agent systems. While our approach incorporates two distinct components (modeling and evaluation) with specialized roles, they are applied sequentially, lacking dynamic, real-time inter-agent communication and autonomous behavior adjustment in multi-agent systems [R8, R9].

We reclassified our approach as a "multi-step" agent, which more accurately reflects its nature:

• A modeling step that generates and optimizes new model specifications.
• An evaluation step that assesses generated models reflects on requirements, and provides targeted improvement feedback.

Actions taken: We have now updated this in the paper. Thank you for helping us refine terminology and more precisely position our work.

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[P2] Analyzing computational complexity

Thank you for raising this question. Allow us to clarify the evolutionary process and analyze the computational complexity.

Evolutionary process:

• Each iteration proposes and optimizes one new model specification, conditioned on the top-K previous models
• The population size N=g in the $g^{th}$ iteration
• Only the new model's fitness is evaluated on the validation set
• The selection step retains the top-K models
• Process repeats for G iterations (G=20, K=16 in our experiments)

Computational complexity. We introduce lower-case notation to indicate constant time parameters:

• Evolution/model generation (LLM): $\mathcal{O}(c)$ (LLM inference)
• Parameter optimization: $\mathcal{O}(d)$ (training on $\mathcal{D}_{train}$, in practice, depends on model/dataset complexity)
• Fitness evaluation: $\mathcal{O}(e)$ (on $\mathcal{D}_{val}$)
• Model evaluation (LLM): $\mathcal{O}(f)$ (LLM inference for improvement suggestions)
• Selection: $\mathcal{O}(NlogN)$ (top-K)

The complexity per step is $\mathcal{O}(c+d+e+f+NlogN)$, with total complexity for G generations: $\mathcal{O}(G*(c+d+e+f+NlogN))$. We note that constant complexities vary with different datasets and models. For practical insight, we report average wall-clock times on the Lung Cancer dataset: $c=20s$, $d=12s$, $e=0.004s$, $f=20s$.

Actions taken: Added complexity analysis and wall-clock times in App H.

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[P3] Discussing domains suitable for the algorithm

We appreciate this suggestion and clarify the domain characteristics our algorithm is designed for:

• Continuous-time dynamical systems with a) Markov property and b) continuous time evolution.
• Scenarios where hybrid models are likely to outperform purely mechanistic or neural models: a) partial knowledge of underlying dynamics exists, with some non-negligible unclear aspects, b) limited or unevenly distributed empirical data across the state space.
• Complex model design spaces with multiple hypotheses: where automated exploration can help experts scale model development, review and discover a wider range of potential designs, improving cost- and time-efficiency of model development.

Our experiments focused on systems with partially understood mechanisms and limited observational data. We observed that our algorithm discovered performant hybrid models, leading to more accurate model dynamics, OOD generalization, and improved evolvability with changing conditions.

Actions taken: Included a version of this discussion in S8 and in App A.

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[P4] Highlighting technical novelty

Thank you for your insightful comment. Our work introduces the first automated algorithm for designing hybrid digital twins, with the following key innovations:

• Automated design: We present an evolutionary algorithm that optimizes both model specification and parameterization based on an initial expert prompt. This contrasts with existing hybrid models where experts specify closed-form equations and neural network designs, with only parameters being optimized.
• Hybrid model discovery: Our approach leverages LLMs' capabilities in symbolic manipulation, contextual understanding, and learning to explore a vast combinatorial space of hybrid model specifications. This exploration was previously infeasible with standard techniques, as existing discovery approaches are limited to closed-form equations or neural architecture searches within predefined spaces.

Actions taken: Added this to the RWs section to better position our work and showcase its novelty.

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Limitations

We acknowledge the importance of these discussions. Actions taken: We have integrated an extended discussion in App A and a summarized version in S8:

• Verification of HDTwin models: First, HDTwins are composed of human-interpretable components, enabling higher degrees of expert verification. Second, evolved models should undergo rigorous functional testing (e.g., held-out datasets, robustness/fairness metrics) pre-deployment. Future works could incorporate hallucination mitigation strategies (e.g., RAG, constitutional AI, multi-model consensus) to further improve reliability.
• Variability in evolutionary search: One strategy lies in quantifying variability through multiple independent runs and confidence intervals. Another lies in controlling variability by providing tighter requirements/constraints in the initial prompt or iterative expert model steering to narrow the hypothesis space.
• Ethical implications: We recognize the potential for bias propagation from black-box LLMs. To address this, we recommend rigorous review verification for fairness, bias, privacy, and other ethical concerns.

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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.