Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. In this work we investigate this limitation through the lens of Neural Tangent Kernel (NTK) theory and elucidate how PINNs are biased towards learning functions along the dominant eigen-directions of their limiting NTK. Using this observation, we construct novel architectures that employ spatio-temporal and multi-scale random Fourier features, and justify how such coordinate embedding layers can lead to robust and accurate PINN models. Numerical examples are presented for several challenging cases where conventional PINN models fail, including wave propagation and reaction-diffusion dynamics, illustrating how the proposed methods can be used to effectively tackle both forward and inverse problems involving partial differential equations with multi-scale behavior. All code an data accompanying this manuscript will be made publicly available at \url{https://github.com/PredictiveIntelligenceLab/MultiscalePINNs}.
Time-dependent Partial Differential Equations with given initial conditions are considered in this paper. New differentiation techniques of the unknown solution with respect to time variable are proposed. It is shown that the proposed techniques allow to generate accurate higher order derivatives simultaneously for a set of spatial points. The calculated derivatives can then be used for data-driven solution in different ways. An application for Physics Informed Neural Networks by the well-known DeepXDE software solution in Python under Tensorflow background framework has been presented for thr…
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…
Physics informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely Kolmogorov equations that include the heat equation and Black-Scholes equation of option pricing, as examples. We construct neural networks, whose PINN residual (generalization error) can be made as small as desired. We also prove that the total $L^2$-error can be bounded by the generalization error, which in turn is bounded in terms of the training error, prov…
Physics-informed neural networks (PINNs) as a means of discretizing partial differential equations (PDEs) are garnering much attention in the Computational Science and Engineering (CS&E) world. At least two challenges exist for PINNs at present: an understanding of accuracy and convergence characteristics with respect to tunable parameters and identification of optimization strategies that make PINNs as efficient as other computational science tools. The cost of PINNs training remains a major challenge of Physics-informed Machine Learning (PiML) - and, in fact, machine learning (ML) in general…
Indian Institute of Technology (IIT) Delhi（印度理工学院德里分校）
Deep neural operators are recognized as an effective tool for learning solution operators of complex partial differential equations (PDEs). As compared to laborious analytical and computational tools, a single neural operator can predict solutions of PDEs for varying initial or boundary conditions and different inputs. A recently proposed Wavelet Neural Operator (WNO) is one such operator that harnesses the advantage of time-frequency localization of wavelets to capture the manifolds in the spatial domain effectively. While WNO has proven to be a promising method for operator learning, the dat…
Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs
School of Petroleum Engineering, Yangtze University（扬州大学石油工程学院）；School of Computer Science, Yangtze University（扬州大学计算机科学学院）；State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University)（扬州大学低碳催化与二氧化碳利用国家重点实验室）；Western Research Institute, Yangtze University（扬州大学西部研究院）
Parameterized Physics-informed Neural Networks for Parameterized PDEs
Yonsei University（延世大学）；Arizona State University（亚利桑那州立大学）；LG CNS；Texas A&M University-Commerce（德州农工大学Commerce分校）；Oregon State University（俄勒冈州立大学）；KAIST（韩国科学技术院）
QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEs
Computational Science and Engineering Department, Informatics Institute Istanbul Technical University（信息学院理工学院计算科学与工程系）；Scientific Computing, Rutherford Appleton Laboratory Science and Technology Facilities Council (STFC)（科学与技术设施委员会（STFC）科学计算部）
Physics-informed neural networks (PINNs) encode physical conservation laws and prior physical knowledge into the neural networks, ensuring the correct physics is represented accurately while alleviating the need for supervised learning to a great degree. While effective for relatively short-term time integration, when long time integration of the time-dependent PDEs is sought, the time-space domain may become arbitrarily large and hence training of the neural network may become prohibitively expensive. To this end, we develop a parareal physics-informed neural network (PPINN), hence decomposin…
ETH Zürich（苏黎世联邦理工学院）；ETH AI Center（苏黎世联邦理工学院人工智能中心）
We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator learning. These bounds guarantee that PINNs and (physics-informed) DeepONets or FNOs will efficiently approximate the underlying solution or solution operator of generic partial differential equations (PDEs). Our framework utilizes existing neural network approximation results to obtain bounds on more involved learning architectures for PDEs. We illustrate …
Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encounted in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs. As the number of data grows, PINNs generate a se…
There has been a growing interest in the use of Deep Neural Networks (DNNs) to solve Partial Differential Equations (PDEs). Despite the promise that such approaches hold, there are various aspects where they could be improved. Two such shortcomings are (i) their computational inefficiency relative to classical numerical methods, and (ii) the non-interpretability of a trained DNN model. In this work we present ASPINN, an anisotropic extension of our earlier work called SPINN--Sparse, Physics-informed, and Interpretable Neural Networks--to solve PDEs that addresses both these issues. ASPINNs gen…
Northeastern University（东北大学）；University of Notre Dame（圣母大学）；MIT（麻省理工学院）
Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (PINNs) to solve PDEs as a basis for data-driven modeling and inverse analysis. However, the majority of existing PINN methods, based on fully-connected NNs, pose intrinsic limitations to low-dimensional spatiotemporal parameterizations. Moreover, since the initial/boundary conditions (I/BCs) are softly imposed via penalty, the solution quality heavily relies on h…
CP-PINNs: Data-Driven Changepoints Detection in PDEs Using Online Optimized Physics-Informed Neural Networks
Stony Brook University（石溪大学）；Center of Excellence in Wireless and Information Technology（无线与信息技术卓越中心）；Institute for Advanced Computational Science（高级计算科学研究所）
Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks
Tsinghua University（清华大学）；Bauhaus-Universität Weimar（魏玛包豪斯大学）；CNPC Engineering Technology RD Company Limited（中国石油集团工程技术研究院有限公司）；Queensland University of Technology（昆士兰科技大学）；Leibniz University Hannover（汉诺威莱布尼茨大学）
University of Bonn（波恩大学）；Delft University of Technology（代尔夫特理工大学）；RWTH Aachen University（亚琛工业大学）
Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed-form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the solution of PDEs based on a novel technique that combines the advantages of two recently emerging machine learning based approaches. First, physics-informed neural networks (PINNs) learn continuous solutions of PDEs and can be trained with little to no ground truth data. However, PINNs do not generalize well to unseen domains. Second, convolutional neural network…
Dalian University of Technology（大连理工大学）；Harbin Institute of Technology（哈尔滨工业大学）；JD Explore Academy（京东探索研究院）
Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs), and have been widely used in a variety of PDE problems. However, there still remain some challenges in the application of PINNs: 1) the mechanism of PINNs is unsuitable (at least cannot be directly applied) to exploiting a small size of (usually very few) extra informative samples to refine the networks; and 2) the efficiency of training PINNs often becomes low for some complicated PDEs. In this paper, we propose the generative adversarial physics-informed n…
Universidad Politecnica de Madrid（马德里理工大学）；Hess Corporation（赫斯公司）
Physics-Informed Neural Networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs). PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that deve…
University of Nevada, Las Vegas（内华达大学拉斯维加斯分校）；University of Arkansas（阿肯色大学）；Lehigh University（里海大学）
In this paper, we show a physics-informed neural network solver for the time-dependent surface PDEs. Unlike the traditional numerical solver, no extension of PDE and mesh on the surface is needed. We show a simplified prior estimate of the surface differential operators so that PINN's loss value will be an indicator of the residue of the surface PDEs. Numerical experiments verify efficacy of our algorithm.
Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs
Dept. of Aerospace Eng. Sciences University of Colorado Boulder（航空航天工程科学系，科罗拉多大学博尔德分校）；Center for Systems and Control Delft University of Technology（系统与控制中心，代尔夫特理工大学）
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…
Defense Innovation Institute, 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）；Human Resource Center（人力资源中心）
Learning solutions of partial differential equations (PDEs) with Physics-Informed Neural Networks (PINNs) is an attractive alternative approach to traditional solvers due to its flexibility and ease of incorporating observed data. Despite the success of PINNs in accurately solving a wide variety of PDEs, the method still requires improvements in terms of computational efficiency. One possible improvement idea is to optimize the generation of training point sets. Residual-based adaptive sampling and quasi-uniform sampling approaches have been each applied to improve the training effects of PINN…
Compared with conventional numerical approaches to solving partial differential equations (PDEs), physics-informed neural networks (PINN) have manifested the capability to save development effort and computational cost, especially in scenarios of reconstructing the physics field and solving the inverse problem. Considering the advantages of parameter sharing, spatial feature extraction and low inference cost, convolutional neural networks (CNN) are increasingly used in PINN. However, some challenges still remain as follows. To adapt convolutional PINN to solve different PDEs, considerable effo…
