  We introduce a class of Sparse, Physics-based, and partially Interpretable
Neural Networks (SPINN) for solving ordinary and partial differential equations
(PDEs). By reinterpreting a traditional meshless representation of solutions of
PDEs we develop a class of sparse neural network architectures that are
partially interpretable. The SPINN model we propose here serves as a seamless
bridge between two extreme modeling tools for PDEs, namely dense neural network
based methods like Physics Informed Neural Networks (PINNs) and traditional
mesh-free numerical methods, thereby providing a novel means to develop a new
class of hybrid algorithms that build on the best of both these viewpoints. A
unique feature of the SPINN model that distinguishes it from other neural
network based approximations proposed earlier is that it is (i) interpretable,
in a particular sense made precise in the work, and (ii) sparse in the sense
that it has much fewer connections than typical dense neural networks used for
PDEs. Further, the SPINN algorithm implicitly encodes mesh adaptivity and is
able to handle discontinuities in the solutions. In addition, we demonstrate
that Fourier series representations can also be expressed as a special class of
SPINN and propose generalized neural network analogues of Fourier
representations. We illustrate the utility of the proposed method with a
variety of examples involving ordinary differential equations, elliptic,
parabolic, hyperbolic and nonlinear partial differential equations, and an
example in fluid dynamics.
