arXiv subjects

Fei Lu

Publications and source records attributed to Fei Lu.

At least 37 records · Page 2Linked to original sources

arXiv2024

Small noise analysis for Tikhonov and RKHS regularizations

Regularization plays a pivotal role in ill-posed machine learning and inverse problems. However, the fundamental comparative analysis of various regularization norms remains open. We establish a small noise analysis framework to assess the effects of norms in Tikhonov and RKHS regularizations, in the context of ill-posed linear inverse problems with Gaussian noise. This framework studies the convergence rates of regularized estimators in the small noise limit and reveals the potential instability of the conventional L2-regularizer. We solve such instability by proposing an innovative class of adaptive fractional RKHS regularizers, which covers the L2 Tikhonov and RKHS regularizations by adjusting the fractional smoothness parameter. A surprising insight is that over-smoothing via these fractional RKHSs consistently yields optimal convergence rates, but the optimal hyper-parameter may decay too fast to be selected in practice.

stat.ML↗

arXiv2024

Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery

Despite the recent popularity of attention-based neural architectures in core AI fields like natural language processing (NLP) and computer vision (CV), their potential in modeling complex physical systems remains under-explored. Learning problems in physical systems are often characterized as discovering operators that map between function spaces based on a few instances of function pairs. This task frequently presents a severely ill-posed PDE inverse problem. In this work, we propose a novel neural operator architecture based on the attention mechanism, which we coin Nonlocal Attention Operator (NAO), and explore its capability towards developing a foundation physical model. In particular, we show that the attention mechanism is equivalent to a double integral operator that enables nonlocal interactions among spatial tokens, with a data-dependent kernel characterizing the inverse mapping from data to the hidden parameter field of the underlying operator. As such, the attention mechanism extracts global prior information from training data generated by multiple systems, and suggests the exploratory space in the form of a nonlinear kernel map. Consequently, NAO can address ill-posedness and rank deficiency in inverse PDE problems by encoding regularization and achieving generalizability. We empirically demonstrate the advantages of NAO over baseline neural models in terms of generalizability to unseen data resolutions and system states. Our work not only suggests a novel neural operator architecture for learning interpretable foundation models of physical systems, but also offers a new perspective towards understanding the attention mechanism.

cs.LG↗

arXiv2024

MathOdyssey: Benchmarking Mathematical Problem-Solving Skills in Large Language Models Using Odyssey Math Data

Large language models (LLMs) have significantly advanced natural language understanding and demonstrated strong problem-solving abilities. Despite these successes, most LLMs still struggle with solving mathematical problems due to the intricate reasoning required. This paper investigates the mathematical problem-solving capabilities of LLMs using the newly developed "MathOdyssey" dataset. The dataset includes diverse mathematical problems at high school and university levels, created by experts from notable institutions to rigorously test LLMs in advanced problem-solving scenarios and cover a wider range of subject areas. By providing the MathOdyssey dataset as a resource to the AI community, we aim to contribute to the understanding and improvement of AI capabilities in complex mathematical problem-solving. We conduct benchmarking on open-source models, such as Llama-3 and DBRX-Instruct, and closed-source models from the GPT series and Gemini models. Our results indicate that while LLMs perform well on routine and moderately difficult tasks, they face significant challenges with Olympiad-level problems and complex university-level questions. Our analysis shows a narrowing performance gap between open-source and closed-source models, yet substantial challenges remain, particularly with the most demanding problems. This study highlights the ongoing need for research to enhance the mathematical reasoning of LLMs. The dataset, results, and code are publicly available.

cs.CL↗

arXiv2024

A Novel Approach to Evaluating Battery Charger Controller Design with Nonlinear PID Controller in an Extendable CHIL Setup

The design and development of power electronics converters pose a multitude of challenges. The evaluation of power electronics converters, particularly when operating at high power levels, presents a significant task, offering designers a deeper understanding of the functionality. Several methodologies have been devised to conduct hardware-in-the-loop (HIL) tests, which are classified into two main categories: controller hardware-in-the-loop (CHIL) and power hardware-in-the-loop (PHIL) tests. This paper explores the advantages and drawbacks of these two approaches and introduces a straightforward and cost-effective CHIL method for initial proof of concept and risk-free controller training. This method is based on the interaction between MATLAB/Simulink and Python. To assess the operation of the proposed system, a modeled battery pack (96S1P) is charged using a modeled battery charger converter, employing a non-linear PID controller. In this scenario, the controller benefits from the CC-CV technique for charging the battery pack. The results are presented in the final section.

eess.SY↗

arXiv2024

Gallium Nitride (GaN) based High-Power Multilevel H-Bridge Inverter for Wireless Power Transfer of Electric Vehicles

