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Abstract The degree of rate control (DRC) quantitatively identifies the kinetically relevant (sometimes known as rate‐limiting) steps of a complex reaction network. This concept relies on derivatives which are commonly implemented numerically, for example, with finite differences (FDs). Numerical derivatives are tedious to implement, and can be problematic, and unstable or unreliable. In this study, we demonstrate the use of automatic differentiation (AD) in the evaluation of the DRC. AD libraries are increasingly available through modern machine learning frameworks. Compared with the FDs, AD provides solutions with higher accuracy with lower computational cost. We demonstrate applications in steady‐state and transient kinetics. Furthermore, we illustrate a hybrid local‐global sensitivity analysis method, the distributed evaluation of local sensitivity analysis, to assess the importance of kinetic parameters over an uncertain space. This method also benefits from AD to obtain high‐quality results efficiently.
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Why shallow networks struggle to approximate and learn high frequencies
Abstract In this work, we present a comprehensive study combining mathematical and computational analysis to explain why a two-layer neural network struggles to handle high frequencies in both approximation and learning, especially when machine precision, numerical noise and computational cost are significant factors in practice. Specifically, we investigate the following fundamental computational issues: (1) the minimal numerical error achievable under finite precision, (2) the computational cost required to attain a given accuracy and (3) the stability of the method with respect to perturbations. The core of our analysis lies in the conditioning of the representation and its learning dynamics. Explicit answers to these questions are provided, along with supporting numerical evidence.
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- PAR ID:
- 10617590
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Information and Inference: A Journal of the IMA
- Volume:
- 14
- Issue:
- 3
- ISSN:
- 2049-8772
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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