Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving differential equations, yet their application to nuclear scattering has been hindered by the oscillatory, non-decaying nature of scattering wave functions. In this work, I demonstrate that exterior complex scaling (ECS) transforms scattering boundary conditions into exponentially decaying waves suitable for neural network solutions, enabling PINNs to solve nuclear reaction problems for the first time. I develop a driven-equation formulation where the source term is confined to the real axis, avoiding the need to analytically continue nuclear potentials into the complex plane. The method is validated on nucleon-nucleus scattering (n+$^{40}$Ca at $E_{\text{lab}}=20$~MeV) with 21 partial waves, achieving phase shift accuracy of $Δδ\lesssim 0.1^\circ$ for the strongly absorbed channels ($\ell \leq 4$) and $Δδ\leq 0.60^\circ$ for all channels up to $\ell = 10$, when compared to conventional solvers. I further demonstrate the approach on heavy-ion scattering ($^6$Li+$^{208}$Pb at 40~MeV) with 41 partial waves and strong Coulomb effects, where an auto-adaptive anchor warm-down for weak-source channels yields a mean S-matrix accuracy of $|Δ|S_\ell|| \approx 3 \times 10^{-3}$ across the full angular momentum range, including the absorption-to-transparency transition region. This work establishes the foundation for extending PINNs to inverse problems where end-to-end differentiability enables direct fitting of optical potential parameters, coupled-channel reactions, and few-body scattering where traditional grid methods face exponential scaling.
Differential equations are used in a wide variety of disciplines, describing the complex behavior of the physical world. Analytic solutions to these equations are often difficult to solve for, limiting our current ability to solve complex differential equations and necessitating sophisticated numerical methods to approximate solutions. Trained neural networks act as universal function approximators, able to numerically solve differential equations in a novel way. In this work, methods and applications of neural network algorithms for numerically solving differential equations are explored, wit…
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）
Massachusetts Institute of Technology（麻省理工学院）；Sandia National Laboratories（桑迪亚国家实验室）
Physics-informed machine learning (PIML) has emerged as a promising new approach for simulating complex physical and biological systems that are governed by complex multiscale processes for which some data are also available. In some instances, the objective is to discover part of the hidden physics from the available data, and PIML has been shown to be particularly effective for such problems for which conventional methods may fail. Unlike commercial machine learning where training of deep neural networks requires big data, in PIML big data are not available. Instead, we can train such networ…
Purdue University（普渡大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）
Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations in a forward and inverse manner using deep neural networks. However, training these networks can be challenging for multiscale problems. While statistical methods can be employed to scale the regression loss on data, it is generally challenging to scale the loss terms for equations. This paper proposes a method for scaling the mean squared loss terms in the objective function used to train PINNs. Instead of using automatic differentiation to calculate the temporal derivative, we use backward …
Distributed physics-informed neural networks via domain decomposition for fast flow reconstruction
College of Control Science and Engineering（控制科学与工程学院）；School of Mathematical Sciences（数学科学学院）；Department of Mathematics（数学系）；State Key Laboratory of Industrial Control Technology（工业控制技术国家重点实验室）
Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks
Department of Information Engineering and Computer Science, University of Trento（特伦托大学信息工程与计算机科学系）；Division of Applied Mathematics, Brown University（布朗大学应用数学部）；Center for Biomedical Engineering, Brown University（布朗大学生物医学工程中心）；Institute of Mathematics, TU Berlin（柏林工业大学数学研究所）；ETH Zurich（苏黎世联邦理工学院）
Fourier Feature Pyramids for Physics-Informed Neural Networks
Department of Computing and Mathematical Sciences, California Institute of Technology（计算与数学科学部，加州理工学院）；Cahill Center for Astronomy and Astrophysics, California Institute of Technology（天文与天体物理中心，加州理工学院）；Google DeepMind（谷歌DeepMind）
Virginia Tech（弗吉尼亚理工大学）；University of Michigan（密歇根大学）
Physics-informed Neural Networks (PINNs) are gaining attention in the engineering and scientific literature for solving a range of differential equations with applications in weather modeling, healthcare, manufacturing, etc. Poor scalability is one of the barriers to utilizing PINNs for many real-world problems. To address this, a Self-scalable tanh (Stan) activation function is proposed for the PINNs. The proposed Stan function is smooth, non-saturating, and has a trainable parameter. During training, it can allow easy flow of gradients to compute the required derivatives and also enable syst…
