Xiaoliang Wan

Li Zeng, Xiaoliang Wan, Tao Zhou

In this work, we propose adaptive deep learning approaches based on normalizing flows for solving fractional Fokker-Planck equations (FPEs). The solution of a FPE is a probability density function (PDF). Traditional mesh-based methods are ineffective because of the unbounded computation domain, a large number of dimensions and the nonlocal fractional operator. To this end, we represent the solution with an explicit PDF model induced by a flow-based deep generative model, simplified KRnet, which constructs a transport map from a simple distribution to the target distribution. We consider two methods to approximate the fractional Laplacian. One method is the Monte Carlo approximation. The other method is to construct an auxiliary model with Gaussian radial basis functions (GRBFs) to approximate the solution such that we may take advantage of the fact that the fractional Laplacian of a Gaussian is known analytically. Based on these two different ways for the approximation of the fractional Laplacian, we propose two models, MCNF and GRBFNF, to approximate stationary FPEs and MCTNF to approximate time-dependent FPEs. To further improve the accuracy, we refine the training set and the approximate solution alternately. A variety of numerical examples is presented to demonstrate the effectiveness of our adaptive deep density approaches.

Xiu Yang, Xiaoliang Wan, Lin Lin et al.

Compressive sensing has become a powerful addition to uncertainty quantification when only limited data is available. In this paper we provide a general framework to enhance the sparsity of the representation of uncertainty in the form of generalized polynomial chaos expansion. We use alternating direction method to identify new sets of random variables through iterative rotations such that the new representation of the uncertainty is sparser. Consequently, we increases both the efficiency and accuracy of the compressive sensing-based uncertainty quantification method. We demonstrate that the previously developed iterative method to enhance the sparsity of Hermite polynomial expansion is a special case of this general framework. Moreover, we use Legendre and Chebyshev polynomials expansions to demonstrate the effectiveness of this method with applications in solving stochastic partial differential equations and high-dimensional (O(100)) problems.

Yani Feng, Kejun Tang, Xiaoliang Wan et al.

We present a dimension-reduced KRnet map approach (DR-KRnet) for high-dimensional Bayesian inverse problems, which is based on an explicit construction of a map that pushes forward the prior measure to the posterior measure in the latent space. Our approach consists of two main components: data-driven VAE prior and density approximation of the posterior of the latent variable. In reality, it may not be trivial to initialize a prior distribution that is consistent with available prior data; in other words, the complex prior information is often beyond simple hand-crafted priors. We employ variational autoencoder (VAE) to approximate the underlying distribution of the prior dataset, which is achieved through a latent variable and a decoder. Using the decoder provided by the VAE prior, we reformulate the problem in a low-dimensional latent space. In particular, we seek an invertible transport map given by KRnet to approximate the posterior distribution of the latent variable. Moreover, an efficient physics-constrained surrogate model without any labeled data is constructed to reduce the computational cost of solving both forward and adjoint problems involved in likelihood computation. With numerical experiments, we demonstrate the accuracy and efficiency of DR-KRnet for high-dimensional Bayesian inverse problems.

Xiaoliang Wan, Haijun Yu

In this work, we consider a non-standard preconditioning strategy for the numerical approximation of the classical elliptic equations with log-normal random coefficients. In \cite{Wan_model}, a Wick-type elliptic model was proposed by modeling the random flux through the Wick product. Due to the lower-triangular structure of the uncertainty propagator, this model can be approximated efficiently using the Wiener chaos expansion in the probability space. Such a Wick-type model provides, in general, a second-order approximation of the classical one in terms of the standard deviation of the underlying Gaussian process. Furthermore, when the correlation length of the underlying Gaussian process goes to infinity, the Wick-type model yields the same solution as the classical one. These observations imply that the Wick-type elliptic equation can provide an effective preconditioner for the classical random elliptic equation under appropriate conditions. We use the Wick-type elliptic model to accelerate the Monte Carlo method and the stochastic Galerkin finite element method. Numerical results are presented and discussed.

