My research focuses on using physics-informed machine learning (PIML) to solve complex fluid dynamics problems in astronomy. Specifically, we tackle the time-dependent 2D compressible Navier-Stokes equations to model the evolution of rotating accretion flows within protoplanetary disks, influenced by planetary gravitational forces. These forces create sharp, time-dependent features that are challenging to solve with traditional methods.

We developed an enhanced PIML approach that trains neural networks using physical laws, without relying on large sets of labeled data. This method includes innovations such as a self-scalable activation function, adaptive time-marching strategy, biased loss function, and spatial-temporal adaptive sampling. It effectively captured sharp spatial features and provided reliable solutions with less computational effort. This work offers a promising avenue for solving similar complex fluid dynamics problems in various scientific fields.

On the forefront of scientific computing, Deep Learning (DL), specifically using Deep Neural Networks (DNNs), has emerged as a powerful tool for solving Partial Differential Equations (PDEs). We leverage recent advancements in function approximation using sparsity-based techniques and random sampling to develop an efficient high-dimensional PDE solver based on deep learning (DL). Our approach demonstrates both theoretically and numerically that it competes with a novel, stable, and accurate compressive spectral collocation method. We introduce a new practical existence theorem, establishing the existence of a class of trainable deep neural networks (DNNs) with suitable bounds on the network architecture and a sufficient condition on sample complexity, with logarithmic scaling in dimension. This ensures that the resulting networks stably and accurately approximate a diffusion-reaction PDE with high probability. 

Work in progress: Physics-informed deep learning and compressive collocation for high-dimensional diffusion-reaction equations: \\practical existence theory and numerics.

We propose a new method for the numerical solution of high-dimensional PDEs with periodic boundary conditions called compressive Fourier collocation. Combining techniques from sparse high-dimensional approximation and Monte Carlo sampling, our method requires a number of collocation points that scales mildly (i.e., logarithmically) with respect to the dimension of the PDE domain. Focusing on the case of high-dimensional diffusion equations, we provide recovery guarantees based on the recently proposed framework of sparse recovery in subsampled bounded Riesz systems and carry out numerical experiments up to dimension 20. We currently focus on implementing our technique on even higher dimensional domains and comparing it to recently proposed methods based on deep learning.

Publication: Weiqi Wang, Simone Brugiapaglia, Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions, IMA J. Numer. Anal. (2024), pp. drad102. https://doi.org/10.1093/imanum/drad102. 

We propose a non-homogeneous Markov chain model to predict the delay over stations. We build our prediction model by assuming the delay evolution over stations follows a non-homogeneous Markov chain. We propose a chi-square Markov property test to verify our assumption that the delay evolution over stations has a first-order Markov property. Each transition matrix in our model can characterize the unique pattern of delay evolution between two adjacent stations that a train travels through. We propose a Gaussian-kernel-based method to recover the transition matrices for the Markov chain when the training data are limited. This recovery method captures the delay transition probabilities from the existing data. We show that this recovery method achieves a higher prediction accuracy than the other naive approaches.

Publication: Xu, J., Wang, W., Gao, Z., Luo, H., & Wu, Q. (2023). A novel Markov model for near-term railway delay prediction. Computers & Industrial Engineering, 109302.  https://doi.org/10.48550/arXiv.2205.10682.