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A persistent adjoint method with dynamic time-scaling and an application to mass action kinetics

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Abstract

In this article, we consider an optimization problem where the objective function is evaluated at the fixed-point of a contraction mapping parameterized by a control variable, and optimization takes place over this control variable. Since the derivative of the fixed-point with respect to the parameter can usually not be evaluated exactly, an adjoint dynamical system can be used to estimate gradients. Using this estimation procedure, the optimization algorithm alternates between derivative estimation and an approximate gradient descent step. We analyze a variant of this approach involving dynamic time-scaling, where after each parameter update the adjoint system is iterated until a convergence threshold is passed. We prove that, under certain conditions, the algorithm can find approximate stationary points of the objective function. We demonstrate the approach in the settings of an inverse problem in chemical kinetics, and learning in attractor networks.

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Funding

This research was supported, in part, under the National Science Foundation Grants CNS-0958379, CNS-0855217, and ACI-1126113; the City University of New York High Performance Computing Center at the College of Staten Island; the United State Department of Energy contract DE-SC0012704; and DOE Office of Science under grant ASCR KJ040201.

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  1. Computational Science Initiative, Brookhaven National Laboratory, Upton, NY, 11973, USA

    Thomas Flynn

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  1. Thomas Flynn
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Correspondence to Thomas Flynn.

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Flynn, T. A persistent adjoint method with dynamic time-scaling and an application to mass action kinetics. Numer Algor 89, 87–113 (2022). https://doi.org/10.1007/s11075-021-01107-8

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