学术报告报告主题1：Multiple Relaxation Structure-preserving Schemes and Their Applications
报 告 人：李东方 教授（华中科技大学）
报告时间：2026年7月16日（星期四）上午09:30-10:10
报告摘要：
A novel family of high-order structure-preserving methods is proposed. The methods are developed by applying the multiple relaxation idea to the different Runge-Kutta methods. It is shown that the multiple relaxation Runge-Kutta methods can achieve high-order accuracy in time and preserve multiple original invariants at the discrete level. Several numerical experiments are carried out to support the theoretical results and illustrate the effectiveness and efficiency of the proposed methods.
报告人介绍：
李东方，华中科技大学数学与统计学院教授，博导，华中卓越学者，国家级青年人才。主持国家级课题7项。主要从事微分方程数值解、机器学习和信号处理等领域的研究工作。尤其在微分方程保结构算法和分数阶微分方程的高效数值算法和理论上取得一些有意义的进展。
报告主题2：Nonlinear Transformation Based Infinite-Dimensional Variational Inference for Statistical Inverse Problems
报 告 人：贾骏雄 教授（西安交通大学）
报告时间：2026年7月16日（星期四）上午10:10-10:50
报告摘要：
Inverse problems for PDEs are common in scientific disciplines and can be formulated as statistical inference problems via Bayes’ theorem. For large-scale problems, developing discretization-invariant algorithms is crucial, achievable by formulating methods in infinite-dimensional spaces. Restricting the variational family to the pushforward of a prior measure's nonlinear transformation yields various variational inference methods. Overcoming singularity issues in infinite-dimensional function spaces, we develop two methods: infinite-dimensional Stein variational gradient descent (iSVGD) and functional normalizing flows (FNF). The transformations in both iSVGD and FNF involve a sequence of identity operator perturbations. In iSVGD, perturbation mappings are in a reproducing kernel Hilbert space, while in FNF, they are constructed with designed neural network architectures. We apply these algorithms to an inverse problem of the steady-state Darcy flow equation. Numerical results validate the theoretical analysis, show the algorithms' efficiency, and confirm their discretization-invariant properties.
报告人介绍：
贾骏雄，西安交通大学数学与统计学院教授。其长期致力于偏微分方程反问题的无穷维贝叶斯分析方法研究，系统建立了无穷维PAC-Bayes先验分布学习理论，发展了一系列无穷维变分推断理论与算法，并严格证明了变分贝叶斯反演的最优相合速率估计，代表性成果发表于JASA、Math Comp.、SIAM 系列、JMLR、JFA、IP等领域权威期刊。他曾获国家自然科学基金青年基金（B类）资助，并作为联合负责人主持“数学与智能+”交叉重点专项。
报告主题3：A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients
报 告 人：欧阳督 博士（清华大学）
报告时间：2026年7月16日（星期四）上午11:10-11:50
报告摘要：
High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data-generating dynamics and therefore do not apply to black-box environments. We introduce two types of simulators as data-generating mechanisms, and take a ``representing-then-learning approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations. Our representation of derivatives relies on the zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories. This fully model-free approach generates targets for the gradient and Hessian networks using only function evaluations. We provide a statistical learning analysis of the proposed approach, including a bias--variance tradeoff for ZODs. Assuming a standard contraction property of the underlying operator, we establish a non-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias. Crucially, we derive the sample complexity of the learned representations in (weighted) Sobolev space, characterizing the error up to second-order derivatives. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions.
报告人介绍：
欧阳督于 2026 年6月获得清华大学数学博士学位。他的研究兴趣包括拟蒙特卡罗方法、随机控制、强化学习和数值分析。相关研究成果发表于SIAM Journal on Numerical Analysis、SIAM Journal on Control and Optimization等期刊。
报告地点：1号楼306
邀 请 人：何志坚 教授
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数学学院
2026年7月13日