Partial differential equations (PDEs) govern nearly every physical process in science and engineering, but solving them at scale remains prohibitively expensive. Generative AI has transformed language, vision, and protein science, but learned PDE solvers have not undergone a comparable shift. Existing paradigms each capture part of the problem. Physics-informed neural networks embed residual structure, although they are often difficult to optimize in stiff, multiscale, or large-domain regimes. Neural operators amortize across instances, although they commonly inherit a snapshot-prediction view of solving and can degrade over long rollouts. Diffusion-based solvers model uncertainty, although they are often built on a solver template that still centers on state regression. We argue that the core issue is the abstraction used to train learned solvers. Many models are asked to predict states, while many scientific settings require modeling how uncertainty moves through constrained dynamics. The relevant object is transport over physically admissible futures. This motivates flow learners: models that parameterize transport vector fields and generate trajectories through integration, echoing the continuous dynamics that define PDE evolution. This physics-to-physics alignment supports continuous-time prediction, native uncertainty quantification, and new opportunities for physics-aware solver design. We explain why transport-based learning offers a stronger organizing principle for learned PDE solving and outline the research agenda that follows from this shift.
Multiphysics Bench: Benchmarking and Investigating Scientific Machine Learning for Multiphysics PDEs
xLeaF Lab, Hong Kong University of Science and Technology (Guangzhou)（xLeaF实验室，香港科技大学（广州））；National University of Singapore（新加坡国立大学）；AI for Science, Shanghai AI Lab（AI for Science，上海人工智能实验室）
Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques such as deep neural networks and statistical learning are applied to classical problems of applied mathematics. In this paper, we develop a novel numerical algorithm that incorporates machine learning and artificial intelligence to solve PDEs. Based on the Legendre-Galerkin framework, we propose the {\it unsupervised machine learning} algorithm to learn {\it …
BENO: Boundary-embedded Neural Operators for Elliptic PDEs
National Engineering Research Center for Software Engineering, Peking University（北京大学软件工程国家工程研究中心）；Department of Engineering, Westlake University（西湖大学工学院）；Department of Astronautics and Aeronautics, Stanford University（斯坦福大学航空航天系）
RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations
Department of Civil and Environmental Engineering, Hong Kong Polytechnic University（香港理工大学土木与环境工程系）；Department of Statistics and Data Science, Yale University（耶鲁大学统计与数据科学系）
Neural Measures for learning distributions of Random PDEs
DMAM, University of Crete, Greece（DMAM，希腊克里特大学）；IACM, Foundation for Research and Technology - Hellas, Greece（IACM，希腊基础研究与技术研究所）；MPS, University of Sussex, United Kingdom（MPS，英国 Sussex 大学）
Symbolic regression corresponds to an ensemble of techniques that allow to uncover an analytical equation from data. Through a closed form formula, these techniques provide great advantages such as potential scientific discovery of new laws, as well as explainability, feature engineering as well as fast inference. Similarly, deep learning based techniques has shown an extraordinary ability of modeling complex patterns. The present paper aims at applying a recent end-to-end symbolic regression technique, i.e. the equation learner (EQL), to get an analytical equation for wind speed forecasting. …
National Innovation Institute of Defense Technology, Chinese Academy of Military Science（中国人民解放军军事科学院国防科技创新研究院）
Physics-informed neural networks (PINNs) have been widely used to solve various scientific computing problems. However, large training costs limit PINNs for some real-time applications. Although some works have been proposed to improve the training efficiency of PINNs, few consider the influence of initialization. To this end, we propose a New Reptile initialization based Physics-Informed Neural Network (NRPINN). The original Reptile algorithm is a meta-learning initialization method based on labeled data. PINNs can be trained with less labeled data or even without any labeled data by adding p…
This paper addresses distributional offline continuous-time reinforcement learning (DOCTR-L) with stochastic policies for high-dimensional optimal control. A soft distributional version of the classical Hamilton-Jacobi-Bellman (HJB) equation is given by a semilinear partial differential equation (PDE). This `soft HJB equation' can be learned from offline data without assuming that the latter correspond to a previous optimal or near-optimal policy. A data-driven solution of the soft HJB equation uses methods of Neural PDEs and Physics-Informed Neural Networks developed in the field of Scientifi…
