The discovery of governing differential equations from data is an open frontier in machine learning. The sparse identification of nonlinear dynamics (SINDy) \citep{brunton_discovering_2016} framework enables data-driven discovery of interpretable models in the form of sparse, deterministic governing laws. Recent works have sought to adapt this approach to the stochastic setting, though these adaptations are severely hampered by the curse of dimensionality. On the other hand, Bayesian-inspired deep learning methods have achieved widespread success in high-dimensional probabilistic modeling via computationally efficient approximate inference techniques, suggesting the use of these techniques for efficient stochastic equation discovery. Here, we introduce HyperSINDy, a framework for modeling stochastic dynamics via a deep generative model of sparse governing equations whose parametric form is discovered from data. HyperSINDy employs a variational encoder to approximate the distribution of observed states and derivatives. A hypernetwork \citep{ha_hypernetworks_2016} transforms samples from this distribution into the coefficients of a differential equation whose sparse form is learned simultaneously using a trainable binary mask \citep{louizos_learning_2018}. Once trained, HyperSINDy generates stochastic dynamics via a differential equation whose coefficients are driven by a Gaussian white noise. In experiments, HyperSINDy accurately recovers ground truth stochastic governing equations, with learned stochasticity scaling to match that of the data. Finally, HyperSINDy provides uncertainty quantification that scales to high-dimensional systems. Taken together, HyperSINDy offers a promising framework for model discovery and uncertainty quantification in real-world systems, integrating sparse equation discovery methods with advances in statistical machine learning and deep generative modeling.
A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables
VENI, VINDy, VICI: a generative reduced-order modeling framework with uncertainty quantification
MOX -- Department of Mathematics, Politecnico di Milano, Milan, Italy（米兰理工学院数学系）；Institute of Engineering and Computational Mechanics, University of Stuttgart, Stuttgart, Germany（斯图加特大学工程与计算力学研究所）；Department of Civil Engineering, Politecnico di Milano, Milan, Italy（米兰理工学院土木工程系）；Department of Mechanical Engineering, University of Washington, Seattle, USA（华盛顿大学机械工程系）；Department of Applied Mathematics and Electrical and Computer Engineering, University of Washington, Seattle, USA（华盛顿大学应用数学与电气与计算机工程系）
Physics-Informed Generative Solver: Bridging Data-Driven Priors and Conservation Laws for Stable Spatiotemporal Field Reconstruction
School of Advanced Manufacturing Engineering, Nanjing University（南京大学先进制造工程学院）；National Laboratory of Solid State Microstructures, Nanjing University（南京大学固态微结构国家实验室）；Suzhou Acoustics Industry Technology Research Institute Co., Ltd.（苏州声学工业技术研究所有限公司）；School of Mechanical and Electric Engineering, Soochow University（苏州大学机械与电子工程学院）；Shishan Laboratory, Nanjing University（仙山实验室）
Generative Discovery of Partial Differential Equations by Learning from Math Handbooks
Eastern Institute of Technology（东方理工）；Tsinghua University（清华大学）；Ocean University of China（中国海洋大学）；Imperial College London（伦敦帝国理工学院）；University of Oxford（牛津大学）；University of Manchester（曼彻斯特大学）；Peking University（北京大学）；Lingnan University（岭南大学）
Generating Physically Plausible Parachute Dynamics with Deep Generative Modeling
PhD Candidate, Mechanical and Aerospace Engineering, Engineering Quadrangle, Princeton, NJ 08544（普林斯顿大学机械与航空航天工程系博士研究生）；Group Supervisor, Dynamics, Modeling, Flight Mechanics & UQ, 4800 Oak Grove Dr, M/S 198-326, Pasadena, CA 91011, AIAA Member（加州帕萨迪纳AIAA会员，动力学、建模、飞行力学及不确定性量化组负责人）；Assistant Professor, Mechanical and Aerospace Engineering, Engineering Quadrangle, Princeton, NJ 08544（普林斯顿大学机械与航空航天工程系助理教授）；Assistant Professor, Center for Statistics and Machine Learning, Engineering Quadrangle, Princeton, NJ 08544（普林斯顿大学统计与机器学习中心助理教授）
Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems
German Research Center for Artificial Intelligence (DFKI)（德国人工智能研究中心）；Department of Computer Science, University of Kaiserslautern–Landau (RPTU)（凯泽斯劳滕-兰道大学计算机科学系）；Department of Mathematics, Imperial College London（伦敦帝国理工学院数学系）
RODE-Net: Learning Ordinary Differential Equations with Randomness from Data
Academy for Advanced Interdisciplinary Studies（前沿交叉学科研究院）；Peking University（北京大学）；School of Mathematical Science（数学科学学院）；National School of Development（国家发展研究院）；Theory Lab（理论实验室）；Huawei Hong Kong Research Center（华为香港研究中心）；Beijing International Center for Mathematical Research（北京国际数学研究中心）；Institute for Artificial Intelligence（人工智能研究院）；Center for Data Science（数据科学中心）
Sparsely-Supervised Data Assimilation via Physics-Informed Schrödinger Bridge
Department of Industrial \& Systems Engineering, KAIST, Daejeon, Repulic of Korea；Department of Mathematical Sciences, KAIST, Daejeon, Repulic of Korea
Embedded Variational Neural Stochastic Differential Equations for Learning Heterogeneous Dynamics
National Institute of Technology, Rourkela, India（印度国立理工学院鲁尔克拉分校）；Poverty Alleviation Research Centre, National Institute of Technology, Rourkela, India（印度国立理工学院鲁尔克拉分校扶贫研究中心）
