<pre>Operator learning has the potential to supplement traditional numerical methods, especially when speed is desired more than accuracy. <br>This includes the architectures DeepONets, Fourier neural operators and Koopman autoencoders.<br>First, this dissertation provides the background material for operator learning. <br>Then, it studies some general best practices for operator learning.<br>Then, it studies the loss functions and operator forms for Koopman autoencoders. <br>Finally, it studies the use of an adversarial addition to neural operators that have an autoencoder structure.</pre><p></p>

<p dir="ltr">Neural operator learning is emerging as a prominent technique in scientific machine learn- ing for modeling complex nonlinear systems with multi-physics and multi-scale applications. A common drawback of such operators is that they are data-hungry and the results are highly dependent on the quality and quantity of the training data provided to the models. Moreover, obtaining high-quality data in sufficient quantity can be computationally prohibitive. Faster surrogate models are required to overcome this drawback which can be learned from datasets of variable fidelity and also quantify the uncertainty. In this work, we propose a Multi-Level Stacked Deep Operator Network (MLSDON) which can learn from datasets of different fidelity and is not dependent on the input function. Through various experiments, we demonstrate that the MLSDON can approximate the high-fidelity solution operator with better accuracy compared to a Vanilla DeepONet when sufficient high-fidelity data is unavailable. We also extend MLSDON to build robust confidence intervals by making conformalized predictions. This technique guarantees trajectory coverage of the predictions irrespective of the input distribution. Various numerical experiments are conducted to demonstrate the applicability of MLSDON to multi-fidelity, multi-scale, and multi-physics problems.</p>