Physics-informed neural networks (PINNs) have attracted a lot of attention in scientific computing as their functional representation of partial differential equation (PDE) solutions offers flexibility and accuracy features. However, their training cost has limited their practical use as a real alternative to classic numerical methods. Thus, we propose to incorporate multi-resolution hash encoding into PINNs to improve the training efficiency, as such encoding offers a locally-aware (at multi resolution) coordinate inputs to the neural network. Borrowed from the neural representation field community (NeRF), we investigate the robustness of calculating the derivatives of such hash encoded neural networks with respect to the input coordinates, which is often needed by the PINN loss terms. We propose to replace the automatic differentiation with finite-difference calculations of the derivatives to address the discontinuous nature of such derivatives. We also share the appropriate ranges for the hash encoding hyperparameters to obtain robust derivatives. We test the proposed method on three problems, including Burgers equation, Helmholtz equation, and Navier-Stokes equation. The proposed method admits about a 10-fold improvement in efficiency over the vanilla PINN implementation.
Coordinate Encoding on Linear Grids for Physics-Informed Neural Networks
Graduate School of Engineering, Gifu University, Japan（名古屋大学工学研究科，日本）；International Center for Synchrotron Radiation Innovation（同步辐射创新国际中心）；Smart, Tohoku University, Japan（Smart，东京东京大学，日本）；Unprecedented-scale Data Analytics Center, Tohoku University, Japan（大规模数据解析中心，东京东京大学，日本）；Graduate School of Information Sciences, Tohoku University, Japan（信息科学研究生院，东京东京大学，日本）；Center for Basic Research on Materials, National Institute for Materials Science, Japan（材料基础研究中心，日本材料研究所）；Center for Advanced Intelligence Project, RIKEN, Japan（先进智能项目中心，日本理化学研究所）
Symmetry-Reduced Physics-Informed Learning of Tensegrity Dynamics
Department of Mathematics, University of Kentucky（数学系，肯塔基大学）；Department of Mechanical and Aerospace Engineering, University of Houston（机械与航空航天工程系，休斯顿大学）
Neural networks behave as hash encoders: An empirical study
University of Sydney（悉尼大学）；University of Science and Technology of China（中国科学技术大学）
The input space of a neural network with ReLU-like activations is partitioned into multiple linear regions, each corresponding to a specific activation pattern of the included ReLU-like activations. We demonstrate that this partition exhibits the following encoding properties across a variety of deep learning models: (1) { \it determinism}: almost every linear region contains at most one training example . We can therefore represent almost every training example by a unique activation pattern, which is parameterized by a { \it neural code}; and (2) { \it categorization}: according to the neura…
Transformers with Physics-Informed Encodings and Simulation-Based Inference for Robust Detection of Eccentric Binary Black Holes in Pulsar Timing Array Data
Department of Physics, National University of Singapore（新加坡国立大学物理系）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）
Shape-informed cardiac mechanics surrogates in data-scarce regimes via geometric encoding and generative augmentation
MOX Laboratory, Department of Mathematics, Politecnico di Milano（米兰Politecnico大学数学系MOX实验室）；Division 2.2 Process Simulation, Bundesanstalt für Materialforschung und -prüfung (BAM)（材料研究所（BAM）第2.2过程模拟部）；Department of Mathematics, University of Trento（特伦托大学数学系）
Learning Physics-Informed Surrogate Model of Linear Elastic Displacement Fields from Geometry
PROMES-CNRS, Université Perpignan Via Domitia (UPVD)（法国国家科学研究中心佩皮尼昂大学多姆itia分校能源与过程系统优化实验室）；LAAS-CNRS, Université de Toulouse (UT)（法国国家科学研究中心图卢兹大学航空航天系统实验室）
A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries
Graduate School of Science and Technology, Keio University（Keio大学科学技术研究生院）；Department of Mechanical Engineering, Keio University（Keio大学机械工程系）；Access e.V.
