Qifeng Liao

Qifeng Liao, Jinglai Li

In Bayesian inverse problems sampling the posterior distribution is often a challenging task when the underlying models are computationally intensive. To this end, surrogates or reduced models are often used to accelerate the computation. However, in many practical problems, the parameter of interest can be of high dimensionality, which renders standard model reduction techniques infeasible. In this paper, we present an approach that employs the ANOVA decomposition method to reduce the model with respect to the unknown parameters, and the reduced basis method to reduce the model with respect to the physical parameters. Moreover, we provide an adaptive scheme within the MCMC iterations, to perform the ANOVA decomposition with respect to the posterior distribution. With numerical examples, we demonstrate that the proposed model reduction method can significantly reduce the computational cost of Bayesian inverse problems, without sacrificing much accuracy.

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.

Zhihang Xu, Yingzhi Xia, Qifeng Liao

Bayesian inverse problems are often computationally challenging when the forward model is governed by complex partial differential equations (PDEs). This is typically caused by expensive forward model evaluations and high-dimensional parameterization of priors. This paper proposes a domain-decomposed variational auto-encoder Markov chain Monte Carlo (DD-VAE-MCMC) method to tackle these challenges simultaneously. Through partitioning the global physical domain into small subdomains, the proposed method first constructs local deterministic generative models based on local historical data, which provide efficient local prior representations. Gaussian process models with active learning address the domain decomposition interface conditions. Then inversions are conducted on each subdomain independently in parallel and in low-dimensional latent parameter spaces. The local inference solutions are post-processed through the Poisson image blending procedure to result in an efficient global inference result. Numerical examples are provided to demonstrate the performance of the proposed method.

Kejun Tang, Qifeng Liao

This work proposes a systematic model reduction approach based on rank adaptive tensor recovery for partial differential equation (PDE) models with high-dimensional random parameters. Since the standard outputs of interest of these models are discrete solutions on given physical grids which are high-dimensional, we use kernel principal component analysis to construct stochastic collocation approximations in reduced dimensional spaces of the outputs. To address the issue of high-dimensional random inputs, we develop a new efficient rank adaptive tensor recovery approach to compute the collocation coefficients. Novel efficient initialization strategies for non-convex optimization problems involved in tensor recovery are also developed in this work. We present a general mathematical framework of our overall model reduction approach, analyze its stability, and demonstrate its efficiency with numerical experiments.

Junjie He, Zhihang Xu, Qifeng Liao

In this paper, we present a deep neural network based adaptive learning (DNN-AL) approach for switched systems. Currently, deep neural network based methods are actively developed for learning governing equations in unknown dynamic systems, but their efficiency can degenerate for switching systems, where structural changes exist at discrete time instants. In this new DNN-AL strategy, observed datasets are adaptively decomposed into subsets, such that no structural changes within each subset. During the adaptive procedures, DNNs are hierarchically constructed, and unknown switching time instants are gradually identified. Especially, network parameters at previous iteration steps are reused to initialize networks for the later iteration steps, which gives efficient training procedures for the DNNs. For the DNNs obtained through our DNN-AL, bounds of the prediction error are established. Numerical studies are conducted to demonstrate the efficiency of DNN-AL.

Yunyu Huang, Yani Feng, Qifeng Liao

In this paper, we study a Bayesian tensor train (TT) decomposition method to recover streaming data by approximating the latent structure in high-order streaming data. Drawing on the streaming variational Bayes method, we introduce the TT format into Bayesian tensor decomposition methods for streaming data, and formulate posteriors of TT cores. Thanks to the Bayesian framework of the TT format, the proposed algorithm (SPTT) excels in recovering streaming data with high-order, incomplete, and noisy properties. The experiments in synthetic and real-world datasets show the accuracy of our method compared to state-of-the-art Bayesian tensor decomposition methods for streaming data.

Xiaolong Wu, Qifeng Liao

Solving high-dimensional Fokker-Planck (FP) equations is a challenge in computational physics and stochastic dynamics, due to the curse of dimensionality (CoD) and the bottleneck of evaluating second-order diffusion terms. Existing deep learning approaches, such as Physics-Informed Neural Networks (PINNs), face computational challenges as dimensionality increases, driven by the $O(D^2)$ complexity of automatic differentiation for second-order derivatives. While recent probability flow approaches bypass this by learning score functions or matching velocity fields, they often involve serial computational operations or depend on sampling efficiency in complex distributions. To address these issues, we propose the Self-Consistent Probability Flow (SCPF) method. We reformulate the second-order FP equation into an equivalent first-order deterministic Probability Flow ODE (PF-ODE) constraint. Unlike score matching or velocity matching, SCPF solves this problem by minimizing the residual of the PF-ODE continuity equation, which avoids explicit Hessian computation. We leverage Continuous Normalizing Flows (CNF) combined with the Hutchinson Trace Estimator (HTE) to reduce the training complexity to linear scale $O(D)$, achieving an effective $O(1)$ wall-clock time on GPUs. To address data sparsity in high dimensions, we apply a generative adaptive sampling strategy and theoretically prove that dynamically aligning collocation points with the evolving probability mass is a necessary condition to bound the approximation error. Experiments on diverse benchmarks -- ranging from anisotropic Ornstein-Uhlenbeck (OU) processes and high-dimensional Brownian motions with time-varying diffusion terms, to Geometric OU processes featuring non-Gaussian solutions -- demonstrate that SCPF effectively mitigates the CoD, maintaining high accuracy and constant computational cost for problems up to 100 dimensions.

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.

Yani Feng, Kejun Tang, Lianxing He et al.

This work proposes a novel tensor train random projection (TTRP) method for dimension reduction, where pairwise distances can be approximately preserved. Our TTRP is systematically constructed through a tensor train (TT) representation with TT-ranks equal to one. Based on the tensor train format, this new random projection method can speed up the dimension reduction procedure for high-dimensional datasets and requires less storage costs with little loss in accuracy, compared with existing methods. We provide a theoretical analysis of the bias and the variance of TTRP, which shows that this approach is an expected isometric projection with bounded variance, and we show that the Rademacher distribution is an optimal choice for generating the corresponding TT-cores. Detailed numerical experiments with synthetic datasets and the MNIST dataset are conducted to demonstrate the efficiency of TTRP.

Ke Li, Kejun Tang, Tianfan Wu et al.

A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization problem, and we design a multi-fidelity neural network framework to solve this optimization problem. Our contribution is to develop a systematical computational procedure for the underlying problem in parallel with domain decomposition. Our analysis shows that the D3M approximation solution converges to the exact solution of underlying PDEs. Our proposed framework establishes a foundation to use variational deep learning in large-scale engineering problems and designs. We present a general mathematical framework of D3M, validate its accuracy and demonstrate its efficiency with numerical experiments.