Solving differential equations using deep neural networks

•A novel framework for solving irregular PDEs using Deep Neural Networks (DNNs) is developed, extended, and analyzed.•The framework demonstrates phenomenal ease of prototyping PDE solutions.•For the first time, irregular shock solutions to a classical fluid problem are analyzed, and compared against finite volume and finite element solutions, demonstrating good accuracy and stability behaviors.•The method demonstrates novel and remarkable computational scalability in parameter exploration.•A novel data-enrichment algorithm for incorporating experimental data into the DNN/PDE framework is presented, which is used to reveal hidden underlying physical subsystems observed in the experiment, but unaccounted for in the theoretical model. Recent work on solving partial differential equations (PDEs) with deep neural networks (DNNs) is presented. The paper reviews and extends some of these methods while carefully analyzing a fundamental feature in numerical PDEs and nonlinear analysis: irregular solutions. First, the Sod shock tube solution to the compressible Euler equations is discussed and analyzed. This analysis includes a comparison of a DNN-based approach with conventional finite element and finite volume methods, and demonstrates that the DNN is competitive in terms of degrees of freedom required for a given accuracy. Further, the DNN-based approach is extended to consider performance improvements and simultaneous parameter space exploration. Next, a shock solution to compressible magnetohydrodynamics (MHD) is solved for, and used in a scenario where experimental data is utilized to enhance a PDE system that is a priori insufficient to validate against the observed/experimental data. This is accomplished by enriching the model PDE system with source terms that are then inferred via supervised training with synthetic experimental data. The resulting DNN framework for PDEs enables straightforward system prototyping and natural integration of large data sets (be they synthetic or experimental), all while simultaneously enabling single-pass exploration of an entire parameter space.