Ordinary differential equations (ODE) are used extensively in science and engineering to model dynamical systems. In this dissertation, we use noisy observations to address two learning tasks regarding dynamical systems. The first one is called ODE parameter estimation, where the shape of the ODEs are known, and we try to learn (estimate) the parameters. The second task is called learning the governing equations, where we learn both the shape and the parameters of the ODEs.

In the first part of this dissertation, we address the ODE parameter estimation problem. We propose and analyze a block coordinate descent proximal algorithm (BCD-prox) for simultaneous filtering and parameter estimation of ODE models. The main idea is to learn the states and the parameters in an alternation, where the states are restricted to change slowly from one iteration to the next. As we show on ODE systems with up to $d=40$ dimensions, as compared to state-of-the-art methods, BCD-prox exhibits increased robustness (to noise, parameter initialization, and hyperparameters), decreased training times, and improved accuracy of both filtered states and estimated parameters. We show how BCD-prox can be used with multistep numerical discretizations, and we establish convergence of BCD-prox under hypotheses that include real systems of interest.

In the second part of this dissertation, we address the problem of learning the governing equations given the noisy observations. While we use powerful neural networks to learn the ODEs, we propose a novel structure that makes our network interpretable. The idea is to use the neural network to learn a set of one-dimensional and multi-dimensional shape functions, whose linear combinations give use the equations. To make our method robust to the noise in the observations, we learn the clean states and the parameters of the network simultaneously using the block coordinate descent proximal algorithm (BCD-prox). As we show in our experiments, our method is robust to its hyperparameter, robust to the noise, and outperforms the state-of-the-art methods by achieving more accurate state predictions.

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