The Vanishing Gradient Problem for Stiff Neural Differential Equations
This work addresses a fundamental optimization barrier for researchers and practitioners training stiff neural ODEs, revealing a universal limitation rather than an incremental improvement.
The paper tackled the vanishing gradient problem in stiff neural differential equations by proving that all A-stable and L-stable numerical integration schemes inherently suppress parameter sensitivities in stiff regimes, with the slowest decay rate being O(|z|^{-1}).
Gradient-based optimization of neural differential equations and other parameterized dynamical systems fundamentally relies on the ability to differentiate numerical solutions with respect to model parameters. In stiff systems, it has been observed that sensitivities to parameters controlling fast-decaying modes become vanishingly small during training, leading to optimization difficulties. In this paper, we show that this vanishing gradient phenomenon is not an artifact of any particular method, but a universal feature of all A-stable and L-stable stiff numerical integration schemes. We analyze the rational stability function for general stiff integration schemes and demonstrate that the relevant parameter sensitivities, governed by the derivative of the stability function, decay to zero for large stiffness. Explicit formulas for common stiff integration schemes are provided, which illustrate the mechanism in detail. Finally, we rigorously prove that the slowest possible rate of decay for the derivative of the stability function is $O(|z|^{-1})$, revealing a fundamental limitation: all A-stable time-stepping methods inevitably suppress parameter gradients in stiff regimes, posing a significant barrier for training and parameter identification in stiff neural ODEs.