arXiv AI August 18, 2026 By Noura Al Helwani, Sophie Moufawad, Nabil Nassif

A Two-Stage Learning PINN Approach for Solving the Inverse Problem of the 1D Porous Medium Equation

arXiv:2608. 16475v1 Announce Type: cross Abstract: The Porous Medium Equation (PME), given by $u_t = \Delta(u^m)$ for $m > 1$, is a degenerate nonlinear parabolic partial differential equation that arises in various physical applications such as fluid flow in porous media, heat transfer in plasmas, and population dynamics.

Summary generated by The Flow from the publisher's feed. The full article lives at arXiv AI.

diffusion

arXiv Machine Learning

Jun 11

Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems

arXiv:2606. 12337v1 Announce Type: cross Abstract: Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative.

By Zhen Zhang, Alessandro Alla, George Em Karniadakis

benchmarks

arXiv Machine Learning

Jun 19

Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System

arXiv:2606. 19754v1 Announce Type: new Abstract: Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems.

By Zhiwen Yu, Derong Yang, Liujian Zhang, Kaixiang Yang, Peilin Zhan, Jianmin Lv, Jane You, C. L. Philip Chen

arXiv Machine Learning

Jul 28

Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs

arXiv:2607. 24726v1 Announce Type: new Abstract: The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing field of scientific machine learning.

By Justin Sirignano, Konstantinos Spiliopoulos, Samuel Cohen

arXiv Machine Learning

Jul 20

A Physics-Informed Neural Network with a Modified Lorentzian Activation for Nonlocal Gradient-Flow Equations in Dynamic Density Functional Theory

arXiv:2607. 15291v1 Announce Type: cross Abstract: We develop a physics-informed neural network (PINN) framework for nonlocal partial differential equations arising in dynamic density functional theory (DDFT).

By Dimitrios Gourzoulidis, Soumaya Elkantassi, Serafim Kalliadasis

arXiv AI

Jul 10

Deep Neural Networks as Discrete Dynamical Systems: Implications for Physics-Informed Learning

arXiv:2601. 00473v4 Announce Type: replace-cross Abstract: We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms.

By Abhisek Ganguly, Santosh Ansumali, Sauro Succi

safety

arXiv Machine Learning

Jun 19

A fast direct solver based neural network for solving PDEs

arXiv:2606. 19895v1 Announce Type: cross Abstract: The matrices arising from large scale $N$-body problems can be efficiently represented using hierarchical matrices, whose key idea is that the admissible off-diagonal sub-matrices can be well approximated by low-rank matrices across a hierarchy of matrix partitions.

By Jashwanth Reddy Kadaru, Vaishnavi Gujjula

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Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems

Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System

Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs

A Physics-Informed Neural Network with a Modified Lorentzian Activation for Nonlocal Gradient-Flow Equations in Dynamic Density Functional Theory

Deep Neural Networks as Discrete Dynamical Systems: Implications for Physics-Informed Learning

A fast direct solver based neural network for solving PDEs