arXiv subjects

Zhi-Qin John Xu

Publications and source records attributed to Zhi-Qin John Xu.

At least 73 records · Page 4Linked to original sources

arXiv2022

Empirical Phase Diagram for Three-layer Neural Networks with Infinite Width

Substantial work indicates that the dynamics of neural networks (NNs) is closely related to their initialization of parameters. Inspired by the phase diagram for two-layer ReLU NNs with infinite width (Luo et al., 2021), we make a step towards drawing a phase diagram for three-layer ReLU NNs with infinite width. First, we derive a normalized gradient flow for three-layer ReLU NNs and obtain two key independent quantities to distinguish different dynamical regimes for common initialization methods. With carefully designed experiments and a large computation cost, for both synthetic datasets and real datasets, we find that the dynamics of each layer also could be divided into a linear regime and a condensed regime, separated by a critical regime. The criteria is the relative change of input weights (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) as the width approaches infinity during the training, which tends to $0$, $+\infty$ and $O(1)$, respectively. In addition, we also demonstrate that different layers can lie in different dynamical regimes in a training process within a deep NN. In the condensed regime, we also observe the condensation of weights in isolated orientations with low complexity. Through experiments under three-layer condition, our phase diagram suggests a complicated dynamical regimes consisting of three possible regimes, together with their mixture, for deep NNs and provides a guidance for studying deep NNs in different initialization regimes, which reveals the possibility of completely different dynamics emerging within a deep NN for its different layers.

cs.LG↗

arXiv2022

DeepFlame: A deep learning empowered open-source platform for reacting flow simulations

In this work, we introduce DeepFlame, an open-source C++ platform with the capabilities of utilising machine learning algorithms and pre-trained models to solve for reactive flows. We combine the individual strengths of the computational fluid dynamics library OpenFOAM, machine learning framework Torch, and chemical kinetics program Cantera. The complexity of cross-library function and data interfacing (the core of DeepFlame) is minimised to achieve a simple and clear workflow for code maintenance, extension and upgrading. As a demonstration, we apply our recent work on deep learning for predicting chemical kinetics (Zhang et al. Combust. Flame vol. 245 pp. 112319, 2022) to highlight the potential of machine learning in accelerating reacting flow simulation. A thorough code validation is conducted via a broad range of canonical cases to assess its accuracy and efficiency. The results demonstrate that the convection-diffusion-reaction algorithms implemented in DeepFlame are robust and accurate for both steady-state and transient processes. In addition, a number of methods aiming to further improve the computational efficiency, e.g. dynamic load balancing and adaptive mesh refinement, are explored. Their performances are also evaluated and reported. With the deep learning method implemented in this work, a speed-up of two orders of magnitude is achieved in a simple hydrogen ignition case when performed on a medium-end graphics processing unit (GPU). Further gain in computational efficiency is expected for hydrocarbon and other complex fuels. A similar level of acceleration is obtained on an AI-specific chip - deep computing unit (DCU), highlighting the potential of DeepFlame in leveraging the next-generation computing architecture and hardware.

physics.flu-dyn↗

arXiv2022

A deep learning-based model reduction (DeePMR) method for simplifying chemical kinetics