This paper presents a design and implementation of a high-power Gallium Nitride (GaN)-based multilevel Hbridge inverter to excite wireless charging coils for the wireless power transfer of electric vehicles (EVs). Compared to the traditional conductive charging, wireless charging technology offers a safer and more convenient way to charge EVs. Due to the increasing demand of fast charging, high-power inverters play a crucial role in exciting the wireless charging coils within a wireless power transfer system. This paper details the system specifications for the wireless charging of EVs, providing theoretical analysis and a control strategy for the modular design of a 75-kW 3-level and 4-level Hbridge inverter. The goal is to deliver a low-distortion excitation voltage to the wireless charging coils. LTspice simulation results, including output voltage, Fast Fourier Transform (FFT) analysis for both 3-level and 4-level H-bridge inverters, are presented to validate the control strategy and demonstrate the elimination of output harmonic components in the modular design. A GaNbased inverter prototype was employed to deliver a 85-kHz power to the wireless charging pads of the wireless power transfer system. Experimental results at two different voltage and power levels, 100V-215W and 150V-489W, validate the successful performance of the GaN inverter in the wireless charging system.

eess.SY↗

arXiv2024

Scalable iterative data-adaptive RKHS regularization

We present iDARR, a scalable iterative Data-Adaptive RKHS Regularization method, for solving ill-posed linear inverse problems. The method searches for solutions in subspaces where the true solution can be identified, with the data-adaptive RKHS penalizing the spaces of small singular values. At the core of the method is a new generalized Golub-Kahan bidiagonalization procedure that recursively constructs orthonormal bases for a sequence of RKHS-restricted Krylov subspaces. The method is scalable with a complexity of $O(kmn)$ for $m$-by-$n$ matrices with $k$ denoting the iteration numbers. Numerical tests on the Fredholm integral equation and 2D image deblurring show that it outperforms the widely used $L^2$ and $l^2$ norms, producing stable accurate solutions consistently converging when the noise level decays.

math.NA↗

arXiv2023

An adaptive RKHS regularization for Fredholm integral equations

Regularization is a long-standing challenge for ill-posed linear inverse problems, and a prototype is the Fredholm integral equation of the first kind with additive Gaussian measurement noise. We introduce a new RKHS regularization adaptive to measurement data and the underlying linear operator. This RKHS arises naturally in a variational approach, and its closure is the function space in which we can identify the true solution. Also, we introduce a small noise analysis to compare regularization norms by sharp convergence rates in the small noise limit. Our analysis shows that the RKHS- and $L^2$-regularizers yield the same convergence rate when their optimal hyper-parameters are selected using the true solution, and the RKHS-regularizer has a smaller multiplicative constant. However, in computational practice, the RKHS regularizer significantly outperforms the $L^2$-and $l^2$-regularizers in producing consistently converging estimators when the noise level decays or the observation mesh refines.

math.NA↗

arXiv2023

The $Λ$-type P-wave bottom baryon states via the QCD sum rules

Our study focuses on the $Λ$-type P-wave bottom baryon states with the spin-parity $J^P=\frac{1}{2}^-$, $\frac{3}{2}^-$. We introduce an explicit P-wave between the two light quarks in the interpolating currents (and the two light quarks are antisymmetric in flavor space, therefore leads to the name $Λ$-type baryon) to investigate the $Λ_b$ and $Ξ_b$ states within the framework of the full QCD sum rules. The predicted masses show that the $Ξ_b(6087)$ and $Ξ_b(6095/6100)$ could to be the P-wave bottom-strange baryon states with the spin-parity $J^P=\frac{1}{2}^-$ and $\frac{3}{2}^-$, respectively, meanwhile, the $Λ_b(5912)$ and $Λ_b(5920)$ could be the P-wave bottom baryon states with the spin-parity $J^P=\frac{1}{2}^-$ and $\frac{3}{2}^-$, respectively. The $Λ_b(5920)$ and $Ξ_b(6095/6100)$ maybe have two remarkable under-structures or Fock components at least.

hep-ph↗

arXiv2023

Analysis of the D-wave $Σ$-type charmed baryon states with the QCD sum rules

We construct the $Σ$-type currents to investigate the D-wave charmed baryon states with the QCD sum rules systematically. The predicted masses $M=3.35^{+0.13}_{-0.18}\,\rm{GeV}$ ($3.33^{+0.13}_{-0.16}\,\rm{GeV}$), $3.34^{+0.14}_{-0.18}\,\rm{GeV}$ ($3.35^{+0.13}_{-0.16}\,\rm{GeV}$) and $3.35^{+0.12}_{-0.13}\,\rm{GeV}$ ($3.35^{+0.12}_{-0.14}\,\rm{GeV}$) for the $Ω_c(0,2,{\frac{1}{2}}^+)$, $Ω_c(0,2,{\frac{3}{2}}^+)$ and $Ω_c(0,2,{\frac{5}{2}}^+)$ states are in excellent agreement with the experimental data $ 3327.1\pm1.2 \mbox{ MeV}$ from the LHCb collaboration, and support assigning the $Ω_c(3327)$ to be the $Σ$-type D-wave $Ω_c$ state with the spin-parity $J^P={\frac{1}{2}}^+$, ${\frac{3}{2}}^+$ or ${\frac{5}{2}}^+$.

