arXiv subjects

Lihong Feng

Publications and source records attributed to Lihong Feng.

At least 19 recordsLinked to original sources

arXiv2026

Surrogate modeling for convection-dominated parametric problems based on error learning

Convection-dominated problems are known for their slow Kolmogorov $n$-width decays and are challenging for model order reduction (MOR). In this work, we propose a hybrid surrogate modeling approach and a non-intrusive variant that overcome some drawbacks of linear MOR methods. The proposed hybrid surrogate model is a projection-based reduced-order model (ROM), corrected by the error learned from a deep neural network. With the aid of deep learning, the model component of the surrogate model can be kept in a small reduced dimension. The neural network component and the model component are sequentially but separately built during the offline stage. At the online stage, they are easily coupled to output the solution predictions. Due to the intrusive nature of the hybrid-ROM, the numerically discretized operators of the original model must be available. For problems solved using black-box solvers, where the details of the numerical discretization are not accessible, we further propose a non-intrusive variant of the hybrid surrogate. Compared to the existing MOR methods with nonlinear manifolds, the proposed hybrid ROM is more easily built and is also easily assembled for online prediction. In contrast to the surrogate modeling approaches purely based on deep-learning, the proposed non-intrusive variant has a lighter neural network structure with much fewer parameters to be learned. We test the proposed methods on two nonlinear convection parametric problems. The first is the 1D inviscid Burgers' equation with one parameter, and the second is the 2D inviscid Burgers' equation with two parameters. Since both methods are based on error correction, their online predictions exhibit higher accuracy yet with largely reduced prediction time, compared to state-of-the-art methods.

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arXiv2025

Subspace-Distance-Enabled Active Learning for Efficient Data-Driven Model Reduction of Parametric Dynamical Systems

In situations where the solution of a high-fidelity dynamical system needs to be evaluated repeatedly, over a vast pool of parametric configurations and in absence of access to the underlying governing equations, data-driven model reduction techniques are preferable. We propose a novel active learning approach to build a parametric data-driven reduced-order model (ROM) by greedily picking the most important parameter samples from the parameter domain. As a result, during the ROM construction phase, the number of high-fidelity solutions dynamically grow in a principled fashion. The high-fidelity solution snapshots are expressed in several parameter-specific linear subspaces, with the help of proper orthogonal decomposition (POD), and the relative distance between these subspaces is used as a guiding mechanism to perform active learning. For successfully achieving this, we provide a distance measure to evaluate the similarity between pairs of linear subspaces with different dimensions, and also show that this distance measure is a metric. The usability of the proposed subspace-distance-enabled active learning (SDE-AL) framework is demonstrated by augmenting two existing non-intrusive reduced-order modeling approaches, and providing their active-learning-driven (ActLearn) extensions, namely, SDE-ActLearn-POD-KSNN, and SDE-ActLearn-POD-NN. Furthermore, we report positive results for two parametric physical models, highlighting the efficiency of the proposed SDE-AL approach.

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arXiv2025

Spatially-Enhanced Temporal Fusion Transformer: Interpretable Multi-Output Prediction for Parametric Dynamical Systems with Time-Varying Inputs

We explore the promising performance of a transformer model in predicting outputs of parametric dynamical systems with external time-varying input signals. The outputs of such systems vary not only with physical parameters but also with external time-varying input signals. Accurately catching the dynamics of such systems is challenging. We have adapted and extended an existing transformer model, called temporal fusion transformer (TFT), for single-output prediction to a multiple-output transformer, named as Spatially-Enhanced Temporal Fusion Transformer (SE-TFT), which is able to predict multiple output responses of these systems. The SE-TFT generalizes the interpretability of the original TFT model. The generalized interpretable attention weight matrix explores not only the temporal correlations in the sequence, but also the interactions between the multiple outputs, providing explanation for the spatial correlation in the output domain. This proposed SE-TFT accurately predicts the sequence of multiple outputs, regardless of the nonlinearity of the system and the dimensionality of the parameter space.

