SPINN: Sparse, Physics-based, and Interpretable Neural Networks for PDEs

We introduce a class of Sparse, Physics-based, and Interpretable Neural Networks (SPINN) for solving ordinary and partial differential equations. By reinterpreting a traditional meshless representation of solutions of PDEs as a special sparse deep neural network, we develop a class of sparse neural network architectures that are interpretable. The SPINN model we propose here serves as a seamless bridge between two extreme modeling tools for PDEs, dense neural network based methods and traditional mesh-based and 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 we propose that distinguishes it from other neural network based approximations proposed earlier is that our method is both fully interpretable and sparse in the sense that it has much fewer connections than a dense neural network of the same size. Further, we demonstrate that Fourier series representations can 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.