Anisotropic, Sparse and Interpretable Physics-Informed Neural Networks for PDEs

TL;DR

This paper introduces ASPINN, an anisotropic, sparse, and interpretable neural network architecture for solving PDEs that is more efficient and transparent than traditional DNN approaches, bridging classical methods and modern AI.

Contribution

The paper presents ASPINN, a novel anisotropic neural network architecture that improves efficiency and interpretability in solving PDEs, building on and surpassing previous SPINN models.

Findings

ASPINN outperforms generic DNNs in efficiency.

Fewer nodes are needed in ASPINN compared to SPINN.

ASPINN provides visual interpretability of model weights.

Abstract

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.