Neural ODE Transformers: Analyzing Internal Dynamics and Adaptive Fine-tuning

We sincerely thank the reviewer for their thorough review and encouraging comments. Below, we address your concerns comprehensively.

Modern architecture

Following the reviewer's suggestion, we have expanded our evaluation to include the Llama-1B architecture. In our comparative analysis, conducted with a training set of 20B tokens, our model demonstrates superior performance to Llama-1B, achieving a validation perplexity of 9.68 compared to Llama-1B's 10.17. Additionally, our model also shows stronger performance in downstream reading comprehension tasks.

Our initial choice of GPT as a baseline was motivated by two points:

• Resource constraints when we start working on this idea of neural ODE for transformer
• It would be more reasonable to validate DiffEqFormer's capabilities before scaling up

We chose GPT specifically because it provides well-established benchmarks for smaller models, while newer architectures like Llama and Gemma primarily focus on large-scale implementations. However, we acknowledge that our current training setting (20B tokens) represents a relatively small scale compared to state-of-the-art Llama-1B implementations, which are trained on substantially larger datasets.

Downstream task evaluation

To demonstrate the emergent capabilities of LLMs, we provide the downstream evaluation (zero-shot and five-shot) in benchmarks including MMLU in Appendix C.7. Here are some notes on our model performance.

In GPT-large Configuration, our model improves performance in summary as

• Reading Comprehension:

  • Lambada OpenAI (Zero-shot): +1.15% (37.24 → 38.39)
  • Lambada Standard (Zero-shot): +1.07% (29.30 → 30.37)
  • Lambada Standard (Five-shot): +1.77% (23.05 → 24.82)

• Reasoning:

  • Winogrande (Zero-shot): +1.82% (50.75 → 52.57)
  • Winogrande (Five-shot): +4.34% (49.33 → 53.67)

• Scientific Knowledge:

  • SciQ (Zero-shot): +1.50% (45.70 → 47.20)

In Llama-1B Configuration, our model has a notable improvements in reading comprehension tasks:

• Lambada OpenAI (Zero-shot): +1.90% (41.84 → 43.74)
• Lambada OpenAI (Five-shot): +3.94% (30.47 → 34.41)
• Lambada Standard (Zero-shot): +3.97% (27.58 → 31.55)
• Lambada Standard (Five-shot): +4.18%s (24.04 → 28.22)

Clarification for Section 5.1

We thank the reviewer for raising the clarity issue in Section 5.1. We would like to clarify that when $V=-I$, particles in the system repel each other. Regarding our discussion of induction heads, we have revised the main text by providing a clearer description of their behavior, which consists of two key components: pattern matching (associated with QK pairs) and copying (associated with OV pairs). The simultaneous presence of positive eigenvalues in both matrices encourages the formation of induction heads according to Elhage et al. (2021). We have updated the manuscript to reflect these clarifications.

Computation aspects

Regarding computation aspect, we provide additional ablation on GPU memory allocation as the same concern from other reviewers. We would like to note that our model may incur additional memory allocation, roughly less than 10% more GPU memory consumption.

Randomized time steps

We appreciate the suggestion regarding random time points for enhancing generalization during fine-tuning. We though about this approach but haven't looked carefully into it. However, we note potential challenges like adaptive step-size ODE solvers, which typically require careful step-size selection. Unlike diffusion models that use score-matching between arbitrary time points without ODE solutions, our model, or neural ODEs in general, may face stability issues with uneven step sizes. Nevertheless, we agree this direction is interesting for future investigation.

Misc

We included the attention-based explanation to compare with our Lyapunov exponent approach. We fixed the suggested typos.

We sincerely thank the reviewer for their valuable feedback. Below, we address your concerns in detail.

Large-Scale Model Performance

Regarding large-scale models, we conducted additional investigations with recently acquired computational resources by scaling our model to 1B parameters using the Llama architecture. Our model demonstrates better performance compared to both the baseline Llama-1B and GPT-large. This improved performance stems from using more suitable optimization parameters—specifically, we found that our model benefits from a lower $\beta_2$ value in the Adam optimizer, which enables more flexible adaptation of time-dependent weights. These results confirm that our model shows comparable or better performance to GPT/Llama, though we acknowledge that further large-scale investigations would be valuable.

Downstream task performance

We conducted downstream task evaluations using lm-evaluation-harness in our updated manuscript. Please refer to Appendix C.7 for details.

Our model shows similar performance across multiple benchmarks compared to baselines. It's worth noting the our model presents strong results in reading comprehension tasks. The following is a quick summary of improvements in our models.