Harnessing Scale and Physics: A Multi-Graph Neural Operator Framework for PDEs on Arbitrary Geometries
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Beihang University（北京航空航天大学）；The Hong Kong University of Science and Technology（香港科技大学）
Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs
School of Mathematics and Statistics（数学与统计学学院）；Department of Engineering Mechanics（工程力学系）；Department of Aeronautical Engineering（航空工程系）；MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University（复杂系统动力学与控制国家重点实验室，西北工业大学）；College of Computer Science and Technology（计算机科学与技术学院）；National University of Defense Technology（国防科技大学）；Department of Aerospace Engineering（航空航天工程系）
Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems
Centre for Industrial Mathematics (ZeTeM), University of Bremen（不来梅大学工业数学中心（ZeTeM））；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；Robert Bosch GmbH（罗伯特·博世有限公司）
In this paper, we introduce SciANN, a Python package for scientific computing and physics-informed deep learning using artificial neural networks. SciANN uses the widely used deep-learning packages Tensorflow and Keras to build deep neural networks and optimization models, thus inheriting many of Keras's functionalities, such as batch optimization and model reuse for transfer learning. SciANN is designed to abstract neural network construction for scientific computations and solution and discovery of partial differential equations (PDE) using the physics-informed neural networks (PINN) archite…
RocketML Inc.（RocketML 公司）；New York University (NYU)（纽约大学）；Iowa State University（爱荷华州立大学）；RocketML Inc（RocketML 公司）
Recent progress in scientific machine learning (SciML) has opened up the possibility of training novel neural network architectures that solve complex partial differential equations (PDEs). Several (nearly data free) approaches have been recently reported that successfully solve PDEs, with examples including deep feed forward networks, generative networks, and deep encoder-decoder networks. However, practical adoption of these approaches is limited by the difficulty in training these models, especially to make predictions at large output resolutions ($\geq 1024 \times 1024$). Here we report on…
University of Oxford（牛津大学）；Yonsei University（延世大学）；Cornell University（康奈尔大学）
Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are especially useful to learn solution operators associated with partial differential equations (PDEs) from pairs of forcing functions and solutions when numerical solvers are not available or the underlying physics is poorly understood. In this work, we attempt to provide theoretical foundations to understand the amount of training data needed to learn time-dependent PDEs. Given input-output pairs from a parabolic PDE in…
Qu & Co B.V.（Qu & Co 公司）；Czech Technical University（捷克技术大学）；University of Exeter（埃克塞特大学）
Quantum computing promises to speed up some of the most challenging problems in science and engineering. Quantum algorithms have been proposed showing theoretical advantages in applications ranging from chemistry to logistics optimization. Many problems appearing in science and engineering can be rewritten as a set of differential equations. Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime, where deep and wide quantum circuits can be used to solve large linear systems like partial differential equations (PDEs) …
University of Naples Federico II（那不勒斯费德里科二世大学）；SISSA – International School for Advanced Studies（国际高等研究学院（SISSA））；University of Colorado Boulder（科罗拉多大学博尔德分校）
Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related adv…
Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems. However, both classical solvers and recent deep learning-based surrogate models are typically extremely computationally intensive, because of their local evolution: they need to update the state of each discretized cell at each time step during inference. Here we develop Latent Evolution of PDEs (LE-PDE), a simple, fast and scalable method to accelerate the simulat…
Elucidating the Conformal Structure of the Brinkman Penalisation Method for Geometry-Adapted, Structure-Preserving Operator Learning of Hamiltonian PDEs
University of Bath（巴斯大学）；RIKEN（理化学研究所）；Kyushu University（九州大学）
Brown University（布朗大学）；Shanghai Normal University（上海师范大学）