Xiaoliang Wan, Tao Zhou, Yuancheng Zhou

We introduce an adaptive sampling method for the Deep Ritz method aimed at solving partial differential equations (PDEs). Two deep neural networks are used. One network is employed to approximate the solution of PDEs, while the other one is a deep generative model used to generate new collocation points to refine the training set. The adaptive sampling procedure consists of two main steps. The first step is solving the PDEs using the Deep Ritz method by minimizing an associated variational loss discretized by the collocation points in the training set. The second step involves generating a new training set, which is then used in subsequent computations to further improve the accuracy of the current approximate solution. We treat the integrand in the variational loss as an unnormalized probability density function (PDF) and approximate it using a deep generative model called bounded KRnet. The new samples and their associated PDF values are obtained from the bounded KRnet. With these new samples and their associated PDF values, the variational loss can be approximated more accurately by importance sampling. Compared to the original Deep Ritz method, the proposed adaptive method improves accuracy, especially for problems characterized by low regularity and high dimensionality. We demonstrate the effectiveness of our new method through a series of numerical experiments.

Xiaodong Feng, Haojiong Shangguan, Tao Tang et al.

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based formulation where the neural network is defined on a spatiotemporal domain, our methodology utilizes finite element basis functions in the time direction where the space-dependent coefficients are defined as the output of a neural network. We then apply the Galerkin or collocation projection in the time direction to obtain a system of PDEs for the space-dependent coefficients which is approximated in the framework of PINN. The advantages of such a hybrid formulation are twofold: statistical errors are avoided for the integral in the time direction, and the neural network's output can be regarded as a set of reduced spatial basis functions. To further alleviate the difficulties from high dimensionality and low regularity, we have developed an adaptive sampling strategy that refines the training set. More specifically, we use an explicit density model to approximate the distribution induced by the PDE residual and then augment the training set with new time-dependent random samples given by the learned density model. The effectiveness and efficiency of our proposed method have been demonstrated through a series of numerical experiments.

Xiaodong Feng, Haojiong Shangguan, Tao Tang et al.

Evolution equations, including both ordinary differential equations (ODEs) and partial differential equations (PDEs), play a pivotal role in modeling dynamic systems. However, achieving accurate long-time integration for these equations remains a significant challenge. While physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDEs, they often suffer from temporal error accumulation, which limits their effectiveness in capturing long-time behaviors. To alleviate this issue, we propose integral regularization PINNs (IR-PINNs), a novel approach that enhances temporal accuracy by incorporating an integral-based residual term into the loss function. This method divides the entire time interval into smaller sub-intervals and enforces constraints over these sub-intervals, thereby improving the resolution and correlation of temporal dynamics. Furthermore, IR-PINNs leverage adaptive sampling to dynamically refine the distribution of collocation points based on the evolving solution, ensuring higher accuracy in regions with sharp gradients or rapid variations. Numerical experiments on benchmark problems demonstrate that IR-PINNs outperform original PINNs and other state-of-the-art methods in capturing long-time behaviors, offering a robust and accurate solution for evolution equations.

Yueyang Wang, Kejun Tang, Xili Wang et al.

The committor functions are central to investigating rare but important events in molecular simulations. It is known that computing the committor function suffers from the curse of dimensionality. Recently, using neural networks to estimate the committor function has gained attention due to its potential for high-dimensional problems. Training neural networks to approximate the committor function needs to sample transition data from straightforward simulations of rare events, which is very inefficient. The scarcity of transition data makes it challenging to approximate the committor function. To address this problem, we propose an efficient framework to generate data points in the transition state region that helps train neural networks to approximate the committor function. We design a Deep Adaptive Sampling method for TRansition paths (DASTR), where deep generative models are employed to generate samples to capture the information of transitions effectively. In particular, we treat a non-negative function in the integrand of the loss functional as an unnormalized probability density function and approximate it with the deep generative model. The new samples from the deep generative model are located in the transition state region and fewer samples are located in the other region. This distribution provides effective samples for approximating the committor function and significantly improves the accuracy. We demonstrate the effectiveness of the proposed method through both simulations and realistic examples.

Kejun Tang, Jiayu Zhai, Xiaoliang Wan et al.