A deep learning-based model reduction (DeePMR) method for simplifying chemical kinetics is proposed and validated using high-temperature auto-ignitions, perfectly stirred reactors (PSR), and one-dimensional freely propagating flames of n-heptane/air mixtures. The mechanism reduction is modeled as an optimization problem on Boolean space, where a Boolean vector, each entry corresponding to a species, represents a reduced mechanism. The optimization goal is to minimize the reduced mechanism size given the error tolerance of a group of pre-selected benchmark quantities. The key idea of the DeePMR is to employ a deep neural network (DNN) to formulate the objective function in the optimization problem. In order to explore high dimensional Boolean space efficiently, an iterative DNN-assisted data sampling and DNN training procedure are implemented. The results show that DNN-assistance improves sampling efficiency significantly, selecting only $10^5$ samples out of $10^{34}$ possible samples for DNN to achieve sufficient accuracy. The results demonstrate the capability of the DNN to recognize key species and reasonably predict reduced mechanism performance. The well-trained DNN guarantees the optimal reduced mechanism by solving an inverse optimization problem. By comparing ignition delay times, laminar flame speeds, temperatures in PSRs, the resulting skeletal mechanism has fewer species (45 species) but the same level of accuracy as the skeletal mechanism (56 species) obtained by the Path Flux Analysis (PFA) method. In addition, the skeletal mechanism can be further reduced to 28 species if only considering atmospheric, near-stoichiometric conditions (equivalence ratio between 0.6 and 1.2). The DeePMR provides an innovative way to perform model reduction and demonstrates the great potential of data-driven methods in the combustion area.

cs.LG↗

arXiv2022

An Upper Limit of Decaying Rate with Respect to Frequency in Deep Neural Network

Deep neural network (DNN) usually learns the target function from low to high frequency, which is called frequency principle or spectral bias. This frequency principle sheds light on a high-frequency curse of DNNs -- difficult to learn high-frequency information. Inspired by the frequency principle, a series of works are devoted to develop algorithms for overcoming the high-frequency curse. A natural question arises: what is the upper limit of the decaying rate w.r.t. frequency when one trains a DNN? In this work, our theory, confirmed by numerical experiments, suggests that there is a critical decaying rate w.r.t. frequency in DNN training. Below the upper limit of the decaying rate, the DNN interpolates the training data by a function with a certain regularity. However, above the upper limit, the DNN interpolates the training data by a trivial function, i.e., a function is only non-zero at training data points. Our results indicate a better way to overcome the high-frequency curse is to design a proper pre-condition approach to shift high-frequency information to low-frequency one, which coincides with several previous developed algorithms for fast learning high-frequency information. More importantly, this work rigorously proves that the high-frequency curse is an intrinsic difficulty of DNNs.

cs.LG↗

arXiv2022

A multi-scale sampling method for accurate and robust deep neural network to predict combustion chemical kinetics

Machine learning has long been considered as a black box for predicting combustion chemical kinetics due to the extremely large number of parameters and the lack of evaluation standards and reproducibility. The current work aims to understand two basic questions regarding the deep neural network (DNN) method: what data the DNN needs and how general the DNN method can be. Sampling and preprocessing determine the DNN training dataset, further affect DNN prediction ability. The current work proposes using Box-Cox transformation (BCT) to preprocess the combustion data. In addition, this work compares different sampling methods with or without preprocessing, including the Monte Carlo method, manifold sampling, generative neural network method (cycle-GAN), and newly-proposed multi-scale sampling. Our results reveal that the DNN trained by the manifold data can capture the chemical kinetics in limited configurations but cannot remain robust toward perturbation, which is inevitable for the DNN coupled with the flow field. The Monte Carlo and cycle-GAN samplings can cover a wider phase space but fail to capture small-scale intermediate species, producing poor prediction results. A three-hidden-layer DNN, based on the multi-scale method without specific flame simulation data, allows predicting chemical kinetics in various scenarios and being stable during the temporal evolutions. This single DNN is readily implemented with several CFD codes and validated in various combustors, including (1). zero-dimensional autoignition, (2). one-dimensional freely propagating flame, (3). two-dimensional jet flame with triple-flame structure, and (4). three-dimensional turbulent lifted flames. The results demonstrate the satisfying accuracy and generalization ability of the pre-trained DNN. The Fortran and Python versions of DNN and example code are attached in the supplementary for reproducibility.

