In this paper, we introduce a new deep learning framework for discovering the phase field models from existing image data. The new framework embraces the approximation power of physics informed neural networks (PINN), and the computational efficiency of the pseudo-spectral methods, which we named pseudo-spectral PINN or SPINN. Unlike the baseline PINN, the pseudo-spectral PINN has several advantages. First of all, it requires less training data. A minimum of two snapshots with uniform spatial resolution would be adequate. Secondly, it is computationally efficient, as the pseudo-spectral method is used for spatial discretization. Thirdly, it requires less trainable parameters compared with the baseline PINN. Thus, it significantly simplifies the training process and assures less local minima or saddle points. We illustrate the effectiveness of pseudo-spectral PINN through several numerical examples. The newly proposed pseudo-spectral PINN is rather general, and it can be readily applied to discover other PDE-based models from image data.
Discovering Scaling Exponents with Physics-Informed Müntz-Szász Networks
Axiom Research Group（Axiom研究组）；Department of Applied Mathematics and Computational Science（应用数学与计算科学系）；African Institute for Mathematical Sciences (AIMS), Research and Innovation Centre (RIC)（非洲数学科学研究所（AIMS）、研究与创新中心（RIC））；Graduate School of Engineering, The University of Tokyo（东京大学工学研究院）
Residual Pseudospectra Reveal a Physics-Informed Koopman Backbone for Tropical Pacific Variability and ENSO Prediction
Centro Euromediterraneo sui Cambiamenti Climatic（欧洲-地中海气候变化中心）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；Dipartimento di Scienze Geologiche, Biologiche e Ambientali, Universita’ di Bologna（博洛尼亚大学地质、生物与环境科学系）
The application of deep learning toward discovery of data-driven models requires careful application of inductive biases to obtain a description of physics which is both accurate and robust. We present here a framework for discovering continuum models from high fidelity molecular simulation data. Our approach applies a neural network parameterization of governing physics in modal space, allowing a characterization of differential operators while providing structure which may be used to impose biases related to symmetry, isotropy, and conservation form. We demonstrate the effectiveness of our f…
The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constraints are often imposed as soft penalties during model training and effectively act as domain-specific regularizers of the empirical risk loss. Physics-informed neural networks is an example of this philosophy in which the outputs of deep neural networks are constrained to approximately satisfy a given set of partial differential equations. In this work we review recent advances in scientific machine l…
This work is concerned with discovering the governing partial differential equation (PDE) of a physical system. Existing methods have demonstrated the PDE identification from finite observations but failed to maintain satisfying results against noisy data, partly owing to suboptimal estimated derivatives and found PDE coefficients. We address the issues by introducing a noise-aware physics-informed machine learning (nPIML) framework to discover the governing PDE from data following arbitrary distributions. We propose training a couple of neural networks, namely solver and preselector, in a mul…
Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives
Hong Kong University of Science and Technology（香港科技大学）；University of Maryland College Park（马里兰大学帕克分校）；HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute（港科大深港协同创新研究院）
Physics-constrained robust learning of open-form partial differential equations from limited and noisy data
College of Engineering, Peking University（北京大学工学院）；Ningbo Institute of Digital Twin, Eastern Institute of Technology（东方理工大学宁波数字孪生研究院）；School of Environmental Science and Engineering, Southern University of Science and Technology（南方科技大学环境科学与工程学院）；Eastern Institute for Advanced Study, Eastern Institute of Technology（东方理工大学东方高等研究院）
The improved backward compatible physics-informed neural networks for reducing error accumulation and applications in data-driven higher-order rogue waves