Shape Derivative-Informed Neural Operators with Application to Risk-Averse Shape Optimization
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas, USA（奥登计算工程与科学研究所，德克萨斯大学奥斯汀分校，奥斯汀，德克萨斯，美国）；School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia（数学科学学院，昆士兰科技大学，布里斯班，澳大利亚）；Centre for Data Science, Queensland University of Technology, Brisbane, Australia（数据科学中心，昆士兰科技大学，布里斯班，澳大利亚）；Department of Mathematics, The Ohio State University, Columbus, Ohio, USA（数学系，俄亥俄州立大学，哥伦布，俄亥俄，美国）；Department of Engineering Science and Biomedical Engineering, University of Auckland, Auckland, New Zealand（工程科学与生物医学工程系，奥克兰大学，奥克兰，新西兰）；Walker Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas, USA（沃克机械工程系，德克萨斯大学奥斯汀分校，奥斯汀，德克萨斯，美国）
A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients
Department of Mathematics \& POSTECH MINDS (Mathematical Institute for Data Science), Pohang University of Science；Graduate School of Artificial Intelligence \& POSTECH MINDS (Mathematical Institute for Data Science), Pohang University of Science
LRX-PINN: A Layer-Resolving XNet Physics-Informed Neural Network with Integrated Cauchy Activations for Convection-Dominated Problems
Institute for Advanced Research, Great Bay University（先进研究院，大湾大学）；College of Computer Science and Technology, Dongguan University of Technology（计算机科学与技术学院，东莞理工大学）；Department of Mathematics, Northwestern University（数学系，西北大学）
Deconfined regions in relativistic heavy ion collisions are limited to small volumes surrounded by a confined exterior. Using the geometry of a double layered torus, we keep an outside temperature slightly lower than the inside temperature, so that both regions are in the SU(3) scaling region. Deconfined volume sizes are chosen to be in a range typical for such volumes created at the BNL RHIC. Even with small temperature differences a dependence of the (pseudo) deconfining temperature on a colder surrounding temperature is clearly visible. For temporal lattice sizes Ntau=4, 6 and 8 we find con…
Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving
College of Engineering and Computer Science, VinUniversity（VinUniversity工程与计算机科学学院）；Center for AI Research, VinUniversity（VinUniversity人工智能研究中心）；Group of Materials and Structures, School of Mechanical Engineering, Hanoi University of Science and Technology（河内科技大学机械工程学院材料与结构组）；Department of Mathematics and Informatics, Hanoi University of Science and Technology（河内科技大学数学与信息学系）
Physics-Informed Neural Networks for Predicting Hydrogen Sorption in Geological Formations: Thermodynamically Constrained Deep Learning Integrating Classical Adsorption Theory
University of Oslo（奥斯陆大学）；SINTEF Industry（挪威科技工业研究院）；China University of Petroleum, Beijing（中国石油大学（北京））
GRINN: A Physics-Informed Neural Network for solving hydrodynamic systems in the presence of self-gravity
Jet Propulsion Laboratory, California Institute of Technology（加州理工学院喷气推进实验室）；University of Western Ontario（西安大略大学）；Perimeter Institute For Theoretical Physics（圆周理论物理研究所）
Physics-Informed Neural Networks for Optimal Vaccination Plan in SIR Epidemic Models
Department of Applied Artificial Intelligence, 232 Gonneung-ro, Nowon-gu, Seoul, 01811, Republic of Korea（应用人工智能系）；Department of Mathematics, POSTECH, 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk, 06974, Republic of Korea（数学系）
Resolving gradient pathology in physics-informed epidemiological models
Disease-informed Modeling, Methods & Systems (DIMMS) Lab, Department of Mathematics and Statistics, York University, Toronto, Canada（疾病导向建模、方法与系统实验室，数学与统计学系，约克大学，多伦多，加拿大）
Based on the work of Nuttall and Cohen [Phys. Rev. {\bf 188} (1969) 1542] and Resigno et al{} [Phys. Rev. A {\bf 55} (1997) 4253] we present a rigorous formalism for solving the scattering problem for long-range interactions without using exact asymptotic boundary conditions. The long-range interaction may contain both Coulomb and short-range potentials. The exterior complex scaling method, applied to a specially constructed inhomogeneous Schrödinger equation, transforms the scattering problem into a boundary problem with zero boundary conditions. The local and integral representations for the…
The paper proposes a hybrid method for calculating scattering processes that combines the $J$-matrix method with exterior complex scaling as an absorbing boundary condition. It represents the wave function as a finite sum of oscillator eigenstates in the inner region and on grid in the outer region. The method is validated for one and two dimensional model partial wave equation equations. Finally, the method calculates nuclear $p$-shell scattering.