hep-ph↗

arXiv2023

Identifiability of interaction kernels in mean-field equations of interacting particles

This study examines the identifiability of interaction kernels in mean-field equations of interacting particles or agents, an area of growing interest across various scientific and engineering fields. The main focus is identifying data-dependent function spaces where a quadratic loss functional possesses a unique minimizer. We consider two data-adaptive $L^2$ spaces: one weighted by a data-adaptive measure and the other using the Lebesgue measure. In each $L^2$ space, we show that the function space of identifiability is the closure of the RKHS associated with the integral operator of inversion. Alongside prior research, our study completes a full characterization of identifiability in interacting particle systems with either finite or infinite particles, highlighting critical differences between these two settings. Moreover, the identifiability analysis has important implications for computational practice. It shows that the inverse problem is ill-posed, necessitating regularization. Our numerical demonstrations show that the weighted $L^2$ space is preferable over the unweighted $L^2$ space, as it yields more accurate regularized estimators.

stat.ML↗

arXiv2023

Benchmarking optimality of time series classification methods in distinguishing diffusions

Statistical optimality benchmarking is crucial for analyzing and designing time series classification (TSC) algorithms. This study proposes to benchmark the optimality of TSC algorithms in distinguishing diffusion processes by the likelihood ratio test (LRT). The LRT is an optimal classifier by the Neyman-Pearson lemma. The LRT benchmarks are computationally efficient because the LRT does not need training, and the diffusion processes can be efficiently simulated and are flexible to reflect the specific features of real-world applications. We demonstrate the benchmarking with three widely-used TSC algorithms: random forest, ResNet, and ROCKET. These algorithms can achieve the LRT optimality for univariate time series and multivariate Gaussian processes. However, these model-agnostic algorithms are suboptimal in classifying high-dimensional nonlinear multivariate time series. Additionally, the LRT benchmark provides tools to analyze the dependence of classification accuracy on the time length, dimension, temporal sampling frequency, and randomness of the time series.

stat.ML↗

arXiv2022

Bias-Independent Subthreshold Swing in Nanoscale Cold-Source Field-Effect Transistors by Drain Density-of-States Engineering

We report a strategy to design nanoscale cold-source field-effect transistors (CS-FETs) with bias-independent sub-60 mV/dec subthreshold swing (SS). By first-principles calculations and quantum-transport simulations, we reveal that the energy alignment of density of states (DOS) between the drain and source electrodes is critical to achieving bias-independent SS. By defining "gate window", we propose a device model to demonstrate how similar slopes of the drain DOS falling into the gate window can stabilize the SS under different bias. This study underscores the significance of drain DOS engineering in the design of CS-FETs with bias-independent SS for portable electronic applications.

cond-mat.mes-hall↗

arXiv2022

Stochastic Data-Driven Variational Multiscale Reduced Order Models

Trajectory-wise data-driven reduced order models (ROMs) tend to be sensitive to training data, and thus lack robustness. We propose to construct a robust stochastic ROM closure (S-ROM) from data consisting of multiple trajectories from random initial conditions. The S-ROM is a low-dimensional time series model for the coefficients of the dominating proper orthogonal decomposition (POD) modes inferred from data. Thus, it achieves reduction both space and time, leading to simulations orders of magnitude faster than the full order model. We show that both the estimated POD modes and parameters in the S-ROM converge when the number of trajectories increases. Thus, the S-ROM is robust when the training data size increases. We demonstrate the S-ROM on a 1D Burgers equation with a viscosity $ν= 0.002$ and with random initial conditions. The numerical results verify the convergence. Furthermore, the S-ROM makes accurate trajectory-wise predictions from new initial conditions and with a prediction time far beyond the training range, and it quantifies the spread of uncertainties due to the unresolved scales.

