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Area of Science:

  • Numerical Analysis
  • Machine Learning
  • Dynamical Systems

Background:

  • Gradient-based optimization is crucial for training parameterized dynamical systems, including neural ordinary differential equations (NODEs).
  • Stiff systems often exhibit vanishing gradients, impeding effective parameter learning.
  • Existing research has observed this phenomenon but lacked a universal explanation.

Purpose of the Study:

  • To demonstrate that vanishing gradients in stiff systems are a universal characteristic of A-stable and L-stable numerical integration schemes.
  • To analyze the underlying mathematical reasons for gradient suppression in stiff regimes.
  • To establish a fundamental limitation for training NODEs and other stiff parameterized models.

Main Methods:

  • Analysis of the rational stability function for general stiff integration schemes.
  • Derivation of explicit formulas for common stiff integration methods.
  • Rigorous mathematical proof of the decay rate for parameter sensitivities.

Main Results:

  • Vanishing gradients are shown to be an inherent property of all A-stable and L-stable stiff numerical integration schemes.
  • Parameter sensitivities decay to zero for large stiffness, governed by the derivative of the stability function.
  • The slowest possible decay rate for these sensitivities is proven to be O(|z|-1).

Conclusions:

  • All A-stable and L-stable time-stepping methods inevitably suppress parameter gradients in stiff regimes.
  • This poses a significant barrier for training and parameter identification in stiff neural ordinary differential equations.
  • The findings reveal a fundamental limitation in optimizing stiff parameterized dynamical systems using current numerical methods.