用自洽性直接模拟耦合系统，跳过逐时步计算，速度提升百倍。
Time Without Timesteps: Simulating Coupled Dynamical Systems via Self-Consistency
- 训练神经代理直接映射驱动轨迹与初值到输出轨迹，避免逐时步演算。
- 在振子与神经网络上仅需4-10次迭代，远少于传统方法的1500步。
- 梯度计算不再依赖时间递推，内存不随求解深度增加，适合大规模系统。
传统的动力系统数值模拟依赖于时间上的因果推进：每一步状态由前一步计算得出。本文提出一种新方法，针对每类子系统训练神经代理，直接从完整的驱动轨迹和初始条件映射到完整输出轨迹；通过经典波形松弛思想，将耦合系统组装为各轨迹间自洽的固定点问题：模拟转化为对完整轨迹的迭代求解，而非逐步展开。在耦合范德波尔振子与霍奇金-赫克斯利神经网络中，序列深度等价于求解器迭代次数：仅需4-10次牛顿迭代，而参考积分器需1500步。梯度计算亦摆脱时间递推：转为用GMRES求解线性系统，内存不随求解深度增长。从学习算子中提取的单个标量——其雅可比矩阵的谱半径——可提前预测耦合求解的收敛边界；超过该边界，显式反向传播发散，诺伊曼伴随失效，而隐式梯度仍保持0.04%精度。本文报告了该方法成功与代理误差导致性能退化的场景。
原文摘要 · Abstract (English)
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.
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