Research

My research aims to develop rigorous foundations and algorithms for high-dimensional computational challenges in applied mathematics, statistics, and data science. These challenges are prevalent in fine-scale physical equations, large-scale machine learning datasets, and statistical/stochastic models with complex uncertainties. I use probability theory and numerical analysis with multiscale, randomized algorithms and recent progresses in machine learning to formalize and explore the following directions:

  • How to understand and achieve provably efficient numerical estimation for PDEs, matrices, and probability distributions, especially in high dimensional or heterogeneous settings?
  • How to accelerate and automate these estimations with machine learning rigorously?

Some specific topics are as follows:

  • Multiscale analysis, randomized algorithms: numerical homogenization, randomized numerical linear algebra
  • Gaussian processes, kernel methods: scalable algorithms, uncertainty quantification, learning theory
  • Bayesian modeling, stochastic sampling: gradient flows, optimal transport, information geometry, diffusion models

In particular, my recent research focuses on the simulation, sampling, control, and statistical learning of high-dimensional probability distributions, with applications in scientific computing and machine learning. A core objective is to develop mathematical understanding of how dynamics of probability measures and their approximations behave in high-dimensional spaces, and how these insights can be used to systematically design probabilistic methods that automate and accelerate scientific applications. Applications I have been studying include imaging, molecular dynamics, chemistry, and forecasting.

Publications

Preprints

  • Stable Derivative Free Gaussian Mixture Variational Inference for Bayesian Inverse Problems
    Baojun Che, Yifan Chen, Zhenghao Huan, Daniel Zhengyu Huang, Weijie Wang
    Submitted. [arXiv] [Code]

  • Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias
    Yifan Chen, Xiaoou Cheng, Jonathan Niles-Weed, Jonathan Weare
    Submitted. [arXiv] [Slide]

  • Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities
    José A. Carrillo, Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Dongyi Wei
    Submitted. [arXiv]

  • Sequential-in-time Training of Nonlinear Parametrizations for Solving Time-dependent Partial Differential Equations
    Huan Zhang, Yifan Chen, Eric Vanden-Eijnden, Benjamin Peherstorfer
    Submitted. [arXiv]

  • Sampling via Gradient Flows in the Space of Probability Measures
    Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Sebastian Reich, Andrew M. Stuart
    Submitted. [arXiv] [Code] [Slide]

Conference Publications

  • Probabilistic Forecasting with Stochastic Interpolants and Follmer Processes
    Yifan Chen, Mark Goldstein, Mengjian Hua, Michael S. Albergo, Nicholas M. Boffi, Eric Vanden-Eijnden
    ICML, 2024. [Proceedings] [arXiv] [Slide] [Code] [Poster]

  • Inpainting crystal structure generations with score-based denoising
    Xinzhe Dai, Peichen Zhong, Bowen Deng, Yifan Chen, Gerbrand Ceder
    ICML Workshop AI4Science, 2024. [Proceedings]

  • Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors
    Zihui Wu, Yu Sun, Yifan Chen, Bingliang Zhang, Yisong Yue, Katherine L. Bouman
    NeurIPS, 2024. [Proceedings] [arXiv] [Code]

Journal Publications

  • Randomly Pivoted Cholesky: Practical Approximation of A Kernel Matrix with Few Entry Evaluations
    Yifan Chen, Ethan N. Epperly, Joel A. Tropp, Robert J. Webber
    Communications on Pure and Applied Mathematics, 2024. [Journal] [arXiv] [Code] [Slide]

  • Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation
    Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M. Stuart
    Statistics and Computing, 2024. [Journal] [arXiv] [Code]

  • Efficient, Multimodal, and Derivative-Free Bayesian Inference with Fisher-Rao Gradient Flows
    Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Sebastian Reich, Andrew M. Stuart
    Inverse Problems, 2024. [Journal] [arXiv] [Code] [Slide]

  • Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs
    Pau Batlle, Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M. Stuart
    Jounal of Computational Physics, 2024. [Journal] [arXiv] [Code]

  • Provable Probabilistic Imaging using Score-Based Generative Priors
    Yu Sun, Zihui Wu, Yifan Chen, Berthy T. Feng, Katherine L. Bouman
    IEEE Transactions on Computational Imaging, 2024. [Journal] [arXiv] [Code]

  • Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes
    Yifan Chen, Houman Owhadi, Florian Schaefer
    Mathematics of Computation, 2024. [Journal] [arXiv] [Code] [Slide] [Slide+]

  • Exponentially Convergent Multiscale Methods for 2D High Frequency Heterogeneous Helmholtz Equations
    Yifan Chen, Thomas Y. Hou, Yixuan Wang
    SIAM Multiscale Modeling and Simulation, 2023. [Journal] [arXiv] [Slide] [Slide+]

  • Exponentially Convergent Multiscale Finite Element Method
    Yifan Chen, Thomas Y. Hou, Yixuan Wang
    Communications on Applied Mathematics and Computation, 2023. [Journal] [arXiv] [Slide] [Slide+]

  • Multiscale Elliptic PDEs Upscaling and Function Approximation via Subsampled Data
    Yifan Chen, Thomas Y. Hou
    SIAM Multiscale Modeling and Simulation, 2022. [Journal] [arXiv] [Code]

  • Solving and Learning Nonlinear PDEs with Gaussian Processes
    Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M. Stuart
    Jounal of Computational Physics, 2021. [Journal] [arXiv] [Code] [Slide] [Slide+]

  • Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions
    Yifan Chen, Thomas Y. Hou, Yixuan Wang
    SIAM Multiscale Modeling and Simulation, 2021. [Journal] [arXiv] [Slide] [Slide+]

  • Consistency of Empirical Bayes and Kernel Flow For Hierarchical Parameter Estimation
    Yifan Chen, Houman Owhadi, Andrew M. Stuart
    Mathematics of Computation, 2021. [Journal] [arXiv] [Code] [Slide] [Longer Slide] [Short Video] [Longer Video]

  • Function Approximation via The Subsampled Poincare Inequality
    Yifan Chen, Thomas Y. Hou
    Discrete & Continuous Dynamical Systems - A, 2020. [Journal] [arXiv]

  • Optimal Transport Natural Gradient for Statistical Manifolds with Continuous Sample Space
    Yifan Chen, Wuchen Li
    Information Geometry, 2020. [Journal] [arXiv] [Code] [Slide]

  • Run-and-Inspect Method for Nonconvex Optimization and Global Optimality Bounds for R-Local Minimizers
    Yifan Chen, Yuejiao Sun, Wotao Yin
    Mathematical Programming, 2019. [Journal] [arXiv] [Slide]

  • The Quadratic Wasserstein Metric for Earthquake Location
    Jing Chen, Yifan Chen, Hao Wu, Dinghui Yang
    Journal of Computational Physics, 2018. [Journal] [arXiv] [Slide]

Thesis

  • On Multiscale and Statistical Numerical Methods for PDEs and Inverse Problems
    Yifan Chen
    Ph.D. Thesis, The W.P. Carey and Co. Prize in Applied Mathematics, 2023 [CaltechTHESIS] [Slide]

Notes

  • Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance
    Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Sebastian Reich, Andrew M. Stuart
    [arXiv]

Professional Experience

Referee Service