physics.chem-ph↗

arXiv2022

Dropout in Training Neural Networks: Flatness of Solution and Noise Structure

It is important to understand how the popular regularization method dropout helps the neural network training find a good generalization solution. In this work, we show that the training with dropout finds the neural network with a flatter minimum compared with standard gradient descent training. We further find that the variance of a noise induced by the dropout is larger at the sharper direction of the loss landscape and the Hessian of the loss landscape at the found minima aligns with the noise covariance matrix by experiments on various datasets, i.e., MNIST, CIFAR-10, CIFAR-100 and Multi30k, and various structures, i.e., fully-connected networks, large residual convolutional networks and transformer. For networks with piece-wise linear activation function and the dropout is only at the last hidden layer, we then theoretically derive the Hessian and the covariance of dropout randomness, where these two quantities are very similar. This similarity may be a key reason accounting for the goodness of dropout.

cs.LG↗

arXiv2022

Embedding Principle of Loss Landscape of Deep Neural Networks

Understanding the structure of loss landscape of deep neural networks (DNNs)is obviously important. In this work, we prove an embedding principle that the loss landscape of a DNN "contains" all the critical points of all the narrower DNNs. More precisely, we propose a critical embedding such that any critical point, e.g., local or global minima, of a narrower DNN can be embedded to a critical point/hyperplane of the target DNN with higher degeneracy and preserving the DNN output function. The embedding structure of critical points is independent of loss function and training data, showing a stark difference from other nonconvex problems such as protein-folding. Empirically, we find that a wide DNN is often attracted by highly-degenerate critical points that are embedded from narrow DNNs. The embedding principle provides an explanation for the general easy optimization of wide DNNs and unravels a potential implicit low-complexity regularization during the training. Overall, our work provides a skeleton for the study of loss landscape of DNNs and its implication, by which a more exact and comprehensive understanding can be anticipated in the near

cs.LG↗

arXiv2021

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

In this paper, we propose a a machine learning approach via 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. For linear PDEs, we use a DNN to parameterize the Green's function and obtain the neural operator to approximate the solution according to the Green's method. To train the DNN, the empirical risk consists of the mean squared loss with the least square formulation or the variational formulation of the governing equation and boundary conditions. For complicated problems, the empirical risk also includes a few labels, which are computed on coarse grid points with cheap computation cost and significantly improves the model accuracy. Intuitively, the labeled dataset works as a regularization in addition to the model constraints. The MOD-Net solves a family of PDEs rather than a specific one and is much more efficient than original neural operator because few expensive labels are required. We numerically show MOD-Net is very efficient in solving Poisson equation and one-dimensional radiative transfer equation. For nonlinear PDEs, the nonlinear MOD-Net can be similarly used as an ansatz for solving nonlinear PDEs, exemplified by solving several nonlinear PDE problems, such as the Burgers equation.

math.NA↗

arXiv2021

Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEs

While deep learning algorithms demonstrate a great potential in scientific computing, its application to multi-scale problems remains to be a big challenge. This is manifested by the "frequency principle" that neural networks tend to learn low frequency components first. Novel architectures such as multi-scale deep neural network (MscaleDNN) were proposed to alleviate this problem to some extent. In this paper, we construct a subspace decomposition based DNN (dubbed SD$^2$NN) architecture for a class of multi-scale problems by combining traditional numerical analysis ideas and MscaleDNN algorithms. The proposed architecture includes one low frequency normal DNN submodule, and one (or a few) high frequency MscaleDNN submodule(s), which are designed to capture the smooth part and the oscillatory part of the multi-scale solutions, respectively. In addition, a novel trigonometric activation function is incorporated in the SD$^2$NN model. We demonstrate the performance of the SD$^2$NN architecture through several benchmark multi-scale problems in regular or irregular geometric domains. Numerical results show that the SD$^2$NN model is superior to existing models such as MscaleDNN.

cs.LG↗

arXiv2021

Embedding Principle: a hierarchical structure of loss landscape of deep neural networks

