Closed-Form Continuous-Time Networks (CfC) are continuous-time neural networks whose differential equation dynamics have analytically solvable closed-form solutions — eliminating the numerical ODE solver overhead of standard Neural ODEs while retaining the continuous-time benefits of time-varying dynamics, with mathematically guaranteed Lyapunov stability and 1-2 orders of magnitude faster inference than numerically-solved neural ODE variants, making them practical for real-time edge deployment on time-series and control tasks.

The Problem with Numerical ODE Solving in Production

Standard Neural ODEs (Chen et al., 2018) use off-the-shelf ODE solvers (Dormand-Prince, Euler, Runge-Kutta 4) to integrate the learned dynamics. This creates significant operational challenges:

• Variable compute cost: Adaptive solvers take more steps for stiff dynamics, making inference time unpredictable — unacceptable for real-time control systems
• Backpropagation complexity: Requires either storing all intermediate solver states (memory O(N_steps)) or the adjoint method (additional backward ODE integration)
• Numerical stability: Stiff systems require small step sizes, dramatically increasing cost
• Hardware unfriendly: Dynamic computation graphs from adaptive solvers map poorly to specialized accelerators (TPUs, FPGAs)

CfC networks solve all of these by designing the ODE system to have an analytically known solution.

Mathematical Foundation

CfC is derived from Liquid Time-Constant (LTC) networks, which model neuron dynamics as:

dx/dt = [-x + f(x, I)] / τ(x, I)

where τ(x, I) is a state- and input-dependent time constant. The LTC system does not have a general closed-form solution — numerical ODE solving is required.

CfC's key innovation: redesign the network architecture so that the ODE system falls into a class with a known analytical solution. The resulting closed-form is:

x(t) = σ(-A) · x₀ · e^(-t/τ) + (1 - σ(-A)) · g(I)

This is essentially a gated interpolation between the initial state x₀ and a steady-state target g(I), controlled by the time elapsed t and a learned time constant τ. This form: 1. Can be evaluated exactly in O(1) operations (no iterative solver) 2. Is guaranteed asymptotically stable by construction (decays to g(I)) 3. Is differentiable with simple, well-conditioned gradients

Time-Varying Dynamics

Unlike standard RNNs which update state discretely at observation times, CfC networks model the continuous evolution of state between observations. Given observations at times t₁, t₂, ..., tₙ (potentially irregular):

• The network advances the state from t₁ to t₂ using the closed-form solution with Δt = t₂ - t₁
• Longer gaps between observations produce greater state decay toward equilibrium
• The model naturally adapts to irregular time sampling without interpolation or padding

This makes CfC networks intrinsically suited for medical time series (irregular lab measurements), event-based sensors, and network traffic logs.

Stability Guarantees

The closed-form structure provides Lyapunov stability: the state x(t) is guaranteed to converge to the equilibrium g(I) as t → ∞, with convergence rate determined by τ. This means:

• Long sequences do not produce gradient explosion
• Predictions are bounded and physically interpretable
• No gradient clipping or careful initialization required

Performance vs. Neural ODEs

Benchmark comparison on long time-series tasks:

• Inference speed: 10-100x faster than Runge-Kutta Neural ODEs (no solver overhead)
• Accuracy: Matches or exceeds LTC and Neural ODE performance on IMDB sentiment, gesture recognition, and vehicle trajectory tasks
• Parameter efficiency: Fewer parameters needed due to principled inductive bias from the ODE structure

CfC networks have been deployed on embedded ARM processors for real-time human activity recognition, demonstrating that the combination of analytical tractability and strong inductive bias makes them the practical choice for continuous-time sequence modeling on resource-constrained hardware.