“…As a learnable model parameterized by θ ∈ R n , a standard neural ordinary differential equation (NODE) ẋ = φ θ (x, t) is particularly adept at representing complex and nonlinear dynamics (Chen et al, 2018;Liufu et al, 2024), where x ∈ R d is the state at time t, ẋ = dx/dt denotes the time derivative of x, and φ(x, t) is a vector field with φ ∈ (R d × R → R d ) being a function of x and t. Its strength lies in processing timevariant data and adaptively learning from it. This modeling flexibility renders NODEs great potential for the intricate nature of dynamic systems (Hua et al, 2023;Wang et al, 2023;Jin et al, 2024), enabling a more nuanced understanding of complex dynamic systems. Despite these strengths, the standard NODE encounters expressivity limitations, failing to model functions like NOT operations (Kidger, 2021;Xu et al, 2023).…”