MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs

In this paper, we propose a model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solve PDEs based on operator representation with regularization from data. In this work, we use a deep neural network to parameterize the Green's function. The empirical risk consists of the mean square of the governing equation, boundary conditions, and a few labels, which are numerically computed by traditional schemes on coarse grid points with cheap computation cost. With only the labeled dataset or only the model constraints, it is insufficient to accurately train a MOD-Net for complicate problems. Intuitively, the labeled dataset works as a regularization in addition to the model constraints. The MOD-Net is much efficient than original neural operator because the MOD-Net also uses the information of governing equation and the boundary conditions of the PDE rather than purely the expensive labels. Since the MOD-Net learns the Green's function of a PDE, it solves a type of PDEs but not a specific case. We numerically show MOD-Net is very efficient in solving Poisson equation and one-dimensional Boltzmann equation. For non-linear PDEs, where the concept of the Green's function does not apply, the non-linear MOD-Net can be similarly used as an ansatz for solving non-linear PDEs

Optimistic Estimate Uncovers the Potential of Nonlinear Models

We propose an optimistic estimate to evaluate the best possible fitting performance of nonlinear models. It yields an optimistic sample size that quantifies the smallest possible sample size to fit/recover a target function using a nonlinear model. We estimate the optimistic sample sizes for matrix factorization models, deep models, and deep neural networks (DNNs) with fully-connected or convolutional architecture. For each nonlinear model, our estimates predict a specific subset of targets that can be fitted at overparameterization, which are confirmed by our experiments. Our optimistic estimate reveals two special properties of the DNN models -- free expressiveness in width and costly expressiveness in connection. These properties suggest the following architecture design principles of DNNs: (i) feel free to add neurons/kernels; (ii) restrain from connecting neurons. Overall, our optimistic estimate theoretically unveils the vast potential of nonlinear models in fitting at overparameterization. Based on this framework, we anticipate gaining a deeper understanding of how and why numerous nonlinear models such as DNNs can effectively realize their potential in practice in the near future

Linear Stability Hypothesis and Rank Stratification for Nonlinear Models

Models with nonlinear architectures/parameterizations such as deep neural networks (DNNs) are well known for their mysteriously good generalization performance at overparameterization. In this work, we tackle this mystery from a novel perspective focusing on the transition of the target recovery/fitting accuracy as a function of the training data size. We propose a rank stratification for general nonlinear models to uncover a model rank as an "effective size of parameters" for each function in the function space of the corresponding model. Moreover, we establish a linear stability theory proving that a target function almost surely becomes linearly stable when the training data size equals its model rank. Supported by our experiments, we propose a linear stability hypothesis that linearly stable functions are preferred by nonlinear training. By these results, model rank of a target function predicts a minimal training data size for its successful recovery. Specifically for the matrix factorization model and DNNs of fully-connected or convolutional architectures, our rank stratification shows that the model rank for specific target functions can be much lower than the size of model parameters. This result predicts the target recovery capability even at heavy overparameterization for these nonlinear models as demonstrated quantitatively by our experiments. Overall, our work provides a unified framework with quantitative prediction power to understand the mysterious target recovery behavior at overparameterization for general nonlinear models