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Abstract:
Adjoint methods compute gradients of objective functionals at the cost of one forward and one backward solve. In the deterministic setting, symplectic integrators make these gradients exact for the discrete dynamics. In this talk, we extend this geometric picture to dynamical systems driven by geometric rough paths. From a rough Type-II variational principle, we derive the rough Hamilton's equations and the pathwise conservation laws of rough adjoint systems. These conservation laws yield sensitivities with respect to initial conditions and parameters. We then construct rough variational integrators, equivalent to Rough Symplectic Partitioned Runge--Kutta methods, which preserve discrete analogues of these conservation laws. Consequently, discretizing with these methods commutes with forming the adjoint equations, so the computed gradients are exact for the discrete forward map. Numerical experiments on a stochastic Lorenz 96 model show that the adjoint conservation laws are preserved to machine precision, giving more accurate gradients and more reliable optimisation than non-symplectic methods.
Tuesday, October 13, 2026
11:00AM AP&M 7218 and Zoom ID 961 4627 1748
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Randolph E. Bank
Philip E. Gill
Michael Holst
Administrative Contact:
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Rough variational principles and applications to adjoint systems
Ruiao HuUCSD
Abstract:
Adjoint methods compute gradients of objective functionals at the cost of one forward and one backward solve. In the deterministic setting, symplectic integrators make these gradients exact for the discrete dynamics. In this talk, we extend this geometric picture to dynamical systems driven by geometric rough paths. From a rough Type-II variational principle, we derive the rough Hamilton's equations and the pathwise conservation laws of rough adjoint systems. These conservation laws yield sensitivities with respect to initial conditions and parameters. We then construct rough variational integrators, equivalent to Rough Symplectic Partitioned Runge--Kutta methods, which preserve discrete analogues of these conservation laws. Consequently, discretizing with these methods commutes with forming the adjoint equations, so the computed gradients are exact for the discrete forward map. Numerical experiments on a stochastic Lorenz 96 model show that the adjoint conservation laws are preserved to machine precision, giving more accurate gradients and more reliable optimisation than non-symplectic methods.
Tuesday, October 13, 2026
11:00AM AP&M 7218 and Zoom ID 961 4627 1748