Solving partial differential equations (PDEs) is a central task in scientific computing. Recently, neural network approximation of PDEs has received increasing attention due to its flexible meshless discretization and its potential for high-dimensional problems. One fundamental numerical difficulty is that random samples in the training set introduce statistical errors into the discretization of loss functional which may become the dominant error in the final approximation, and therefore overshadow the modeling capability of the neural network. In this work, we propose a new minmax formulation to optimize simultaneously the approximate solution, given by a neural network model, and the random samples in the training set, provided by a deep generative model. The key idea is to use a deep generative model to adjust random samples in the training set such that the residual induced by the approximate PDE solution can maintain a smooth profile when it is being minimized. Such an idea is achieved by implicitly embedding the Wasserstein distance between the residual-induced distribution and the uniform distribution into the loss, which is then minimized together with the residual. A nearly uniform residual profile means that its variance is small for any normalized weight function such that the Monte Carlo approximation error of the loss functional is reduced significantly for a certain sample size. The adversarial adaptive sampling (AAS) approach proposed in this work is the first attempt to formulate two essential components, minimizing the residual and seeking the optimal training set, into one minmax objective functional for the neural network approximation of PDEs.

Li Zeng, Xiaoliang Wan, Tao Zhou

In this paper, we develop an invertible mapping, called B-KRnet, on a bounded domain and apply it to density estimation/approximation for data or the solutions of PDEs such as the Fokker-Planck equation and the Keller-Segel equation. Similar to KRnet, B-KRnet consists of a series of coupling layers with progressively fewer active transformation dimensions, inspired by the triangular structure of the Knothe-Rosenblatt (KR) rearrangement. The main difference between B-KRnet and KRnet is that B-KRnet is defined on a hypercube while KRnet is defined on the whole space, in other words, a new mechanism is introduced in B-KRnet to maintain the exact invertibility. Using B-KRnet as a transport map, we obtain an explicit probability density function (PDF) model that corresponds to the pushforward of a base (uniform) distribution on the hypercube. It can be directly applied to density estimation when only data are available. By coupling KRnet and B-KRnet, we define a deep generative model on a high-dimensional domain where some dimensions are bounded and other dimensions are unbounded. A typical case is the solution of the stationary kinetic Fokker-Planck equation, which is a PDF of position and momentum. Based on B-KRnet, we develop an adaptive learning approach to approximate partial differential equations whose solutions are PDFs or can be treated as PDFs. A variety of numerical experiments is presented to demonstrate the effectiveness of B-KRnet.

Kejun Tang, Xiaoliang Wan, Chao Yang

In this work we propose a deep adaptive sampling (DAS) method for solving partial differential equations (PDEs), where deep neural networks are utilized to approximate the solutions of PDEs and deep generative models are employed to generate new collocation points that refine the training set. The overall procedure of DAS consists of two components: solving the PDEs by minimizing the residual loss on the collocation points in the training set and generating a new training set to further improve the accuracy of current approximate solution. In particular, we treat the residual as a probability density function and approximate it with a deep generative model, called KRnet. The new samples from KRnet are consistent with the distribution induced by the residual, i.e., more samples are located in the region of large residual and less samples are located in the region of small residual. Analogous to classical adaptive methods such as the adaptive finite element, KRnet acts as an error indicator that guides the refinement of the training set. Compared to the neural network approximation obtained with uniformly distributed collocation points, the developed algorithms can significantly improve the accuracy, especially for low regularity and high-dimensional problems. We demonstrate the effectiveness of the proposed DAS method with numerical experiments.

Xiaoliang Wan, Kejun Tang

In this work, we have proposed augmented KRnets including both discrete and continuous models. One difficulty in flow-based generative modeling is to maintain the invertibility of the transport map, which is often a trade-off between effectiveness and robustness. The exact invertibility has been achieved in the real NVP using a specific pattern to exchange information between two separated groups of dimensions. KRnet has been developed to enhance the information exchange among data dimensions by incorporating the Knothe-Rosenblatt rearrangement into the structure of the transport map. Due to the maintenance of exact invertibility, a full nonlinear update of all data dimensions needs three iterations in KRnet. To alleviate this issue, we will add augmented dimensions that act as a channel for communications among the data dimensions. In the augmented KRnet, a fully nonlinear update is achieved in two iterations. We also show that the augmented KRnet can be reformulated as the discretization of a neural ODE, where the exact invertibility is kept such that the adjoint method can be formulated with respect to the discretized ODE to obtain the exact gradient. Numerical experiments have been implemented to demonstrate the effectiveness of our models.

