AI 中文总结
本文提出一种通过自洽性模拟耦合动力系统的方法，训练神经代理模型将完整驱动轨迹映射为输出轨迹，在耦合系统测试中仅需4-10次牛顿迭代，梯度计算内存与求解器深度无关，可通过雅可比谱半径预测收敛性。
AI 中文摘要
动力系统的数值模拟通常组织为沿时间的因果推进：每个状态由前一个状态计算得到。我们为耦合系统探索一种不同的公式化方法。针对每个子系统类型，我们训练一个神经代理模型，将完整的驱动轨迹和初始条件直接映射到完整的输出轨迹；遵循经典波形松弛方法，通过在这些轨迹之间强制自洽性来组装耦合系统：模拟成为针对完整轨迹的不动点问题，而非逐步展开的过程。在耦合范德波尔振荡器和霍奇金-赫胥黎神经元网络上，序列深度成为求解器迭代次数：参考积分器需1500步，而我们的方法仅需4-10次牛顿迭代。梯度同样失去了时间递归性：它成为由GMRES求解的线性系统，内存消耗与求解器深度无关。从学习到的算子中测得的单个标量，即其雅可比矩阵的谱半径，可提前预测耦合求解是否收敛；超出该边界时，展开的反向传播会发散且诺伊曼伴随法失效，而隐式梯度仍保持0.04%的精度。我们报告了该方法成功的场景以及代理误差导致性能下降的场景。
英文摘要
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.
