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This paper introduces a novel approach to simulating coupled dynamical systems by training neural surrogates that map full driving trajectories and initial conditions directly to output trajectories, bypassing traditional stepwise time integration. By enforcing self-consistency among these trajectories, the authors reformulate the simulation as a fixed-point problem, significantly reducing the computational burden compared to conventional methods. Key results demonstrate that the method achieves convergence with fewer iterations and provides a predictive measure of convergence based on the spectral radius of the Jacobian, highlighting both the strengths and limitations of the approach in practical scenarios.
A single scalar from a learned operator can predict convergence in coupled dynamical systems, revealing the limits of traditional backpropagation methods.
Numerical simulation of dynamical systems is usually organized as a causal march through time: each state is computed from the previous one. We explore a different formulation for coupled systems. For each subsystem type we train a neural surrogate mapping a full driving trajectory and initial condition directly to a full output trajectory; following classical waveform relaxation, coupled systems are assembled by enforcing self-consistency among these trajectories: simulation becomes a fixed-point problem over complete trajectories rather than a stepwise rollout. On coupled van der Pol oscillators and Hodgkin-Huxley neuron networks, sequential depth becomes the number of solver iterations: 4-10 Newton iterations where the reference integrator takes 1500 steps. The gradient likewise loses its time recursion: it becomes a linear system solved by GMRES at memory independent of solver depth. A single scalar measured from the learned operator, the spectral radius of its Jacobian, predicts in advance where the coupled solve will converge; past that boundary, unrolled backpropagation diverges and a Neumann adjoint fails, while the implicit gradient remains correct to 0.04%. We report where the approach succeeds and where surrogate error degrades it.
