  There has been a growing interest in the use of Deep Neural Networks (DNNs)
to solve Partial Differential Equations (PDEs). Despite the promise that such
approaches hold, there are various aspects where they could be improved. Two
such shortcomings are (i) their computational inefficiency relative to
classical numerical methods, and (ii) the non-interpretability of a trained DNN
model. In this work we present ASPINN, an anisotropic extension of our earlier
work called SPINN--Sparse, Physics-informed, and Interpretable Neural
Networks--to solve PDEs that addresses both these issues. ASPINNs generalize
radial basis function networks. We demonstrate using a variety of examples
involving elliptic and hyperbolic PDEs that the special architecture we propose
is more efficient than generic DNNs, while at the same time being directly
interpretable. Further, they improve upon the SPINN models we proposed earlier
in that fewer nodes are require to capture the solution using ASPINN than using
SPINN, thanks to the anisotropy of the local zones of influence of each node.
The interpretability of ASPINN translates to a ready visualization of their
weights and biases, thereby yielding more insight into the nature of the
trained model. This in turn provides a systematic procedure to improve the
architecture based on the quality of the computed solution. ASPINNs thus serve
as an effective bridge between classical numerical algorithms and modern DNN
based methods to solve PDEs. In the process, we also streamline the training of
ASPINNs into a form that is closer to that of supervised learning algorithms.
