This dissertation is on the usage of machine learning for the study of dynamical systems, particularly chaotic dynamical systems. Chapter 1 provides a brief intro- duction to the fields of chaotic dynamical systems and machine learning as well as a small overview of chaptes 2-4 In chapter 2 we study the usage of machine learning methods to forecast the spread of COVID-19. We consider the Susceptible-Infected-Confirmed-Recovered- Deceased (SICRD) compartmental model, with the goal of estimating the unknown infected compartment I, and several unknown parameters. We apply a variation of a “Physics Informed Neural Network” (PINN), which uses knowledge of the system to aid learning. First, we ensure estimation is possible by verifying the model’s identifiability. Then, we propose a wavelet transform to process data for the network training. Finally, our central result is a novel modification of the PINN’s loss function to reduce the number of simultaneously considered unknowns. We find that our modified network is capable of stable, efficient, and accurate estimation, while the unmodified network consistently yields incorrect values. The modified network is also shown to be efficient enough to be applied to a model with time-varying parameters. We present an application of our model results in ranking states by estimated relative testing efficiency. Our findings suggest the effectiveness of our modified PINN network, especially in this case of multiple unknown variables. In chapter 3 we introduce the Discrete-Temporal Sobolev Network (DTSN), a neural network loss function that assists dynamical system forecasting by minimiz- ing variational differences between the network output and the training data via a temporal Sobolev norm. This approach is entirely data-driven, architecture agnos- tic, and does not require derivative information from the estimated system. The DTSN is particularly well suited to chaotic dynamical systems as it minimizes noise in the network output which is crucial for such sensitive systems. For our test cases we consider discrete approximations of the Lorenz-63 system and the Chua circuit. For the network architectures we use the Long Short-Term Memory (LSTM) and the Transformer. The performance of the DTSN is compared with the standard MSE loss for both architectures, as well as with the Physics Informed Neural Net- work (PINN) loss for the LSTM. The DTSN loss is shown to substantially improve accuracy for both architectures, while requiring less information than the PINN and without noticeably increasing computational time, thereby demonstrating its potential to improve neural network forecasting of dynamical systems. In chapter 4 we present a new method of performing extended dynamic mode decomposition (EDMD) for systems which admit a symbolic representation. EDMD generates an estimate Km of the Koopman operator K for a system by defining a dictionary of observables on the space and estimating K restricted to be invariant on the span of this dictionary. One of the most important questions of the EDMD is what should be chosen for the choice of dictionary? We consider a class of chaotic dynamical systems with a known or estimable generating partition. For these systems we construct an effective dictionary from indicators of the ”cylinder sets” which have great significance in defining the ”symbolic system” which uses the generating partition. We prove strong operator topology convergence for both the projection onto the span of our dictionary and for Km. We also prove practical finite step estimation bounds for the projection and Km as well. Finally we demonstrate some numerical applications of the algorithm to two example systems, the dyadic map and the logistic map. Finally chapter 5 briefly recaps the results of chapters 2-4 and discusses directions for potential future research