We study neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schrödinger-type dynamics on a learned latent graph. On stable branches, the local stationary problem defines a differentiable implicit graph layer. To learn the graph itself, we optimize over the stratified moduli space of weighted graphs and equip each stratum with a non-degenerate Kähler-Hessian metric that keeps natural-gradient descent and face crossing well posed. We then show that a multilayer stationary network is equivalent to an exact global stationary problem on a supra-graph, and that it admits a penalized global relaxation whose stationary states converge to the exact one as the penalty parameter tends to infinity. Reverse-mode differentiation is recovered as the adjoint of the exact global system, and the penalized adjoint converges to it in the same limit. Finally, under finite-dimensional strong-monotonicity and admissible-lift assumptions, the corresponding represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures with unitary connection. The resulting structural identifications yield complexity bounds controlled by sparse graph or supra-graph geometry rather than dense ambient connectivity.

翻译：我们研究了一种神经网络架构，其中每个隐藏层由学习到的潜在图上耗散薛定谔型动力学的稳态定义。在稳定分支上，局部稳态问题定义了一个可微的隐式图层。为了学习图本身，我们在加权图的分层模空间上进行优化，并为每个层配备一个非退化凯勒-海森度量，以保证自然梯度下降和面交叉的适定性。然后，我们证明多层稳态网络等价于超图上的精确全局稳态问题，并且它允许一个惩罚全局松弛，其稳态在惩罚参数趋于无穷时收敛到精确解。逆向模式微分作为精确全局系统的伴随恢复，并且惩罚伴随在相同极限下收敛到它。最后，在有限维强单调性和可容许提升假设下，对应的表示假设类在解析前馈网络、图稳态网络、超图稳态系统和具有单位连接的层状架构中一致。由此产生的结构识别给出了由稀疏图或超图几何而非稠密环境连接性控制的复杂度界。