DOSnet as a Non-Black-Box PDE Solver: When Deep Learning Meets Operator Splitting
This work addresses the problem of solving complex differential equations in science and engineering by providing a more interpretable and efficient AI-enhanced solver, though it is incremental as it builds on existing operator splitting methods.
The authors tackled the challenge of creating accurate, efficient, and interpretable neural network solvers for evolutionary PDEs by designing DOSnet, a deep operator-splitting network that integrates physics-based operator splitting into its architecture. Experimental results on nonlinear Schrödinger equations showed that DOSnet achieved better accuracy and lower computational complexity than numerical schemes and baseline DNNs.
Deep neural networks (DNNs) recently emerged as a promising tool for analyzing and solving complex differential equations arising in science and engineering applications. Alternative to traditional numerical schemes, learning-based solvers utilize the representation power of DNNs to approximate the input-output relations in an automated manner. However, the lack of physics-in-the-loop often makes it difficult to construct a neural network solver that simultaneously achieves high accuracy, low computational burden, and interpretability. In this work, focusing on a class of evolutionary PDEs characterized by having decomposable operators, we show that the classical ``operator splitting'' numerical scheme of solving these equations can be exploited to design neural network architectures. This gives rise to a learning-based PDE solver, which we name Deep Operator-Splitting Network (DOSnet). Such non-black-box network design is constructed from the physical rules and operators governing the underlying dynamics contains learnable parameters, and is thus more flexible than the standard operator splitting scheme. Once trained, it enables the fast solution of the same type of PDEs. To validate the special structure inside DOSnet, we take the linear PDEs as the benchmark and give the mathematical explanation for the weight behavior. Furthermore, to demonstrate the advantages of our new AI-enhanced PDE solver, we train and validate it on several types of operator-decomposable differential equations. We also apply DOSnet to nonlinear Schrödinger equations (NLSE) which have important applications in the signal processing for modern optical fiber transmission systems, and experimental results show that our model has better accuracy and lower computational complexity than numerical schemes and the baseline DNNs.