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arXiv2024

Discrete empirical interpolation in the tensor t-product framework

The discrete empirical interpolation method (DEIM) is a well-established approach, widely used for state reconstruction using sparse sensor/measurement data, nonlinear model reduction, and interpretable feature selection. We introduce the tensor t-product Q-DEIM (t-Q-DEIM), an extension of the DEIM framework for dealing with tensor-valued data. The proposed approach seeks to overcome one of the key drawbacks of DEIM, viz., the need for matricizing the data, which can distort any structural and/or geometric information. Our method leverages the recently developed tensor t-product algebra to avoid reshaping the data. In analogy with the standard DEIM, we formulate and solve a tensor-valued least-squares problem, whose solution is achieved through an interpolatory projection. We develop a rigorous, computable upper bound for the error resulting from the t-Q-DEIM approximation. Using five different tensor-valued datasets, we numerically illustrate the better approximation properties of t-Q-DEIM and the significant computational cost reduction it offers.

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arXiv2024

Data-Augmented Predictive Deep Neural Network: Enhancing the extrapolation capabilities of non-intrusive surrogate models

Numerically solving a large parametric nonlinear dynamical system is challenging due to its high complexity and the high computational costs. In recent years, machine-learning-aided surrogates are being actively researched. However, many methods fail in accurately generalizing in the entire time interval $[0, T]$, when the training data is available only in a training time interval $[0, T_0]$, with $T_0<T$. To improve the extrapolation capabilities of the surrogate models in the entire time domain, we propose a new deep learning framework, where kernel dynamic mode decomposition (KDMD) is employed to evolve the dynamics of the latent space generated by the encoder part of a convolutional autoencoder (CAE). After adding the KDMD-decoder-extrapolated data into the original data set, we train the CAE along with a feed-forward deep neural network using the augmented data. The trained network can predict future states outside the training time interval at any out-of-training parameter samples. The proposed method is tested on two numerical examples: a FitzHugh-Nagumo model and a model of incompressible flow past a cylinder. Numerical results show accurate and fast prediction performance in both the time and the parameter domain.

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arXiv2024

Estimates of the Kolmogorov n-width for nonlinear transformations with application to distributed-parameter control systems

This paper aims at characterizing the approximability of bounded sets in the range of nonlinear operators in Banach spaces by finite-dimensional linear varieties. In particular, the class of operators we consider includes the endpoint maps of nonlinear distributed-parameter control systems. We describe the relationship between the Kolmogorov n-width of a bounded subset and the width of its image under an essentially nonlinear transformation. We propose explicit estimates of the n-width in the space of images in terms of the affine part of the corresponding operator and the width of its nonlinear perturbation. These $n$-width estimates enable us to describe the reachable sets for infinite-dimensional bilinear control systems, with applications to controlling the Euler-Bernoulli beam using a contraction force and to a single-input Schr\"odinger equation.

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arXiv2023

Fast and Reliable Reduced-Order Models for Cardiac Electrophysiology

Mathematical models of the human heart are increasingly playing a vital role in understanding the working mechanisms of the heart, both under healthy functioning and during disease. The aim is to aid medical practitioners diagnose and treat the many ailments affecting the heart. Towards this, modelling cardiac electrophysiology is crucial as the heart's electrical activity underlies the contraction mechanism and the resulting pumping action. The governing equations and the constitutive laws describing the electrical activity in the heart are coupled, nonlinear, and involve a fast moving wave front, which is generally solved by the finite element method. The simulation of this complex system as part of a virtual heart model is challenging due to the necessity of fine spatial and temporal resolution of the domain. Therefore, efficient surrogate models are needed to predict the dynamics under varying parameters and inputs. In this work, we develop an adaptive, projection-based surrogate model for cardiac electrophysiology. We introduce an a posteriori error estimator that can accurately and efficiently quantify the accuracy of the surrogate model. Using the error estimator, we systematically update our surrogate model through a greedy search of the parameter space. Furthermore, using the error estimator, the parameter search space is dynamically updated such that the most relevant samples get chosen at every iteration. The proposed adaptive surrogate modelling technique is tested on three benchmark models to illustrate its efficiency, accuracy, and ability of generalization.

