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Electrostatic forces play many important roles in molecular biology, but are hard to model due to the complicated interactions between biomolecules and the surrounding solvent, a fluid composed of water and dissolved ions. Continuum model…

Numerical Analysis · Mathematics 2015-12-29 Jaydeep P. Bardhan , Matthew G. Knepley

Using Physics-Informed Neural Networks (PINNs) to solve a specific boundary value problem is becoming more popular as an alternative to traditional methods. However, depending on the specific problem, they could be computationally expensive…

High Energy Astrophysical Phenomena · Physics 2023-07-06 Jorge F. Urbán , Petros Stefanou , Clara Dehman , José A. Pons

This paper introduces a novel approach to solve inverse problems by leveraging deep learning techniques. The objective is to infer unknown parameters that govern a physical system based on observed data. We focus on scenarios where the…

Machine Learning · Computer Science 2023-10-02 Sidney Besnard , Frédéric Jurie , Jalal M. Fadili

Accurate modeling of boundary conditions is crucial in computational physics. The ever increasing use of neural networks as surrogates for physics-related problems calls for an improved understanding of boundary condition treatment, and its…

Machine Learning · Computer Science 2021-07-06 Antonio Alguacil , Wagner Gonçalves Pinto , Michael Bauerheim , Marc C. Jacob , Stéphane Moreau

Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE). In PINNs, the residual form of the PDE of interest and its boundary conditions are lumped into a composite objective…

Computational Physics · Physics 2022-05-24 Shamsulhaq Basir , Inanc Senocak

Physics-informed neural networks (PINNs) often struggle with multi-scale PDEs featuring sharp gradients and nontrivial boundary conditions, as the physics residual and boundary enforcement compete during optimization. We present a…

Numerical Analysis · Mathematics 2025-12-01 Naseem Abbas , Vittorio Colao , Davide Macri , William Spataro

In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…

Optimization and Control · Mathematics 2024-09-06 Michael Hintermüller , Denis Korolev

Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the…

Numerical Analysis · Mathematics 2021-03-05 Zhuochao Tang , Zhuojia Fu

We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations; and a PDE…

Numerical Analysis · Mathematics 2024-03-11 Marius Zeinhofer , Rami Masri , Kent-André Mardal

Physics-informed neural networks (PINNs) have been increasingly employed due to their capability of modeling complex physics systems. To achieve better expressiveness, increasingly larger network sizes are required in many problems. This…

Machine Learning · Computer Science 2023-02-28 Ziyue Liu , Xinling Yu , Zheng Zhang

Modeling and simulating a power distribution network (PDN) for printed circuit boards (PCBs) with irregular board shapes and multi-layer stackup is computationally inefficient using full-wave simulations. This paper presents a new concept…

Machine Learning · Computer Science 2021-06-22 Ling Zhang , Jack Juang , Zurab Kiguradze , Bo Pu , Shuai Jin , Songping Wu , Zhiping Yang , Chulsoon Hwang

In recent years, Physics-Informed Neural Networks (PINNs) have emerged as a powerful and robust framework for solving nonlinear differential equations across a wide range of scientific and engineering disciplines, including biology,…

Fluid Dynamics · Physics 2025-11-19 Ritik Pal , Soubhik Mukherjee , Urmi Dutta , Arghya Choudhury

Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems, whose basic concept is to embed physical laws to constrain/inform neural networks, with the need of less data for training…

Fluid Dynamics · Physics 2020-11-24 Chengping Rao , Hao Sun , Yang Liu

We present a deep learning framework for quantifying and propagating uncertainty in systems governed by non-linear differential equations using physics-informed neural networks. Specifically, we employ latent variable models to construct…

Machine Learning · Statistics 2019-06-26 Yibo Yang , Paris Perdikaris

Physically informed neural networks (PINNs) are a promising emerging method for solving differential equations. As in many other deep learning approaches, the choice of PINN design and training protocol requires careful craftsmanship. Here,…

Machine Learning · Statistics 2023-10-09 Inbar Seroussi , Asaf Miron , Zohar Ringel

Physics-informed neural networks (PINNs) offer a unified framework for solving both forward and inverse problems of differential equations, yet their performance and physical consistency strongly depend on how governing laws are…

Machine Learning · Computer Science 2026-03-31 Guojie Li , Liu Hong

The identification of material parameters occurring in constitutive models has a wide range of applications in practice. One of these applications is the monitoring and assessment of the actual condition of infrastructure buildings, as the…

Machine Learning · Computer Science 2023-06-14 David Anton , Henning Wessels

In this paper, we investigate several techniques for modeling the one-dimensional advection equation for a specific class of problems with discontinuous initial and boundary conditions using physics-informed neural networks (PINNs). To…

Numerical Analysis · Mathematics 2026-01-30 Omid Khosravi , Mehdi Tatari

Physics-Informed Neural Networks (PINNs) solve partial differential equations (PDEs) by embedding governing equations and boundary/initial conditions into the loss function. However, enforcing Dirichlet boundary conditions accurately…

Machine Learning · Computer Science 2025-01-15 Nahil Sobh , Rini Jasmine Gladstone , Hadi Meidani

We apply physics-informed neural networks (PINNs) to first-order two-scale periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp…

Materials Science · Physics 2023-01-25 Celal Soyarslan , Marc Pradas