arXiv Machine Learning By Jashwanth Reddy Kadaru, Vaishnavi Gujjula

A fast direct solver based neural network for solving PDEs

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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.

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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
arXiv Machine Learning
Jul 21

One-shot acceleration of transient PDE solvers via online-learned preconditioners

arXiv:2509. 08765v4 Announce Type: replace-cross Abstract: Data-driven acceleration of scientific computing workflows has been a high-profile aim of machine learning (ML) for science, with numerical simulation of transient partial differential equations (PDEs) being one of the main applications.

By Mikhail Khodak, Min Ki Jung, Brian Wynne, Edmond Chow, Egemen Kolemen
arXiv Machine Learning
Sep 15

Linearized PINN with pretrained nonlinear layers

The paper introduces a linearized Physics-Informed Neural Network (lPINN), a reduced‑order neural basis approach for solving forward and inverse differential equations. In an offline phase, lPINN learns continuous, differentiable neural basis functions from numerical solutions, which are then frozen for new problem instances; the online solution is obtained by minimizing the governing‑equation residual with additional constraints. Experiments on advection‑diffusion, Burgers', and nonlinear pendulum equations show that lPINN achieves lower solution and parameter errors than vanilla PINNs while reducing online inference times by up to three orders of magnitude, and its continuous representation generalizes to finer meshes without retraining.

By Wenhao Chen, Alexandre M. Tartakovsky