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arXiv2023

Parametric Dynamic Mode Decomposition for nonlinear parametric dynamical systems

A non-intrusive model order reduction (MOR) method that combines features of the dynamic mode decomposition (DMD) and the radial basis function (RBF) network is proposed to predict the dynamics of parametric nonlinear systems. In many applications, we have limited access to the information of the whole system, which motivates non-intrusive model reduction. One bottleneck is capturing the dynamics of the solution without knowing the physics inside the "black-box" system. DMD is a powerful tool to mimic the dynamics of the system and give a reliable approximation of the solution in the time domain using only the dominant DMD modes. However, DMD cannot reproduce the parametric behavior of the dynamics. Our contribution focuses on extending DMD to parametric DMD by RBF interpolation. Specifically, a RBF network is first trained using snapshot matrices at limited parameter samples. The snapshot matrix at any new parameter sample can be quickly learned from the RBF network. DMD will use the newly generated snapshot matrix at the online stage to predict the time patterns of the dynamics corresponding to the new parameter sample. The proposed framework and algorithm are tested and validated by numerical examples including models with parametrized and time-varying inputs.

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arXiv2023

Accurate error estimation for model reduction of nonlinear dynamical systems via data-enhanced error closure

Accurate error estimation is crucial in model order reduction, both to obtain small reduced-order models and to certify their accuracy when deployed in downstream applications such as digital twins. In existing a posteriori error estimation approaches, knowledge about the time integration scheme is mandatory, e.g., the residual-based error estimators proposed for the reduced basis method. This poses a challenge when automatic ordinary differential equation solver libraries are used to perform the time integration. To address this, we present a data-enhanced approach for a posteriori error estimation. Our new formulation enables residual-based error estimators to be independent of any time integration method. To achieve this, we introduce a corrected reduced-order model which takes into account a data-driven closure term for improved accuracy. The closure term, subject to mild assumptions, is related to the local truncation error of the corresponding time integration scheme. We propose efficient computational schemes for approximating the closure term, at the cost of a modest amount of training data. Furthermore, the new error estimator is incorporated within a greedy process to obtain parametric reduced-order models. Numerical results on three different systems show the accuracy of the proposed error estimation approach and its ability to produce ROMs that generalize well.

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arXiv2023

Active-Learning-Driven Surrogate Modeling for Efficient Simulation of Parametric Nonlinear Systems

When repeated evaluations for varying parameter configurations of a high-fidelity physical model are required, surrogate modeling techniques based on model order reduction are desired. In absence of the governing equations describing the dynamics, we need to construct the parametric reduced-order surrogate model in a non-intrusive fashion. In this setting, the usual residual-based error estimate for optimal parameter sampling associated with the reduced basis method is not directly available. Our work provides a non-intrusive optimality criterion to efficiently populate the parameter snapshots, thereby, enabling us to effectively construct a parametric surrogate model. We consider separate parameter-specific proper orthogonal decomposition (POD) subspaces and propose an active-learning-driven surrogate model using kernel-based shallow neural networks, abbreviated as ActLearn-POD-KSNN surrogate model. To demonstrate the validity of our proposed ideas, we present numerical experiments using two physical models, namely Burgers' equation and shallow water equations. Both the models have mixed -- convective and diffusive -- effects within their respective parameter domains, with each of them dominating in certain regions. The proposed ActLearn-POD-KSNN surrogate model efficiently predicts the solution at new parameter locations, even for a setting with multiple interacting shock profiles.

