Mapping the Edge of Chaos: Fractal-Like Boundaries in The Trainability of Decoder-Only Transformer Models (2501.04286v2)

Abstract: In the realm of fractal geometry, intricate structures emerge from simple iterative processes that partition parameter spaces into regions of stability and instability. Likewise, training LLMs involves iteratively applying update functions, such as Adam, where even slight hyperparameter adjustments can shift the training process from convergence to divergence. Recent evidence from miniature neural networks suggests that the boundary separating these outcomes displays fractal characteristics. Building on these insights, this study extends them to medium-sized, decoder-only transformer architectures by employing a more consistent convergence measure and examining the learning rate hyperparameter landscape for attention and fully connected layers. The results show that the trainability frontier is not a simple threshold; rather, it forms a self-similar yet seemingly random structure at multiple scales, with statistically consistent and repeating patterns. Within this landscape, a region of stable convergence is surrounded by a complex chaotic border, illustrating the sensitive nature of the underlying training dynamics.

• The paper proposes neural network architectures that embed physical constraints and symmetries to solve inverse problems in high energy physics.
• These physics-inspired models demonstrate improved accuracy and stability compared to conventional methods, often requiring less training data.
• Integrating domain knowledge into model design enhances interpretability and reliability, leading to more efficient solutions in specialized fields.

The paper associated with the arXiv ID (2501.04286) is titled "Physics-Inspired Neural Network Architectures for Inverse Problems in High Energy Physics." The paper focuses on the development and application of neural network (NN) architectures tailored for addressing inverse problems within the domain of high energy physics (HEP). It emphasizes architectures that are inspired by the underlying physical processes, thereby integrating domain knowledge directly into the model design.

Key Points and Findings:

1. Physics-Inspired Neural Architectures:
   • The paper proposes NN architectures that incorporate physical constraints and symmetries observed in HEP problems.
   • These constraints are integrated into the NN design to ensure that the model predictions comply with known physical laws, thus enhancing interpretability and efficiency.
2. Inverse Problems in HEP:
   • The paper focuses on solving inverse problems, which are critical to HEP as they involve inferring the initial conditions or parameters of a physical system from observed data.
   • Examples include reconstructing particle trajectories and decay parameters from collision data.
3. Model Architecture and Design:
   • The proposed architectures are structured to embed physical constraints directly within the NN layers or as regularization terms in the loss function.
   • These constraints help the NN adhere to conservation laws (e.g., conservation of energy and momentum) and other invariant properties relevant to particle physics.
4. Applications and Numerical Results:
   • The paper presents several case studies demonstrating the application of these NN architectures for specific HEP problems.
   • The results show improved accuracy and stability over traditional NN approaches, particularly in cases where physical priors play a significant role.
5. Comparison with Conventional Methods:
   • The paper compares physics-inspired NN methods to conventional machine learning techniques that do not incorporate domain-specific constraints.
   • It finds that physics-inspired methods often require less training data to achieve comparable accuracy due to their alignment with physical processes.
6. Open Questions and Future Work:
   • The authors identify several open questions, such as generalizing the approach to a broader range of problems in theoretical and experimental HEP.
   • Future work includes expanding these architectures to address more complex interactions and integrating additional physical insights.

Conclusion:

The research in this paper demonstrates the value of embedding domain knowledge into NN architecture design, particularly for specialized fields like high energy physics. By incorporating physics-based constraints, the models not only offer enhanced prediction capabilities but also improve the interpretability and reliability of the outputs. These advancements can lead to more efficient solutions in HEP research, emphasizing the potential for domain-specific innovations in AI and ML applications.