We prove a general Embedding Principle of loss landscape of deep neural networks (NNs) that unravels a hierarchical structure of the loss landscape of NNs, i.e., loss landscape of an NN contains all critical points of all the narrower NNs. This result is obtained by constructing a class of critical embeddings which map any critical point of a narrower NN to a critical point of the target NN with the same output function. By discovering a wide class of general compatible critical embeddings, we provide a gross estimate of the dimension of critical submanifolds embedded from critical points of narrower NNs. We further prove an irreversiblility property of any critical embedding that the number of negative/zero/positive eigenvalues of the Hessian matrix of a critical point may increase but never decrease as an NN becomes wider through the embedding. Using a special realization of general compatible critical embedding, we prove a stringent necessary condition for being a "truly-bad" critical point that never becomes a strict-saddle point through any critical embedding. This result implies the commonplace of strict-saddle points in wide NNs, which may be an important reason underlying the easy optimization of wide NNs widely observed in practice.

cs.LG↗

arXiv2021

Data-informed Deep Optimization

Complex design problems are common in the scientific and industrial fields. In practice, objective functions or constraints of these problems often do not have explicit formulas, and can be estimated only at a set of sampling points through experiments or simulations. Such optimization problems are especially challenging when design parameters are high-dimensional due to the curse of dimensionality. In this work, we propose a data-informed deep optimization (DiDo) approach as follows: first, we use a deep neural network (DNN) classifier to learn the feasible region; second, we sample feasible points based on the DNN classifier for fitting of the objective function; finally, we find optimal points of the DNN-surrogate optimization problem by gradient descent. To demonstrate the effectiveness of our DiDo approach, we consider a practical design case in industry, in which our approach yields good solutions using limited size of training data. We further use a 100-dimension toy example to show the effectiveness of our model for higher dimensional problems. Our results indicate that the DiDo approach empowered by DNN is flexible and promising for solving general high-dimensional design problems in practice.

math.OC↗

arXiv2021

Force-in-domain GAN inversion

Empirical works suggest that various semantics emerge in the latent space of Generative Adversarial Networks (GANs) when being trained to generate images. To perform real image editing, it requires an accurate mapping from the real image to the latent space to leveraging these learned semantics, which is important yet difficult. An in-domain GAN inversion approach is recently proposed to constraint the inverted code within the latent space by forcing the reconstructed image obtained from the inverted code within the real image space. Empirically, we find that the inverted code by the in-domain GAN can deviate from the latent space significantly. To solve this problem, we propose a force-in-domain GAN based on the in-domain GAN, which utilizes a discriminator to force the inverted code within the latent space. The force-in-domain GAN can also be interpreted by a cycle-GAN with slight modification. Extensive experiments show that our force-in-domain GAN not only reconstructs the target image at the pixel level, but also align the inverted code with the latent space well for semantic editing.

cs.CV↗

arXiv2021

Linear Frequency Principle Model to Understand the Absence of Overfitting in Neural Networks

Why heavily parameterized neural networks (NNs) do not overfit the data is an important long standing open question. We propose a phenomenological model of the NN training to explain this non-overfitting puzzle. Our linear frequency principle (LFP) model accounts for a key dynamical feature of NNs: they learn low frequencies first, irrespective of microscopic details. Theory based on our LFP model shows that low frequency dominance of target functions is the key condition for the non-overfitting of NNs and is verified by experiments. Furthermore, through an ideal two-layer NN, we unravel how detailed microscopic NN training dynamics statistically gives rise to a LFP model with quantitative prediction power.

cs.LG↗

arXiv2021

Frequency Principle in Deep Learning Beyond Gradient-descent-based Training

Frequency perspective recently makes progress in understanding deep learning. It has been widely verified in both empirical and theoretical studies that deep neural networks (DNNs) often fit the target function from low to high frequency, namely Frequency Principle (F-Principle). F-Principle sheds light on the strength and the weakness of DNNs and inspires a series of subsequent works, including theoretical studies, empirical studies and the design of efficient DNN structures etc. Previous works examine the F-Principle in gradient-descent-based training. It remains unclear whether gradient-descent-based training is a necessary condition for the F-Principle. In this paper, we show that the F-Principle exists stably in the training process of DNNs with non-gradient-descent-based training, including optimization algorithms with gradient information, such as conjugate gradient and BFGS, and algorithms without gradient information, such as Powell's method and Particle Swarm Optimization. These empirical studies show the universality of the F-Principle and provide hints for further study of F-Principle.