math.NA↗

arXiv2022

NySALT: Nyström-type inference-based schemes adaptive to large time-stepping

Large time-stepping is important for efficient long-time simulations of deterministic and stochastic Hamiltonian dynamical systems. Conventional structure-preserving integrators, while being successful for generic systems, have limited tolerance to time step size due to stability and accuracy constraints. We propose to use data to innovate classical integrators so that they can be adaptive to large time-stepping and are tailored to each specific system. In particular, we introduce NySALT, Nyström-type inference-based schemes adaptive to large time-stepping. The NySALT has optimal parameters for each time step learnt from data by minimizing the one-step prediction error. Thus, it is tailored for each time step size and the specific system to achieve optimal performance and tolerate large time-stepping in an adaptive fashion. We prove and numerically verify the convergence of the estimators as data size increases. Furthermore, analysis and numerical tests on the deterministic and stochastic Fermi-Pasta-Ulam (FPU) models show that NySALT enlarges the maximal admissible step size of linear stability, and quadruples the time step size of the Störmer--Verlet and the BAOAB when maintaining similar levels of accuracy.

math.NA↗

arXiv2022

Unsupervised learning of observation functions in state-space models by nonparametric moment methods

We investigate the unsupervised learning of non-invertible observation functions in nonlinear state-space models. Assuming abundant data of the observation process along with the distribution of the state process, we introduce a nonparametric generalized moment method to estimate the observation function via constrained regression. The major challenge comes from the non-invertibility of the observation function and the lack of data pairs between the state and observation. We address the fundamental issue of identifiability from quadratic loss functionals and show that the function space of identifiability is the closure of a RKHS that is intrinsic to the state process. Numerical results show that the first two moments and temporal correlations, along with upper and lower bounds, can identify functions ranging from piecewise polynomials to smooth functions, leading to convergent estimators. The limitations of this method, such as non-identifiability due to symmetry and stationarity, are also discussed.

stat.ML↗

arXiv2022

Nonparametric learning of kernels in nonlocal operators

Nonlocal operators with integral kernels have become a popular tool for designing solution maps between function spaces, due to their efficiency in representing long-range dependence and the attractive feature of being resolution-invariant. In this work, we provide a rigorous identifiability analysis and convergence study for the learning of kernels in nonlocal operators. It is found that the kernel learning is an ill-posed or even ill-defined inverse problem, leading to divergent estimators in the presence of modeling errors or measurement noises. To resolve this issue, we propose a nonparametric regression algorithm with a novel data adaptive RKHS Tikhonov regularization method based on the function space of identifiability. The method yields a noisy-robust convergent estimator of the kernel as the data resolution refines, on both synthetic and real-world datasets. In particular, the method successfully learns a homogenized model for the stress wave propagation in a heterogeneous solid, revealing the unknown governing laws from real-world data at microscale. Our regularization method outperforms baseline methods in robustness, generalizability and accuracy.

stat.ML↗

arXiv2022

Data adaptive RKHS Tikhonov regularization for learning kernels in operators

We present DARTR: a Data Adaptive RKHS Tikhonov Regularization method for the linear inverse problem of nonparametric learning of function parameters in operators. A key ingredient is a system intrinsic data-adaptive (SIDA) RKHS, whose norm restricts the learning to take place in the function space of identifiability. DARTR utilizes this norm and selects the regularization parameter by the L-curve method. We illustrate its performance in examples including integral operators, nonlinear operators and nonlocal operators with discrete synthetic data. Numerical results show that DARTR leads to an accurate estimator robust to both numerical error due to discrete data and noise in data, and the estimator converges at a consistent rate as the data mesh refines under different levels of noises, outperforming two baseline regularizers using $l^2$ and $L^2$ norms.

stat.ML↗

arXiv2021

Shock trace prediction by reduced models for a viscous stochastic Burgers equation

Viscous shocks are a particular type of extreme events in nonlinear multiscale systems, and their representation requires small scales. Model reduction can thus play an important role in reducing the computational cost for an efficient prediction of shocks. Yet, reduced models typically aim to approximate large-scale dominating dynamics, which do not resolve the small scales by design. To resolve this representation barrier, we introduce a new qualitative characterization of the space-time locations of shocks, named as the ``shock trace'', via a space-time indicator function based on an empirical resolution-adaptive threshold. Different from the exact shocks, the shock traces can be captured within the representation capacity of the large scales, which facilitates the forecast of the timing and locations of the shocks utilizing reduced models. Within the context of a viscous stochastic Burgers equation, we show that a data-driven reduced model, in the form of nonlinear autoregression (NAR) time series models, can accurately predict the random shock traces, with relatively low rates of false predictions. The NAR model significantly outperforms the corresponding Galerkin truncated model in the scenario of either noiseless or noisy observations. The results illustrate the importance of the data-driven closure terms in the NAR model, which account for the effects of the unresolved small scale dynamics on the resolved ones due to nonlinear interactions.

math.NA↗