Trainability and gradient stability for general multi-layer neural oscillators

Develop a rigorous trainability theory for the multi-layer neural oscillator architecture defined by the second-order neural ODEs ddot{y}^ℓ(t) = σ(w^ℓ ⊙ y^ℓ(t) + V^ℓ y^{ℓ−1}(t) + b^ℓ) for ℓ = 1,…,L with y^0(t) = u(t) and affine readout z(t) = A y^L(t) + c, by extending existing results on mitigation of exploding and vanishing gradients established for CoRNN, UnICORNN, and GraphCON. In particular, analyze the adjoint system associated with this second-order neural ODE to establish gradient stability and derive bounds on backpropagated gradients during training.

Background

While the paper establishes universality of multi-layer neural oscillators for approximating causal and continuous operators, it does not address trainability or generalization. Prior works have proved gradient stability and mitigation of exploding/vanishing gradients for specific oscillator-based architectures such as CoRNN, UnICORNN, and GraphCON. Extending such analysis to the general second-order neural ODE that defines multi-layer neural oscillators would provide theoretical foundations for training dynamics beyond these special cases.

The authors suggest that analyzing the adjoint system of the general second-order neural ODE may be a viable approach to derive gradient stability results, but this analysis is not carried out in the paper and is explicitly deferred to future work.