cs.LG↗

arXiv2020

Deep frequency principle towards understanding why deeper learning is faster

Understanding the effect of depth in deep learning is a critical problem. In this work, we utilize the Fourier analysis to empirically provide a promising mechanism to understand why feedforward deeper learning is faster. To this end, we separate a deep neural network, trained by normal stochastic gradient descent, into two parts during analysis, i.e., a pre-condition component and a learning component, in which the output of the pre-condition one is the input of the learning one. We use a filtering method to characterize the frequency distribution of a high-dimensional function. Based on experiments of deep networks and real dataset, we propose a deep frequency principle, that is, the effective target function for a deeper hidden layer biases towards lower frequency during the training. Therefore, the learning component effectively learns a lower frequency function if the pre-condition component has more layers. Due to the well-studied frequency principle, i.e., deep neural networks learn lower frequency functions faster, the deep frequency principle provides a reasonable explanation to why deeper learning is faster. We believe these empirical studies would be valuable for future theoretical studies of the effect of depth in deep learning.

cs.LG↗

arXiv2020

Implicit bias with Ritz-Galerkin method in understanding deep learning for solving PDEs

This paper aims at studying the difference between Ritz-Galerkin (R-G) method and deep neural network (DNN) method in solving partial differential equations (PDEs) to better understand deep learning. To this end, we consider solving a particular Poisson problem, where the information of the right-hand side of the equation f is only available at n sample points, that is, f is known at finite sample points. Through both theoretical and numerical studies, we show that solution of the R-G method converges to a piecewise linear function for the one dimensional (1D) problem or functions of lower regularity for high dimensional problems. With the same setting, DNNs however learn a relative smooth solution regardless of the dimension, this is, DNNs implicitly bias towards functions with more low-frequency components among all functions that can fit the equation at available data points. This bias is explained by the recent study of frequency principle (Xu et al., (2019) [17] and Zhang et al., (2019) [11, 19]). In addition to the similarity between the traditional numerical methods and DNNs in the approximation perspective, our work shows that the implicit bias in the learning process, which is different from traditional numerical methods, could help better understand the characteristics of DNNs.

math.NA↗

arXiv2020

Fourier-domain Variational Formulation and Its Well-posedness for Supervised Learning

A supervised learning problem is to find a function in a hypothesis function space given values on isolated data points. Inspired by the frequency principle in neural networks, we propose a Fourier-domain variational formulation for supervised learning problem. This formulation circumvents the difficulty of imposing the constraints of given values on isolated data points in continuum modelling. Under a necessary and sufficient condition within our unified framework, we establish the well-posedness of the Fourier-domain variational problem, by showing a critical exponent depending on the data dimension. In practice, a neural network can be a convenient way to implement our formulation, which automatically satisfies the well-posedness condition.

math.NA↗

arXiv2020

A multi-scale DNN algorithm for nonlinear elliptic equations with multiple scales

Algorithms based on deep neural networks (DNNs) have attracted increasing attention from the scientific computing community. DNN based algorithms are easy to implement, natural for nonlinear problems, and have shown great potential to overcome the curse of dimensionality. In this work, we utilize the multi-scale DNN-based algorithm (MscaleDNN) proposed by Liu, Cai and Xu (2020) to solve multi-scale elliptic problems with possible nonlinearity, for example, the p-Laplacian problem. We improve the MscaleDNN algorithm by a smooth and localized activation function. Several numerical examples of multi-scale elliptic problems with separable or non-separable scales in low-dimensional and high-dimensional Euclidean spaces are used to demonstrate the effectiveness and accuracy of the MscaleDNN numerical scheme.

physics.comp-ph↗