Yiqi Gu

Yaru Liu, Yiqi Gu

In this paper, we propose a new optimization framework, the layer separation (LySep) model, to improve the deep learning-based methods in solving partial differential equations. Due to the highly non-convex nature of the loss function in deep learning, existing optimization algorithms often converge to suboptimal local minima or suffer from gradient explosion or vanishing, resulting in poor performance. To address these issues, we introduce auxiliary variables to separate the layers of deep neural networks. Specifically, the output and its derivatives of each layer are represented by auxiliary variables, effectively decomposing the deep architecture into a series of shallow architectures. New loss functions with auxiliary variables are established, in which only variables from two neighboring layers are coupled. Corresponding algorithms based on alternating directions are developed, where many variables can be updated optimally in closed forms. Moreover, we provide theoretical analyses demonstrating the consistency between the LySep model and the original deep model. High-dimensional numerical results validate our theory and demonstrate the advantages of LySep in minimizing loss and reducing solution error.

Yiqi Gu, John Harlim, Senwei Liang et al.

In this paper, we consider the density estimation problem associated with the stationary measure of ergodic Itô diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take an advantage of the characterization of density function through the stationary solution of a parabolic-type Fokker-Planck PDE, we proceed as follows. First, we employ deep neural networks to approximate the drift and diffusion terms of the SDE by solving appropriate supervised learning tasks. Subsequently, we solve a steady-state Fokker-Plank equation associated with the estimated drift and diffusion coefficients with a neural-network-based least-squares method. We establish the convergence of the proposed scheme under appropriate mathematical assumptions, accounting for the generalization errors induced by regressing the drift and diffusion coefficients, and the PDE solvers. This theoretical study relies on a recent perturbation theory of Markov chain result that shows a linear dependence of the density estimation to the error in estimating the drift term, and generalization error results of nonparametric regression and of PDE regression solution obtained with neural-network models. The effectiveness of this method is reflected by numerical simulations of a two-dimensional Student's t distribution and a 20-dimensional Langevin dynamics.

Qiang Du, Yiqi Gu, Haizhao Yang et al.

Identifying hidden dynamics from observed data is a significant and challenging task in a wide range of applications. Recently, the combination of linear multistep methods (LMMs) and deep learning has been successfully employed to discover dynamics, whereas a complete convergence analysis of this approach is still under development. In this work, we consider the deep network-based LMMs for the discovery of dynamics. We put forward error estimates for these methods using the approximation property of deep networks. It indicates, for certain families of LMMs, that the $\ell^2$ grid error is bounded by the sum of $O(h^p)$ and the network approximation error, where $h$ is the time step size and $p$ is the local truncation error order. Numerical results of several physically relevant examples are provided to demonstrate our theory.