A quantum-inspired multi-level tensor-train monolithic space-time method for nonlinear PDEs
Khalifa University of Science and Technology（哈利法科学技术大学）；Technology Innovation Institute（技术创新研究所）；Emirates Nuclear Energy Company（阿联酋核能公司）；Emirates Nuclear Technology Center（阿联酋核技术中心）
N. R. Rapaka, R. Peddinti, E. Tiunov, N. J. Faraj, A. N. Alkhooori, L. Aolita, Y. Addad, M. K. Riahi
We propose a multilevel tensor-train (TT) framework for solving nonlinear partial differential equations (PDEs) in a global space-time formulation. While space-time TT solvers have demonstrated significant potential for compressed high-dimensional simulations, the literature contains few systematic comparisons with classical time-stepping methods, limited error convergence analyses, and little quantitative assessment of the impact of TT rounding on numerical accuracy. Likewise, existing studies fail to demonstrate performance across a diverse set of PDEs and parameter ranges. In practice, monolithic Newton iterations may stagnate or fail to converge in strongly nonlinear, stiff, or advection-dominated regimes, where poor initial guesses and severely ill-conditioned space-time Jacobians hinder robust convergence. We overcome this limitation by introducing a coarse-to-fine multilevel strategy fully embedded within the TT format. Each level refines both spatial and temporal resolutions while transferring the TT solution through low-rank prolongation operators, providing robust initializations for successive Newton solves. Residuals, Jacobians, and transfer operators are represented directly in TT and solved with the adaptive-rank DMRG algorithm. Numerical experiments for a selection of nonlinear PDEs including Fisher-KPP, viscous Burgers, sine-Gordon, and KdV cover diffusive, convective, and dispersive dynamics, demonstrating that the multilevel TT approach consistently converges where single-level space-time Newton iterations fail. In dynamic, advection-dominated (nonlinear) scenarios, multilevel TT surpasses single-level TT, achieving high accuracy with significantly reduced computational cost, specifically when high-fidelity numerical simulation is required.
A machine learning solver for high-dimensional integrals: Solving Kolmogorov PDEs by stochastic weighted minimization and stochastic gradient descent through a high-order weak approximation scheme of SDEs with Malliavin weights
Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space
Universität Potsdam（波茨坦大学）；Freie Universität Berlin（柏林自由大学）；Brandenburgische Technische Universität Cottbus-Senftenberg（勃兰登堡工业大学科特布斯-森夫滕贝格）
Pseudospectral method for solving PDEs using Matrix Product States
Institute of Fundamental Physics IFF-CSIC（基础物理研究所）；Millennium Institute for Research in Optics（光学研究千年研究所）；Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción（康塞普西翁大学物理与数学科学学院）
Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networks
提出了一种求解高维Hamilton-Jacobi-Bellman（HJB）型椭圆型偏微分方程（PDE）的新的数值方法.的HJB偏微分方程，重新制定为最优控制问题，处理的演员-评论家框架的启发，强化学习，基于神经网络参数化的价值和控制功能。在行动者-批评者框架内，我们采用了政策梯度的方法来改善控制，而对于价值函数，我们推导出一个方差减少的最小二乘时间差分法，使用随机分析。数值离散的随机控制问题，我们采用了自适应步长计划，以提高精度附近的区域边界。最后，通过对$20$空间维数的线性二次型调节器、随机Van der Pol振子、扩散Eikonal方程和完全非线性椭圆型偏微分方程的数值算例，验证了本文方法的有效性.
Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms
State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China（非线性力学国家重点实验室，力学研究所，中国科学院，北京100190，中国）；School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 100049, China（中国科学院大学工程科学学院，北京100049，中国）；School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China（西北工业大学数学与统计学院，西安710072，中国）
School of Mathematical Sciences（数学科学学院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Peking University, Beijing, China（北京大学北京中国）；Beijing Computational Science Research Center（北京计算科学研究中心）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Center for Data Science, Peking University（北京大学数据科学中心）；Beijing Institute of Big Data Research（北京大数据研究院）
Diffusion models with physics-guided inference for solving partial differential equations
Central South University（中南大学）；National Engineering Research Center of High-speed Railway Construction Technology（高速铁路建设技术国家工程研究中心）；Zhejiang University of Technology（浙江工业大学）
