Abstract

The "on-the-fly" (OTF) method is a gradient-based approach to equation discovery. It makes use of (potentially limited) observational data, a data assimilation technique such as continuous data assimilation (also called nudging), and an error metric. From these it computes an asymptotic approximation of a simulated system's error sensitivity with respect to unknown parameters and utilizes any gradient-based optimization algorithm to perform parameter updates. These parameter updates may be done online or offline, and the model may be of a precisely specified form or simply a library of possible terms with potentially nonlinear parameters. We have applied the adjoint method to the same problem and found that, in the case of complete observations, asymptotically approximating the adjoint state leads to the same approximate gradient as OTF. However, in the case of incomplete observations, the full asymptotic approximation of the adjoint state--derived in this thesis--often allows the identification of parameters where OTF fails, yet at a lower computational cost than simulating the full adjoint state. We have also generalized the approach to complex state and parameter spaces with the use of Wirtinger calculus. To aid in testing and developing the algorithm, we have developed a Python package that requires only observational data and a proposed model to implement the method. The package relies on automatic differentiation and can make use of any gradient-based optimization procedure, such as Adam. We present results of digital twin experiments obtained using the package on a variety of systems. In particular, these experiments demonstrate the effectiveness of Adam over standard gradient descent and of the full asymptotic approximation of the adjoint state when observations are incomplete. Code is available at github.com/schilln/otf.

Degree

College and Department

Computational, Mathematical, and Physical Sciences; Mathematics

Rights

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