One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs). We address this problem by taking advantage of recent advances in scientific machine learning and the dynamically orthogonal (DO) and bi-orthogonal (BO) methods for representing stochastic processes. Specifically, we propose two new Physics-Informed Neural Networks (PINNs) for solving time-dependent SPDEs, namely the NN-DO/BO methods, which incorporate the DO/BO constraints into the loss function with an implicit form instead of generating explicit expre…
Artificial intelligence for partial differential equations in computational mechanics: A review
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China（清华大学工程力学系）；Simulation Technology, Institute of Photonics, Leibniz University Hannover, Hannover 30167, Germany（汉诺威莱布尼茨大学光子研究所仿真技术部）；Drilling Mechanical Department, CNPC Engineering Technology RD Company Limited, Beijing 102206, China（中石油工程科技研发有限公司钻探机械部）；School of Mechanical, Medical and Process Engineering, Queensland University of Technology, Brisbane, QLD 4000, Australia（昆士兰科技大学机械、医疗与加工工程学院）；ARC Industrial Transformation Training Centre—Joint Biomechanics, Queensland University of Technology, Brisbane, QLD 4000, Australia（昆士兰科技大学 ARC 工业转型培训中心—联合生物力学）
Adaptive Physics-informed Neural Networks: A Survey
Institute for Artificial Intelligence, University of Stuttgart（斯图加特大学人工智能研究所）；Robert Bosch GmbH（罗伯特·博世有限公司）；IMPRS-IS（国际研究生学院智能系统（IMPRS-IS））；NEC Labs Europe（NEC欧洲实验室）；SimTech（仿真技术研究所（SimTech））
Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation
School of Mathematical Sciences, East China Normal University（东华大学数学科学学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；Institute of Artificial Intelligence, Beihang University（北航人工智能研究所）；Shanghai Zhangjiang Institute of Mathematics（上海张江数学研究所）；Academy for Advanced Interdisciplinary Studies, Northeast Normal University（东北师范大学先进交叉学科研究院）
Osaka University（大阪大学）；The Institute of Scientific and Industrial Research, Osaka University（大阪大学科学与工业研究所）
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high nonlinearity domain. To improve the generalizability, we introduce the novel approach of employing multi-task learning techniques, the uncertainty-weighting loss and the gradients surgery, in the context of learning PDE solutions. The multi-task scheme exploits the benefits of learning shared representations, controlled by cross-stitch modules, between multip…
Brown University（布朗大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from the implementation point of view, PINNs solve inverse problems as easily as forward problems. We pro…
Diffeomorphism Neural Operator for various domains and parameters of partial differential equations
College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics（南京航空航天大学机电学院）；School of Mechanical and Power Engineering, Nanjing Tech University（南京工业大学机械与动力工程学院）
Discrete Solution Operator Learning for Geometry-Dependent PDEs
Institute of Biomechanics and Medical Engineering Applied Mechanics Laboratory (AML) Tsinghua University（生物力学与医学工程研究所应用力学实验室（AML）清华大学）；Department of Aeronautics Imperial College London（航空航天系帝国理工学院伦敦）；School of Mechanical, Medical, and Process Engineering Queensland University of Technology（机械、医学与工艺工程学院昆士兰理工大学）
Kinetic-based regularization: Learning spatial derivatives and PDE applications
Engineering Mechanics Unit（工程力学单元）；Jawaharlal Nehru Centre for Advanced Scientific Research（贾瓦哈拉尔·奈尔中心先进科学研究所）；Fondazione Istituto Italiano di Tecnologia（意大利技术研究院）
School of Mathematics, Jilin University, 2699 Qianjin Street, Changchun, 130012, Jilin, China. Chair for Dynamics, Control, Machine Learning；Numerics (Alexander von Humboldt Professorship), Department of Mathematics, Friedrich–Alexander-Universit\"at Erlangen–N\"urnberg, 91058 Erlangen, Germany.；Institut de Mathematiques de Bourgogne, Université Bourgogne Europe, CNRS, 21000 Dijon, France.；Division of Mathematical Sciences, School of Physical；Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371, Singapore.；School of Mathematical Sciences, Nankai University, Tianjin 300071, China.；Numerics (Alexander von Humboldt Professorship), Department of Mathematics, Friedrich–Alexander-Universit\"at Erlangen–N\"urnberg, 91058 Erlangen, Germany. Departamento de Matem\'aticas, Universidad Aut\'onoma de Madrid, 28049 Madrid, Spain. Chair of Computational Mathematics, Fundaci\'on Deusto. Av. de las Universidades, 24, 48007 Bilbao, Basque Country, Spain.
From Equations to Insights: Unraveling Symbolic Structures in PDEs with LLMs
Department of Computer Science, University of Maryland, College Park, Maryland, USA（计算机科学系，马里兰大学， College Park, Maryland, USA）；Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA（数学系，田纳西大学， Knoxville, Tennessee, USA）；Department of Mathematics, and Department of Computer Science, University of Maryland, College Park, Maryland, USA（数学系和计算机科学系，马里兰大学， College Park, Maryland, USA）