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arXiv2023

Inf-Sup-Constant-Free State Error Estimator for Model Order Reduction of Parametric Systems in Electromagnetics

A reliable model order reduction process for parametric analysis in electromagnetics is detailed. Special emphasis is placed on certifying the accuracy of the reduced-order model. For this purpose, a sharp state error estimator is proposed. Standard a posteriori state error estimation for model order reduction relies on the inf-sup constant. For parametric systems, the inf-sup constant is parameter-dependent. The a posteriori error estimation for systems with very small or vanishing inf-sup constant poses a challenge, since it is inversely proportional to the inf-sup constant, resulting in rather useless, overly pessimistic error estimators. Such systems appear in electromagnetics since the inf-sup constant values are close to zero at points close to resonant frequencies, where they eventually vanish. We propose a novel a posteriori state error estimator which avoids the calculation of the inf-sup constant. The proposed state error estimator is compared with the standard error estimator and a recently proposed one in the literature. It is shown that our proposed error estimator outperforms both existing estimators. Numerical experiments are performed on real-life microwave devices such as narrowband and wideband antennas, two types of dielectric resonator filters as well as a dual-mode waveguide filter. These examples show the capabilities and efficiency of the proposed methodology.

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arXiv2023

Multi-fidelity error estimation accelerates greedy model reduction of complex dynamical systems

Model order reduction usually consists of two stages: the offline stage and the online stage. The offline stage is the expensive part that sometimes takes hours till the final reduced-order model is derived, especially when the original model is very large or complex. Once the reduced-order model is obtained, the online stage of querying the reduced-order model for simulation is very fast and often real-time capable. This work concerns a strategy to significantly speed up the offline stage of model order reduction for large and complex systems. In particular, it is successful in accelerating the greedy algorithm that is often used in the offline stage for reduced-order model construction. We propose multi-fidelity error estimators and replace the high-fidelity error estimator in the greedy algorithm. Consequently, the computational complexity at each iteration of the greedy algorithm is reduced and the algorithm converges more than 3 times faster without incurring noticeable accuracy loss.

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arXiv2021

Fast A Posteriori State Error Estimation for Reliable Frequency Sweeping in Microwave Circuits via the Reduced-Basis Method

We develop a compact, reliable model order reduction approach for fast frequency sweeps in microwave circuits by means of the reduced-basis method. Contrary to what has been previously done, special emphasis is placed on certifying the accuracy of the reduced-order model with respect to the original full-order model in an effective and efficient way. Previous works on model order reduction accuracy certification rely on costly $\textit{a posteriori}$ error estimators, which typically require expensive $\textit{inf-sup}$ constant evaluations of the underlying full-order model. This scenario is often too time-consuming and unaffordable in electromagnetic applications. As a result, less expensive and heuristic error estimators are commonly used instead. Very often, one is interested in knowing about the full state vector, instead of just some output quantities derived from the full state. Therefore, error estimators for the full state vector become relevant. In this work, we detail the frequency behavior of both the electric field and the state error when an approximation to the electric field solution is carried out. Both field quantities share the same frequency behavior. Based on this observation, we focus on the efficient estimation of the electric field state error and propose a fast evaluation of the reduced-order model state error in the frequency band of analysis, minimizing the number of full-order model evaluations. This methodology is of paramount importance to carry out a reliable fast frequency sweep in microwave circuits. Finally, real-life applications will illustrate the capabilities and efficiency of the proposed approach.

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arXiv2021

A Training Set Subsampling Strategy for the Reduced Basis Method

We present a subsampling strategy for the offline stage of the Reduced Basis Method. The approach is aimed at bringing down the considerable offline costs associated with using a finely-sampled training set. The proposed algorithm exploits the potential of the pivoted QR decomposition and the discrete empirical interpolation method to identify important parameter samples. It consists of two stages. In the first stage, we construct a low-fidelity approximation to the solution manifold over a fine training set. Then, for the available low-fidelity snapshots of the output variable, we apply the pivoted QR decomposition or the discrete empirical interpolation method to identify a set of sparse sampling locations in the parameter domain. These points reveal the structure of the parametric dependence of the output variable. The second stage proceeds with a subsampled training set containing a by far smaller number of parameters than the initial training set. Different subsampling strategies inspired from recent variants of the empirical interpolation method are also considered. Tests on benchmark examples justify the new approach and show its potential to substantially speed up the offline stage of the Reduced Basis Method, while generating reliable reduced-order models.