GPT-large Configuration:

• Reading Comprehension

  • Lambada OpenAI (Zero-shot): +1.15% (37.24 → 38.39)
  • Lambada Standard (Zero-shot): +1.07% (29.30 → 30.37)
  • Lambada Standard (Five-shot): +1.77% (23.05 → 24.82)

• Reasoning

  • Winogrande (Zero-shot): +1.82% (50.75 → 52.57)
  • Winogrande (Five-shot): +4.34% (49.33 → 53.67)

• Scientific Knowledge

  • SciQ (Zero-shot): +1.50% (45.70 → 47.20)

Llama-1B Configuration

• Reading Comprehension:

  • Lambada OpenAI: +1.90% (Zero-shot: 41.84 → 43.74), +3.94% (Five-shot: 30.47 → 34.41)
  • Lambada Standard: +3.97% (Zero-shot: 27.58 → 31.55), +4.18% (Five-shot: 24.04 → 28.22)

These results highlight our model's particular strength in reading comprehension tasks across both architectures, with especially significant gains in the Llama-1B configuration.

Computational Efficiency and Parameters

While our model's total parameter count exceeds that of the GPT model, the effective parameters per batched input remain comparable to the baseline. This is because the time-dependent weight calculations $W(t)$ are batch-independent, computed before any batch-input operations. Regarding memory usage, our ablation study (Appendix D.1, Figure 16) demonstrates sub-linear memory scaling with respect to $d_{emb}$ dimension—a 150-dimension $d_{emb}$ increase results in less than 10% additional GPU memory usage. To address computational overhead, we believe promising directions include efficient representations such as sparse structure matrices (Monarch; Dao et al., 2022) and Block Tensor Train methods (Qiu et al., 2024).

We acknowledge that the lack of clear guidelines for selecting optimal $d_{emb}$ values represents a limitation in our current work. Developing systematic approaches for determining these values presents an important direction for future research.

References

• Dao et al. (2022). Monarch: Expressive Structured Matrices for Efficient and Accurate Training
• Qiu et al. (2024). Compute Better Spent: Replacing Dense Layers with Structured Matrices

We sincerely appreciate the reviewer's thoughtful feedback. We have addressed your concerns as follows:

Downstream task evaluation

Following reviewer suggestions, we conducted comprehensive downstream task evaluations using lm-evaluation-harness. Detailed results are available in Appendix C.7.

Our model demonstrates performance comparable to or better than baselines across multiple benchmarks, with particularly strong results in reading comprehension tasks. Here are the key findings:

In GPT-large Configuration, our model improves performance in summary as

Reading Comprehension:

• Lambada OpenAI (Zero-shot): +1.15% (37.24 → 38.39)
• Lambada Standard (Zero-shot): +1.07% (29.30 → 30.37)
• Lambada Standard (Five-shot): +1.77% (23.05 → 24.82)

Reasoning:

• Winogrande (Zero-shot): +1.82% (50.75 → 52.57)
• Winogrande (Five-shot): +4.34% (49.33 → 53.67)

Scientific Knowledge:

• SciQ (Zero-shot): +1.50% (45.70 → 47.20)

In Llama-1B Configuration, our model has a notable improvements in reading comprehension tasks:

• Lambada OpenAI (Zero-shot): +1.90 points (41.84 → 43.74)
• Lambada OpenAI (Five-shot): +3.94 points (30.47 → 34.41)
• Lambada Standard (Zero-shot): +3.97 points (27.58 → 31.55)
• Lambada Standard (Five-shot): +4.18 points (24.04 → 28.22)

These results demonstrate our model's particular strength in reading comprehension tasks across both architectures, with substantial improvements in the Llama-1B configuration.

Computational Resources and Parameter Counts

We agree that our model has a higher total parameter count than the GPT model. However, the number of effective parameters per batched input remains roughly equivalent to the baseline model. This is because the parameters from time-dependent weight calculations $W(t)$ are batch-independent, as these weights are computed prior to any batch-input operations. Regarding memory usage, although we require additional storage for time-dependent weight parameters, our ablation study (Appendix D.1, Figure 16) demonstrates sub-linear memory scaling with respect to $d_{emb}$ dimension—increasing $d_{emb}$ to 150 results in only a <10% increase in GPU memory usage. We would think that approaches using more efficient representations, such as sparse structure matrices (Monarch; Dao et al., 2022) and Block Tensor Train methods (Qiu et al., 2024) can be good directions to address these issues.

Reference:

• Dao et al 2022, Monarch: Expressive Structured Matrices for Efficient and Accurate Training
• Qiu et al, 2024 Compute Better Spent: Replacing Dense Layers with Structured Matrices

Dear Reviewer 4Us1,

As we are approaching the end of the discussion period, we wanted to follow up on our previous responses and ensure they sufficiently addressed your comments.

If you have any remaining questions or concerns, we would be glad to provide additional clarification.

