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Koopman Operator Based Numerical Methods and Neural Network Based Models for Differential Equations
No full textThe study of differential equations is of paramount importance as it facilitates algorithms that are applied to a number of fields. However, most differential equations do not have a closed-form solution. To this end, the dissertation introduces two classes of numerical methods for differential equations, which are built on the Koopman operator and deep neural networks, respectively. \Based on the properties of the Koopman operator, we propose the Koopman method to solve autonomous nonlinear dynamical systems described by ordinary differential equations (ODEs). The solution is represented as a linear combination of the multiplication of Koopman eigenfunctions and eigenvalues, where the eigenpairs are approximated with the spectral-collocation (i.e., pseudo-spectral) method. Adaptivity is added to the Koopman method to preserve accuracy over the long term, resulting in the algorithm, adaptive spectral Koopman (ASK) method. Although ASK achieves high accuracy, it is computationally more expensive for multi-dimensional systems compared with conventional time integration schemes. To accelerate ASK, the sparse-grid-based ASK (SASK) method is developed to use the Smolyak sparse grid for eigenpair approximation, which significantly reduces the number of collocation points. We demonstrate that SASK can solve partial differential equations (PDEs) based-on their semi-discrete forms. Unlike conventional solvers such as Euler\u27s scheme and Runge-Kutta scheme, ASK and SASK are mesh-free, so they are more flexible when evaluating the solutions. The Koopman method is further extended to random dynamical systems induced by random parameters or random initial values for uncertainty quantification. Furthermore, we transplant the framework onto the stochastic Koopman operator to compute the moments of the stochastic processes generated by stochastic differential equations. Numerical experiments demonstrate both the accuracy and the computational advantage of these Koopman operator based algorithms compared with Monte-Carlo simulation with traditional solvers. Interestingly, the Koopman method can be also applied to unconstrained nonlinear optimization problems by solving the gradient flow (an ODE) that is associated with the objective function. The convergence properties are empirically demonstrated on multiple test functions with comparison with other gradient-based optimization algorithms. \ The second part of the dissertation focuses on solving seismic wave equations using neural networks, since it is a pivotal component in the study of subsurface seismic imaging. The advancement of deep learning enables us to learn the solution to a PDE which is usually a complicated nonlinear mapping. This approach can be faster than traditional numerical methods when numerous instances are to be solved. In our work, we target on the frequency domain wave equation, i.e., the Helmholtz equation. Firstly, the physics-informed learning framework is implemented on a fixed velocity model. It exhibits better generalization than a purely data-driven counterpart. This is because the PDE constraint provides additional regularization to the network during training. Then, inspired by the idea of operator learning, our work leverages the Fourier neural operator (FNO) to effectively learn the frequency domain wavefields under the context of variable velocity models. Also, we proposed a new architecture, the paralleled Fourier neural operator (PFNO), for efficiently solving the Helmholtz equation with multiple source locations and frequencies. In a set of experiments, both FNO and PFNO attain high accuracy on complicated velocity models from the OpenFWI datasets. Furthermore, the cross-dataset generalization test verifies that PFNO adapts to out-of-distribution velocity models. Importantly, these neural network based models admit higher computational efficiency in large-scale inferences than the traditional finite-difference methods. The aforementioned advantages endow the FNO-based solvers with the potential to build powerful models for research on seismic waves
Learning Hierarchical Knowledge: Formal Grammar Analysis and Task Modifiers
No full textHierarchical knowledge is critical to efficient problem solving. In artificial intelligence research, hierarchical reinforcement learning (HRL) and hierarchical task network (HTN) planning are two fields that utilize hierarchical knowledge albeit differently. HRL typically learns hierarchical policies to solve problems that require a significant amount of exploration, whereas HTN planning uses decomposition methods to define task hierarchies, which provide greater expressivity than classical planning. Several HRL systems exhibit different capabilities to solve a type of reinforcement learning problem in which rewards are dependent on the patterns of state traversal. We describe a generalized architecture that subsumes the systems, and categorize them into two variants differing in the use of memory. We construct formal grammars to represent the state traversal of the variants and analyze their expressivity. Traditional HTN planning is inadequate when domain dynamics are not fully known. We extend the HTN formalism with the notion of a task modifier, which flexibly addresses exogenous events. We describe a planning algorithm that integrates task modifiers and interleaves planning and acting. To alleviate the knowledge engineering required to create task modifiers, we describe an algorithm that learns from plan trajectories and acts as a task modifier. In a variety of experiments, we validate the formal grammar analysis results and evaluate the task modifier learning algorithm
