A Survey on Solving and Discovering Differential Equations Using Deep Neural Networks

Ordinary and partial differential equations (DE) are used extensively in scientific and mathematical domains to model physical systems. Current literature has focused primarily on deep neural network (DNN) based methods for solving a specific DE or a family of DEs. Research communities with a history of using DE models may view DNN-based differential equation solvers (DNN-DEs) as a faster and transferable alternative to current numerical methods. However, there is a lack of systematic surveys detailing the use of DNN-DE methods across physical application domains and a generalized taxonomy to guide future research. This paper surveys and classifies previous works and provides an educational tutorial for senior practitioners, professionals, and graduate students in engineering and computer science. First, we propose a taxonomy to navigate domains of DE systems studied under the umbrella of DNN-DE. Second, we examine the theory and performance of the Physics Informed Neural Network (PINN) to demonstrate how the influential DNN-DE architecture mathematically solves a system of equations. Third, to reinforce the key ideas of solving and discovery of DEs using DNN, we provide a tutorial using DeepXDE, a Python package for developing PINNs, to develop DNN-DEs for solving and discovering a classic DE, the linear transport equation.

Vehicle Emissions Prediction with Physics-Aware AI Models: Preliminary Results

Given an on-board diagnostics (OBD) dataset and a physics-based emissions prediction model, this paper aims to develop an accurate and computational-efficient AI (Artificial Intelligence) method that predicts vehicle emissions. The problem is of societal importance because vehicular emissions lead to climate change and impact human health. This problem is challenging because the OBD data does not contain enough parameters needed by high-order physics models. Conversely, related work has shown that low-order physics models have poor predictive accuracy when using available OBD data. This paper uses a divergent window co-occurrence pattern detection method to develop a spatiotemporal variability-aware AI model for predicting emission values from the OBD datasets. We conducted a case study using real-world OBD data from a local public transportation agency. Results show that the proposed AI method has approximately 65% improved predictive accuracy than a non-AI low-order physics model and is approximately 35% more accurate than a baseline model.