Best regards,

Authors

Practical Applications of Adaptive Fine-tuning

Adaptive finetuning can offer a practical solution for deploying models under hardware constraints. Consider this scenario:

• Initial Training:
  • Model is pretrained with 18 time steps (layers)
  • Training performed on high-performance servers
• Deployment Adaptation:
  • Target environment: Less powerful inference servers
  • Adaptively finetune to reduce to 12 layers
  • For example, (18→12) configuration can describe this scenario

While this process shares similarities with knowledge distillation, our adaptive finetuning approach is distinct. The reduced-resource model differs from the pretrained version only by the numerical error introduced by the solver's step size adjustment.

Additionally, time-dependent weights can serve as an effective initialization strategy for large-scale discrete-layer transformers. This approach is particularly valuable when starting from a pretrained time-dependent model trained with limited computational resources. The weights can be transferred and adapted to initialize larger models, providing a resource-efficient pathway for scaling up transformer architectures.

Lyapunov Exponent Analysis

Lyapunov analysis provides a more generalized and principled approach compared to traditional perturbation methods. To calculate the Lyapunov exponent, we must solve an ordinary differential equation (ODE) where the vector field is determined by the Jacobian of the output tokens with respect to the input tokens. This Jacobian matrix is influenced by two key factors:

• The attention weight distribution across input tokens
• The embedding distance between input and output tokens (detailed in Appendix B.2)

This dual dependence demonstrates that the Lyapunov exponent captures more complex dynamics than what can be inferred from attention patterns alone (Jain and Wallace).

We thank the reviewer for their time and thoughtful feedback. Below we address each concern:

Relationship to Lu et al. 2019 and Dynamic Particle Systems

We acknowledge that viewing transformers as dynamic particle systems originated with Lu et al. 2019, a perspective that has influenced both theoretical and practical approaches (e.g., Dutta et al. 2021, Geshkovski et al. 2023a,b). As clarified in Section 3, our work differs from Lu et al.'s MacaronNet in two key aspects. First, while MacaronNet was inspired by ODE solvers to create a transformer architecture with two feed-forward blocks and one attention layer, it remains a discrete-layer neural network. This limits its adaptive fine-tuning capabilities. In contrast, our model is a true neural ODE, functioning as a continuous-layer neural network. This allows us to leverage neural ODE advantages, particularly in adaptive fine-tuning.

Distinction from Geshkovski et al. (2023)

We emphasize that our analysis provides compelling evidence that clustering behavior may not occur in autoregressive transformer models. This conclusion is derived from studying a realistic model that achieves performance comparable to GPT. Our work complements the theoretical analyses presented in Geshkovski et al. (2023a,b) by offering a practical, empirically-grounded approach. The key distinction is that our analysis stems from studying DiffEqFormer—a model that demonstrates real-world effectiveness by matching GPT's performance. The insights gained from our model suggest new directions for theoretical dynamical systems research and can help refine the assumptions used in studies like Geshkovski et al. (2023a,b) to better align with practical implementations.

Model Performance Updates

At the time of submission, our model showed slightly lower performance than GPT-large. We identified that this was due to using identical optimizer settings across different architectures without proper hyperparameter tuning. After conducting further optimization, we adjusted the Adam optimizer by decreasing the $\beta_2$ parameter. This modification led to improved performance:

GPT-large configuration:

• Our model's validation perplexity: 15.43
• GPT baseline perplexity: 15.64

Llama-1B configuration:

• Our model's validation perplexity: 9.68
• Llama baseline perplexity: 10.17

We have updated these results in the paper and conducted additional evaluations on downstream tasks as suggested by reviewers. The new results demonstrate performance gains over the baseline mode in reading comprehension tasks. For detailed results and analysis, please refer to Appendix C.5 and Appendix C.7.

Time-Dependent Weight Design

Our time-dependent weight construction comprises three key components. First, we employ sinusoidal embedding, utilizing Fourier features that align with our continuous-time layer indexing. Second, we incorporate an MLP block to ensure non-linearity. This is a design inspired by score functions in diffusion models. Our motivation stems from the established connection between ODEs and diffusion models. The third and distinguishing component of our approach is the weight projection, which transforms embeddings into constructed weights.

Value of Spectral Analysis and Lyapunov Exponents

While our spectral analysis and Lyapunov exponent studies do not directly translate to model performance improvements, they offer valuable interpretability insights. Our model not only achieves performance comparable to GPT but also provides enhanced interpretability—a crucial advantage given the challenges of understanding large language models. The key benefit lies in our continuous weight design: analyzing the time-dependent weight function $W(t)$ offers a more attractive approach to understanding model properties compared to examining multiple discrete layers in GPT. Our analysis draws from classical dynamical systems theory, providing a novel framework for understanding transformer architectures. This continuous perspective offers a more elegant and mathematically grounded approach to model analysis.

Dear Reviewer Pfg2,

As the discussion period is coming to an end, we wanted to check again if our responses addressed your concerns.

We greatly value your feedback and would be happy to provide any clarifications.

Best regards,

Authors