Method of Adjoints for differentiating through ODEs

Constructing a backward (P)DE which effectively gives us the gradients of the forward (P)DE. A trick in automatic differentiation which happens to be useful in differentiating likelihood (or other functions) of time-evolving systems. e.g. (Errico 1997; Kidger, Chen, and Lyons 2021; Kidger et al. 2020; Li et al. 2020; Rackauckas et al. 2018; Stapor, Fröhlich, and Hasenauer 2018; Cao et al. 2003).

Nick McGreivy, The adjoint method for PDE-constrained optimization: the conclusions of my 9-month struggle to understand the word adjoint. I’ll be talking about how automatic differentiation (AD) can help us better understand the adjoint method, and vice versa.


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Gahungu, Paterne, Christopher W. Lanyon, Mauricio A. Álvarez, Engineer Bainomugisha, Michael Thomas Smith, and Richard David Wilkinson. 2022. Adjoint-Aided Inference of Gaussian Process Driven Differential Equations.” In.
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Papoutsis-Kiachagias, E. M., N. Magoulas, J. Mueller, C. Othmer, and K. C. Giannakoglou. 2015. Noise Reduction in Car Aerodynamics Using a Surrogate Objective Function and the Continuous Adjoint Method with Wall Functions.” Computers & Fluids 122 (November): 223–32.
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Rackauckas, Christopher, Yingbo Ma, Vaibhav Dixit, Xingjian Guo, Mike Innes, Jarrett Revels, Joakim Nyberg, and Vijay Ivaturi. 2018. A Comparison of Automatic Differentiation and Continuous Sensitivity Analysis for Derivatives of Differential Equation Solutions.” arXiv:1812.01892 [Cs], December.
Stapor, Paul, Fabian Fröhlich, and Jan Hasenauer. 2018. Optimization and Uncertainty Analysis of ODE Models Using 2nd Order Adjoint Sensitivity Analysis.” bioRxiv, February, 272005.

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