AI 中文总结
本文提出粗糙第二类变分原理，推导粗糙哈密顿方程及守恒律，构造粗糙伽辽金离散化并证明其等价于RSPRK方法，数值实验验证了其保持伴随守恒律至机器精度，从而在优化中提供更准确稳定的梯度。
AI 中文摘要
我们考虑一个由几何粗糙路径驱动的第二类变分原理，其分裂边界条件自然适用于伴随系统。从这个粗糙变分原理出发，我们推导出粗糙哈密顿方程，建立其逐路径守恒律及相应的哈密顿-雅可比方程。随后，我们将该框架专门化到粗糙伴随系统，获得支撑关于初始条件和参数的伴随敏感性分析的逐路径守恒律和拟守恒律。在离散方面，我们构造了粗糙第二类变分原理的粗糙伽辽金离散化，并证明其生成一个辛流，且具有连续守恒律的离散类比。我们建立了其与一类粗糙辛分块龙格-库塔（RSPRK）方法的等价性，并分析了其收敛性和自然性。最后，我们进行数值实验以验证预测的收敛阶，并证明RSPRK方法能将伴随守恒律保持到机器精度，从而在优化问题中比非辛方法产生更准确和稳定的梯度。
英文摘要
We consider a Type-II variational principle driven by geometric rough path with split boundary conditions naturally suited to adjoint systems. From this rough variational principle we derive rough Hamilton's equations, establish their pathwise conservation laws and associated Hamilton--Jacobi equation. We then specialise the framework to rough adjoint systems, obtaining pathwise conservation and quasi-conservation laws that underpin adjoint sensitivity analysis with respect to initial conditions and parameters. On the discrete side, we construct a rough Galerkin discretisation of the rough Type-II variational principle and show that it generates a symplectic flow with discrete analogues of the continuous conservation laws. We establish it's equivalence to a class of Rough Symplectic Partitioned Runge--Kutta (RSPRK) methods and analyse its convergence and naturality properties. Lastly, we perform numerical experiments to validate the predicted convergence rates and demonstrate that RSPRK methods preserve the adjoint conservation laws to machine precision, yielding more accurate and stable gradients in optimisation problems than non-symplectic alternatives.
Comments1st version, 38 pages, 6 figures, all comments welcome
