  Discovering hidden partial differential equations (PDEs) and operators from
data is an important topic at the frontier between machine learning and
numerical analysis. This doctoral thesis introduces theoretical results and
deep learning algorithms to learn Green's functions associated with linear
partial differential equations and rigorously justify PDE learning techniques.
A theoretically rigorous algorithm is derived to obtain a learning rate, which
characterizes the amount of training data needed to approximately learn Green's
functions associated with elliptic PDEs. The construction connects the fields
of PDE learning and numerical linear algebra by extending the randomized
singular value decomposition to non-standard Gaussian vectors and
Hilbert--Schmidt operators, and exploiting the low-rank hierarchical structure
of Green's functions using hierarchical matrices. Rational neural networks
(NNs) are introduced and consist of neural networks with trainable rational
activation functions. The highly compositional structure of these networks,
combined with rational approximation theory, implies that rational functions
have higher approximation power than standard activation functions. In
addition, rational NNs may have poles and take arbitrarily large values, which
is ideal for approximating functions with singularities such as Green's
functions. Finally, theoretical results on Green's functions and rational NNs
are combined to design a human-understandable deep learning method for
discovering Green's functions from data. This approach complements
state-of-the-art PDE learning techniques, as a wide range of physics can be
captured from the learned Green's functions such as dominant modes, symmetries,
and singularity locations.
