AI 中文总结
该研究针对逆势平均场博弈，提出无雅可比反向传播（JFB-r）方法，通过截断微分实现高效逆问题求解，在保持恢复精度的同时降低内存与运行时间。
AI 中文摘要
我们研究逆势平均场博弈（MFGs），其中未知的空间逆代价（迁移率）映射可从观测到的人口密度中推断。我们使用预处理原始-对偶混合梯度（PDHG）方法求解正向MFG，并开发了无雅可比反向传播（JFB-r），该方法从分离的预热启动中仅记录最终r次迭代，同时保留完整的正向求解。为分析此截断微分方法，我们证明了精确邻近对偶外推PDHG映射是极大单调KKT算子的度量预解式。此预解式视角表明，JFB-r可对有限轨迹代理进行精确微分，且在精确均衡分离点处的局部固定活动集下，随跟踪深度增加收敛到隐式梯度。在多个逆MFG设置中，数值实验显示，中等跟踪深度的JFB-r可达到与完整展开相当的恢复精度，同时减少内存和运行时间。
英文摘要
We study inverse potential mean-field games (MFGs), in which an unknown spatial inverse-cost (mobility) map is inferred from observed population densities. We solve the forward MFG with a preconditioned primal-dual hybrid gradient (PDHG) method and develop Jacobian-free backpropagation (JFB-$r$), which records only the final $r$ iterations from a detached warm start while retaining the full forward solve. To analyze this truncated differentiation method, we show that the exact-proximal dual-extrapolated PDHG map is a metric resolvent of the maximal monotone KKT operator. This resolvent view shows that JFB-$r$ exactly differentiates a finite-trajectory surrogate and, under a locally fixed active set at an exact equilibrium detach point, converges to the implicit gradient as the tracked depth increases. Across several inverse-MFG settings, numerical experiments show that JFB-r at moderate tracked depths can achieve recovery accuracy comparable to full unrolling while reducing memory and runtime.
