Time Without Timesteps: Simulating Coupled Dynamical Systems via Self-Consistency

Abstract: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.

Machine Learning (cs.LG)

arXiv:2609.03358 [cs.LG]

(or arXiv:2609.03358v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.03358 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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