{"id":4,"date":"2025-06-16T10:45:40","date_gmt":"2025-06-16T14:45:40","guid":{"rendered":"https:\/\/sites.nd.edu\/zecheng-zhang\/?page_id=4"},"modified":"2025-11-19T09:21:06","modified_gmt":"2025-11-19T14:21:06","slug":"research","status":"publish","type":"page","link":"https:\/\/sites.nd.edu\/zecheng-zhang\/research\/","title":{"rendered":"Research"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">My reseach focuses:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Multiscale modeling and simulation<\/li>\n\n\n\n<li>Mathematics of machine learning<\/li>\n\n\n\n<li>Scientific machine learning<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Active grants:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>DOE AI For Science (DE-SC0025440)<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Publications and preprints.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Adrien Weihs, Jingmin Sun, Zecheng Zhang, Hayden Schaeffer. A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory. ArXiv preprint (2025).<\/li>\n\n\n\n<li>Erhan Bayraktar, Qi Feng, Zecheng Zhang, Zhaoyu Zhang. Deep Neural Operator Learning for Probabilistic Models. ArXiv preprint (2025).<\/li>\n\n\n\n<li class=\"has-black-color has-text-color has-link-color wp-elements-1\">Zecheng Zhang, Liu Hao, Wenjing Liao, Guang Lin. Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees.&nbsp;Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences (2025).<\/li>\n\n\n\n<li>Zecheng Zhang, Christian Moya, Lu Lu, Guang Lin and Hayden Schaeffer. DeepONet as a Multi-Operator Extrapolation Model: Distributed Pretraining with Physics-Informed Fine-Tuning.\u00a0Journal of Computational Physics (2025).<\/li>\n\n\n\n<li>Hao Liu, Zecheng Zhang, Wenjing Liao, Hayden Schaeffer. Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study.&nbsp;ArXiv preprint (2024).<\/li>\n\n\n\n<li>Yuxuan Liu, Jingmin Sun, Xinjie He, Griffin Pinney, Zecheng Zhang, and Hayden Schaeffer. PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamics.&nbsp;Neurips 2024 Workshop Foundation Models for Science.<\/li>\n\n\n\n<li>Derek Jollie, Jingmin Sun, Zecheng Zhang, and Hayden Schaeffer. Time-Series Forecasting, Knowledge Distillation, and Refinement within a Multimodal PDE Foundation Model.&nbsp;ArXiv preprint (2024).<\/li>\n\n\n\n<li>Jingmin Sun, Zecheng Zhang, Hayden Schaeffer. LeMON: Learning to Learn Multi-Operator Networks.&nbsp;ArXiv preprint (2024).<\/li>\n\n\n\n<li>Jingmin Sun, Yuxuan Liu, Zecheng Zhang, Hayden Schaeffer. Towards a Foundation Model for Partial Differential Equation: Multi-Operator Learning and Extrapolation.&nbsp;<a href=\"https:\/\/arxiv.org\/abs\/2404.12355\">Physics Review E (2024)<\/a>.<\/li>\n\n\n\n<li>Zecheng Zhang. MODNO: Multi Operator Learning With Distributed Neural Operators.&nbsp;Computer Methods in Applied Mechanics and Engineering (2024).<\/li>\n\n\n\n<li>Christian Moya, Amirhossein Mollaali, Zecheng Zhang, Lu Lu and Guang Lin. Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks.&nbsp;Physica D: Nonlinear Phenomena (2025).<\/li>\n\n\n\n<li>Zecheng Zhang, Christian Moya, Lu Lu, Guang Lin and Hayden Schaeffer. D2NO: Efficient Handling of Heterogeneous Input Function Spaces with Distributed Deep Neural Operators.&nbsp;Computer Methods in Applied Mechanics and Engineering (2024).<\/li>\n\n\n\n<li>Guang Lin, Na Ou, Zecheng Zhang, Zhidong Zhang. Restoring the Discontinuous Heat Equation Source Using Sparse Boundary Data and Dynamic Sensor.&nbsp;Inverse Problems (2024).<\/li>\n\n\n\n<li>Yuxuan Liu, Zecheng Zhang, Hayden Schaeffer. PROSE: Predicting Operators and Symbolic Expressions using Multimodal Transformers.&nbsp;<a href=\"https:\/\/arxiv.org\/abs\/2309.16816v1\">Neural Networks (2024)<\/a>.<\/li>\n\n\n\n<li>Zecheng Zhang, Christian Moya, Wing Tat Leung, Guang Lin, Hayden Schaeffer. Bayesian deep operator learning for homogenized to fine-scale maps for multiscale PDE.&nbsp;Siam MMS (2024).<\/li>\n\n\n\n<li>Zecheng Zhang, Wing Tat Leung, Hayden Schaeffer. A discretization-invariant extension and analysis of some deep operator networks.&nbsp;Journal of Computational and Applied Mathematics (2024).<\/li>\n\n\n\n<li>Zecheng Zhang, Wing Tat Leung, Hayden Schaeffer. BelNet: Basis enhanced learning, a mesh-free neural operator.&nbsp;Proceedings Royal Society A: Mathematical, Physical and Engineering Sciences (2023).