The dynamics of many natural phenomena are described by partial differential equations (PDEs). With the rise in computing resources, machine learning methods for PDE recovery directly from observations are emerging. Unlike traditional PDE derivation, a machine learning method requires less mathematical prowess and is widely applicable to various dynamical systems. This dissertation uses two machine learning methods, sparse modeling and physics informed neural networks, to address PDE recovery issues: (1) identifying unknown PDE terms; (2) recovering PDE coefficients from noisy, partial observations. We tackle both spatially-independent and spatially-dependent PDEs, with the latter's recovery elucidating the spatial variations of medium material properties, essential for modern industrial applications.
First, sparse modeling methods are used to identify unknown PDEs. A dictionary with redundant hypothetical PDE terms is computed numerically from observations, and then sparse modeling approaches extract some of the terms from this dictionary to form the identified PDE. For spatially-dependent PDE identification, a dictionary is computed from the observations at each location, and then sparse regression is used to extract active terms from the dictionary to be the PDE terms at this location. The methods are validated on both synthetic datasets and real laser measurements of structural vibrations.
Next, a deep learning method spatially dependent physics informed neural network (SD-PINN) is used to recover spatially-dependent PDEs from noisy and partial observations. The spatially-dependent PDE coefficients for each term at all locations in the region of interest (ROI) are modeled as a low-rank coefficient matrix. The low-rank assumption is from the fact that the physical property of the material at one location is impacted by its surroundings, which results in decreased degrees of freedom for the entries in the coefficient matrix. The coefficient matrix is rewritten as a product of a small number of eigenvalues and eigenvectors, where ``small" refers to the rank of the coefficient matrix. Remarkably, these eigenvalues and eigenvectors contain fewer unknowns than the coefficient matrix itself. Given only noisy observations from a subset of locations within ROI, SD-PINN can efficiently recover these eigenvalues and eigenvectors and thus reconstruct the coefficient matrix, which encapsulates the PDE coefficients for all locations.