Theory, Algorithms, and Software for Physics-Informed Deep LearningLu, LucreatorKarniadakis, GeorgeAdvisorHarrison, MatthewReaderStinis, PanosReaderBrown University. Department of Applied Mathematicssponsor2020, None p.born digitalThesis (Ph. D.)--Brown University, 2020thesesThis dissertation is centered around the broad topic of physics-informed deep learning (DL). Specifically, it covers the following three fronts: Algorithms. Recent developments of DL, i.e., deep neural networks (NNs), provide us with opportunities that cannot be tackled solely through traditional methods for scientific applications. Specifically, I have developed three new algorithms to incorporate physics into DL. 1. Multi-fidelity NNs. Instrumented indentation has been developed to extract mechanical properties of materials. I present multi-fidelity approaches for solving the inverse indentation problem, hence significantly reducing the number of expensive high-fidelity datasets required to achieve a given level of accuracy. 2. Physics-informed NNs (PINNs). I further develop PINNs, which embed partial differential equations (PDEs) into NNs. The PINN algorithm can be applied to different types of PDEs, including fractional PDEs and stochastic PDEs. I propose a new residual-based adaptive refinement method to improve the training efficiency. 3. Learning nonlinear operators. I propose the deep operator network (DeepONet) to learn operators accurately and efficiently. I demonstrate that DeepONet significantly reduces the generalization error. I observe high-order error convergence, and even exponential convergence with respect to the dataset size. Theory. Building the mathematical foundations of DL is especially vital to establishing assurance and thus realizing the potential of DL for science. I theoretically address the question of assurance. 1. Dying ReLU. The dying ReLU refers to the problem when ReLU neurons become inactive. I prove that a deep ReLU network will eventually die in probability as the depth goes to infinite. I also propose a new initialization procedure, which effectively prevents the dying ReLU. 2. Generalization. I study the generalization error of NNs for classification problems. I introduce the cover complexity to measure the difficulty of learning a dataset and the inverse of the modulus of continuity to quantify NN smoothness. A quantitative bound for expected accuracy is also derived. Open-source software. I present a Python library for PINNs, DeepXDE, which is designed to serve both as an education tool to be used in the classroom as well as a research tool for solving general problems in computational science and engineering.Differential equations, PartialDeep Learningcomputational physicsEnglishRPB20200720In CopyrightCollection is open for research.10.26300/mpz0-gr82dissertations