Kejun Tang, Xiaoliang Wan, Qifeng Liao

In this paper we present an adaptive deep density approximation strategy based on KRnet (ADDA-KR) for solving the steady-state Fokker-Planck (F-P) equations. F-P equations are usually high-dimensional and defined on an unbounded domain, which limits the application of traditional grid based numerical methods. With the Knothe-Rosenblatt rearrangement, our newly proposed flow-based generative model, called KRnet, provides a family of probability density functions to serve as effective solution candidates for the Fokker-Planck equations, which has a weaker dependence on dimensionality than traditional computational approaches and can efficiently estimate general high-dimensional density functions. To obtain effective stochastic collocation points for the approximation of the F-P equation, we develop an adaptive sampling procedure, where samples are generated iteratively using the approximate density function at each iteration. We present a general framework of ADDA-KR, validate its accuracy and demonstrate its efficiency with numerical experiments.

Xiaoliang Wan, Shuangqing Wei

In this work, we have proposed a generative model, called VAE-KRnet, for density estimation or approximation, which combines the canonical variational autoencoder (VAE) with our recently developed flow-based generative model, called KRnet. VAE is used as a dimension reduction technique to capture the latent space, and KRnet is used to model the distribution of the latent variable. Using a linear model between the data and the latent variable, we show that VAE-KRnet can be more effective and robust than the canonical VAE. VAE-KRnet can be used as a density model to approximate either data distribution or an arbitrary probability density function (PDF) known up to a constant. VAE-KRnet is flexible in terms of dimensionality. When the number of dimensions is relatively small, KRnet can effectively approximate the distribution in terms of the original random variable. For high-dimensional cases, we may use VAE-KRnet to incorporate dimension reduction. One important application of VAE-KRnet is the variational Bayes for the approximation of the posterior distribution. The variational Bayes approaches are usually based on the minimization of the Kullback-Leibler (KL) divergence between the model and the posterior. For high-dimensional distributions, it is very challenging to construct an accurate density model due to the curse of dimensionality, where extra assumptions are often introduced for efficiency. For instance, the classical mean-field approach assumes mutual independence between dimensions, which often yields an underestimated variance due to oversimplification. To alleviate this issue, we include into the loss the maximization of the mutual information between the latent random variable and the original random variable, which helps keep more information from the region of low density such that the estimation of variance is improved.

Xiaoliang Wan, Shuangqing Wei

In this work, we develop an importance sampling estimator by coupling the reduced-order model and the generative model in a problem setting of uncertainty quantification. The target is to estimate the probability that the quantity of interest (QoI) in a complex system is beyond a given threshold. To avoid the prohibitive cost of sampling a large scale system, the reduced-order model is usually considered for a trade-off between efficiency and accuracy. However, the Monte Carlo estimator given by the reduced-order model is biased due to the error from dimension reduction. To correct the bias, we still need to sample the fine model. An effective technique to reduce the variance reduction is importance sampling, where we employ the generative model to estimate the distribution of the data from the reduced-order model and use it for the change of measure in the importance sampling estimator. To compensate the approximation errors of the reduced-order model, more data that induce a slightly smaller QoI than the threshold need to be included into the training set. Although the amount of these data can be controlled by a posterior error estimate, redundant data, which may outnumber the effective data, will be kept due to the epistemic uncertainty. To deal with this issue, we introduce a weighted empirical distribution to process the data from the reduced-order model. The generative model is then trained by minimizing the cross entropy between it and the weighted empirical distribution. We also introduce a penalty term into the objective function to deal with the overfitting for more robustness. Numerical results are presented to demonstrate the effectiveness of the proposed methodology.