<br>A tutorial and programming code of BelNet and operator learning is&nbsp;<a href=\"https:\/\/www.kaggle.com\/code\/zechengzhang\/bel-theory-vburgers?scriptVersionId=133316819%5C\">here<\/a>. This BelNet tutorial is on Kaggle and we will upload a tutorial on GitHub later.<\/li>\n\n\n\n<li>Guanxun Li,Guang Lin, Zecheng Zhang, Quan Zhou. Fast Tempering for Stochastic Gradient Langevin Dynamics.&nbsp;ArXiv preprint (2023).<\/li>\n\n\n\n<li>Na Ou, Zecheng Zhang, Guang Lin, A replica exchange preconditioned Crank-Nicolson Langevin dynamic MCMC method for Bayesian inverse problems.&nbsp;Journal of Computational Physics (2024).<\/li>\n\n\n\n<li>Yalchin Efendiev, Wing Tat Leung, Wenyuan Li, Zecheng Zhang. Hybrid explicit-implicit learning for multiscale problems with time dependent source.&nbsp;Communications in Nonlinear Science and Numerical Simulation (2023).<\/li>\n\n\n\n<li>Guang Lin, Christian Moya, Zecheng Zhang. On Learning the Dynamical Response of Nonlinear Control Systems with Deep Operator Networks.&nbsp;Engineering Application of Artificial Intelligence (2023).<\/li>\n\n\n\n<li>Guang Lin, Zecheng Zhang, Zhidong Zhang. Theoretical and numerical studies of inverse source problem for the linear parabolic equation with sparse boundary measurements.&nbsp;Inverse Problems (2022).<\/li>\n\n\n\n<li>Guang Lin, Christian Moya, Zecheng Zhang. Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEs.&nbsp;Journal of Computational Physics (2022).<\/li>\n\n\n\n<li>Yalchin Efendiev, Wing Tat Leung, Guang Lin, Zecheng Zhang. Efficient hybrid explicit-implicit learning for multiscale problems.&nbsp;Journal of Computational Physics (2022).<\/li>\n\n\n\n<li>Wing Tat Leung, Guang Lin, Zecheng Zhang. NH-PINN: Neural homogenization based the physics-informed neural network for the multiscale problems.&nbsp;Journal of Computational Physics (2022).<\/li>\n\n\n\n<li>Guang Lin, Yating Wang, Zecheng Zhang. Multi-variance replica exchange stochastic gradient MCMC for inverse and forward Bayesian physics-informed neural network.&nbsp;Journal of Computational Physics (2022).<\/li>\n\n\n\n<li>Liu Liu, Tieyong Zeng, Zecheng Zhang. A deep neural network approach on solving the linear transport model under diffusive scaling.&nbsp;ArXiv preprint (2021).<\/li>\n\n\n\n<li>Eric Chung, Yalchin Efendiev, Sai-Mang Pun, Zecheng Zhang. Computational multiscale methods for parabolic wave approximations in heterogeneous media.&nbsp;Applied Mathematics and Computation (2022).<\/li>\n\n\n\n<li>Eric Chung, Yalchin Efendiev, Wing Tat Leung, Sai-Mang Pun and Zecheng Zhang. Multi-agent reinforcement learning aided sampling algorithms for a class of multiscale inverse problems.&nbsp;Journal of Scientific Computing (2023).<\/li>\n\n\n\n<li>Boris Chetverushkin, Eric Chung, Yalchin Efendiev, Sai-Mang Pun and Zecheng Zhang. Computational multiscale methods for quasi-gas dynamic equations.&nbsp;Journal of Computational Physics (2020).<\/li>\n\n\n\n<li>Eric Chung, Wing Tat Leung, Sai-Mang Pun and Zecheng Zhang. A multi-stage deep learning based algorithm for multiscale model reduction.&nbsp;Journal of Computational and Applied Mathematics (2020).<\/li>\n\n\n\n<li>Eric Chung, Yalchin Efendiev, Wing Tat Leung, Zecheng Zhang. Learning Algorithms for Coarsening Uncertainty Space and Applications to Multiscale Simulations.&nbsp;Mathematics (2020).<br><\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>My reseach focuses: Active grants: Publications and preprints.<\/p>\n","protected":false},"author":5086,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-4","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.nd.edu\/zecheng-zhang\/wp-json\/wp\/v2\/pages\/4","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.nd.edu\/zecheng-zhang\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.nd.edu\/zecheng-zhang\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/zecheng-zhang\/wp-json\/wp\/v2\/users\/5086"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/zecheng-zhang\/wp-json\/wp\/v2\/comments?post=4"}],"version-history":[{"count":6,"href":"https:\/\/sites.nd.edu\/zecheng-zhang\/wp-json\/wp\/v2\/pages\/4\/revisions"}],"predecessor-version":[{"id":71,"href":"https:\/\/sites.nd.edu\/zecheng-zhang\/wp-json\/wp\/v2\/pages\/4\/revisions\/71"}],"wp:attachment":[{"href":"https:\/\/sites.nd.edu\/zecheng-zhang\/wp-json\/wp\/v2\/media?parent=4"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}