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arXiv2021

Multivariate moment matching for model order reduction of quadratic-bilinear systems using error bounds

We propose an adaptive moment-matching framework for model order reduction of quadratic-bilinear descriptor systems. In this framework, an important issue is the selection of those shift frequencies where moment-matching is to be achieved. Often, the choice is random or linked to the linear part of the nonlinear system. In this paper, we extend the use of an existing a posteriori error bound for general linear time invariant systems to quadratic-bilinear systems and develop a greedy-type framework to select a good choice of interpolation points for the construction of the projection matrices. The results are compared with standard quadratic-bilinear projection methods and we observe that the approximations obtained by the proposed method yield high accuracy.

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arXiv2021

A FOM/ROM Hybrid Approach for Accelerating Numerical Simulations

The basis generation in reduced order modeling usually requires multiple high-fidelity large-scale simulations that could take a huge computational cost. In order to accelerate these numerical simulations, we introduce a FOM/ROM hybrid approach in this paper. It is developed based on an a posteriori error estimation for the output approximation of the dynamical system. By controlling the estimated error, the method dynamically switches between the full-order model and the reduced-oder model generated on the fly. Therefore, it reduces the computational cost of a high-fidelity simulation while achieving a prescribed accuracy level. Numerical tests on the non-parametric and parametric PDEs illustrate the efficacy of the proposed approach.

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arXiv2021

Reduced basis method for the nonlinear Poisson-Boltzmann equation regularized by the range-separated canonical tensor format

The Poisson-Boltzmann equation (PBE) is a fundamental implicit solvent continuum model for calculating the electrostatic potential of large ionic solvated biomolecules. However, its numerical solution encounters severe challenges arising from its strong singularity and nonlinearity. In [1,2], the effect of strong singularities was eliminated by applying the range-separated (RS) canonical tensor format [3,4] to construct a solution decomposition scheme for the PBE. The RS tensor format allows to derive a smooth approximation to the Dirac delta distribution in order to obtain a regularized PBE (RPBE) model. However, solving the RPBE is still computationally demanding due to its high dimension $\mathcal{N}$, where $\mathcal{N}$ is always in the millions. In this study, we propose to apply the reduced basis method (RBM) and the (discrete) empirical interpolation method ((D)EIM) to the RPBE in order to construct a reduced order model (ROM) of low dimension $N \ll \mathcal{N}$, whose solution accurately approximates the nonlinear RPBE. The long-range potential can be obtained by lifting the ROM solution back to the $\mathcal{N}$-space while the short-range potential is directly precomputed analytically, thanks to the RS tensor format. The sum of both provides the total electrostatic potential. The main computational benefit is the avoidance of computing the numerical approximation of the singular electrostatic potential. We demonstrate in the numerical experiments, the accuracy and efficacy of the reduced basis (RB) approximation to the nonlinear RPBE (NRPBE) solution and the corresponding computational savings over the classical nonlinear PBE (NPBE) as well as over the RBM being applied to the classical NPBE.

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arXiv2020

An Adaptive Sampling Approach for the Reduced Basis Method

The offline time of the reduced basis method can be very long given a large training set of parameter samples. This usually happens when the system has more than two independent parameters. On the other hand, if the training set includes fewer parameter samples, the greedy algorithm might produce a reduced-order model with large errors at the samples outside of the training set. We introduce a method based on a surrogate error model to efficiently sample the parameter domain such that the training set is adaptively updated starting from a coarse set with a small number of parameter samples. A sharp a posteriori error estimator is evaluated on a coarse training set. Radial Basis Functions are used to interpolate the error estimator over a separate fine training set. Points from the fine training set are added into the coarse training set at every iteration based on a user defined criterion. In parallel, parameter samples satisfying a defined tolerance are adaptively removed from the coarse training set. The approach is shown to avoid high computational costs by using a small training set and to provide a reduced-order model with guaranteed accuracy over a fine training set. Further, we show numerical evidence that the reduced-order model meets the defined tolerance over an independently sampled test set from the parameter domain.

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