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SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs
Journal of Computational Physics ( IF 3.8 ) Pub Date : 2021-07-28 , DOI: 10.1016/j.jcp.2021.110600
Amuthan A. Ramabathiran , Prabhu Ramachandran

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.



中文翻译:

SPINN:用于 PDE 的稀疏、基于物理和部分可解释的神经网络

我们引入了一类用于求解常微分方程和偏微分方程 (PDE) 的稀疏、基于物理和部分可解释的神经网络 (SPINN)。通过重新解释 PDE 解的传统无网格表示,我们开发了一类可部分解释的稀疏神经网络架构。我们在此提出的 SPINN 模型作为 PDE 的两个极端建模工具之间的无缝桥梁,即基于密集神经网络的方法,如物理信息神经网络 (PINNs) 和传统的无网格数值方法,从而提供了一种新的方法来开发新的一类混合算法建立在这两种观点的最佳之上。SPINN 模型的一个独特之处在于它与之前提出的其他基于神经网络的近似不同,它是 (i) 可解释的,在特定意义上在工作中变得精确,并且(ii)稀疏,因为它的连接比用于 PDE 的典型密集神经网络少得多。此外,SPINN 算法隐式编码网格自适应性,并能够处理解决方案中的不连续性。此外,我们证明了傅立叶级数表示也可以表示为一类特殊的 SPINN,并提出了傅立叶表示的广义神经网络类似物。我们通过涉及常微分方程、椭圆、抛物线、双曲线和非线性偏微分方程的各种示例以及流体动力学中的一个示例来说明所提出方法的实用性。此外,SPINN 算法隐式编码网格自适应性,并能够处理解决方案中的不连续性。此外,我们证明了傅立叶级数表示也可以表示为一类特殊的 SPINN,并提出了傅立叶表示的广义神经网络类似物。我们通过涉及常微分方程、椭圆、抛物线、双曲线和非线性偏微分方程的各种示例以及流体动力学中的一个示例来说明所提出方法的实用性。此外,SPINN 算法隐式编码网格自适应性,并能够处理解决方案中的不连续性。此外,我们证明了傅立叶级数表示也可以表示为一类特殊的 SPINN,并提出了傅立叶表示的广义神经网络类似物。我们通过涉及常微分方程、椭圆、抛物线、双曲线和非线性偏微分方程的各种示例以及流体动力学中的一个示例来说明所提出方法的实用性。

更新日期:2021-08-07
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