Feasibility of an artificial vegetation patch as a local scour mitigation method
No full textMany flow-altering countermeasures for reducing scour at piers and abutments have been developed, although most experience reduced effectiveness under varying flow conditions. Current scour prevention methods are reviewed, and a novel flow-altering countermeasure is suggested. This new design is based on the natural flow-retarding properties of submerged vegetated canopies. When rigid artificial vegetation is placed surrounding a pier or abutment, similar effects are expected to occur, potentially shielding the bed from local scour. A preliminary design is suggested based on a literature review studying both natural and artificial vegetation patches. This design may work under a wider range of flow, bed, and skew conditions than many of the other flow-altering countermeasures available at the present
Three Essays in Applied Microeconomics
No full textThis dissertation consists of three chapters, exploring three topics in the field of health economics. In the first chapter, I study how ADHD diagnosis affects children�s medical expenditures based on underlying risk. ADHD is one of the most prevalent childhood mental health disorders, affecting nearly 6 million children and adolescents in the US. ADHD requires clinical diagnosis. Consequently, there are concerns about over- and under-diagnosis. Using a gradient boosted tree model, I estimate underlying ADHD severity or risk in childhood using nationally representative Medical Expenditure Panel Survey (MEPS) data. I then investigate the impact of diagnosis on medical expenditures by risk. Since diagnosis is possibly non-random, I instrument diagnosis with months of birth, after validating key identifying assumptions. To my knowledge, this is the first paper to provide causal estimates of risk-based heterogeneous effects of ADHD diagnosis on medical expenditures. Previous findings suggest that there are adverse health, educational and labor market effects of low- risk diagnosis, and that benefits accrue to high-risk diagnosed individuals only. In my analysis, I find that no-risk diagnosed children incur an additional 2,600 in prescription expenditures compared to undiagnosed children of similar risk. For the highest-risk diagnosed children, ambulatory and prescription expenditures are lower by 2,975 respectively than undiagnosed children with the same risk. Benefits in terms of lower expenditures generally occur at the top-end of the risk distribution, often in the top decile or the top percentile. I argue that for low-risk children, symptoms are manifestations of some other underlying condition(s) and not ADHD. Diagnosis and treatment then potentially lead to adverse health outcomes, instead of ameliorating the true underlying condition(s). For high-risk individuals, diagnosis and treatment alleviate symptoms and possibly, underlying severity. I also find evidence that diagnosis before the age of 7 is more beneficial for high-risk diagnosed children. Finally, consistent with previous studies, I find that diagnosis rates for girls are substantially lower than for boys.In my second chapter, I evaluate a conditional cash transfer program in Bangladesh aimed at secondary-school going girls, the Female Secondary School Stipend Program, and find that the program not only led to an increase in years of schooling, but also led to better health outcomes. Using variations in age-eligibility for the program and gender in a difference-in-differences framework, I find that the program increased female years of schooling by 1.4 years, and led to a decrease in the probability of having any chronic condition by 5 percentage points.In my last chapter, I investigate whether there is any link between deforestation and malaria in under-five children in Sub-Saharan Africa. I combine 12 years of high-resolution satellite data on forest change with individual-level and nationally representative malaria tests for more than 180,000 under-five children in 19 countries in Sub-Saharan Africa, and find that deforestation is associated with a higher probability of testing positive for malaria. For my main analysis, I instrument yearly change in forest cover in a region with the region�s annual change in cropland. My IV estimate suggests that a decrease in forest area amounting to one percent of the entire area of a region is associated with a 5.1 percentage point increase in the probability of testing positive. Additionally, I run a gamut of robustness checks, and also find evidence that this result is mostly driven by the clearing of savannas and woody savannas. There is also considerable heterogeneity in the way the population is affected - older children, and those living in rural areas and low-HDI countries are more vulnerable
