A fast direct solver based neural network for solving PDEs

2026-06-18 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, quick

Summary

A new neural network architecture is introduced for solving partial differential equations (PDEs) and linear problems. This network learns the inverse operation of Hierarchical Off-Diagonal Low-Rank (HODLR) matrices, building upon the fast direct solver developed by Ambikasaran and Darve (2013). The architecture extends its capability to nonlinear solution operators for PDEs by integrating deep sub-networks in place of some linear layers. Comprehensive experiments demonstrate its performance, including solving the Fredholm integral equation of the second kind, nonlinear Schrödinger equation, Burgers' equation, and the steady-state Darcy's flow equation. The study also evaluates the network's generalization across varying parameter values, compares its inference time against classical numerical solvers, and benchmarks it against existing neural operator learning networks.

Key takeaway

For research scientists developing fast PDE solvers, this neural network offers a promising alternative to classical methods. You should consider its HODLR-based inverse learning and deep sub-network extensions for tackling complex nonlinear problems like the Schrödinger or Burgers' equations. Evaluate its inference speed and generalization capabilities against existing neural operators to potentially accelerate your computational simulations.

Key insights

The neural network learns HODLR matrix inversion and extends to solve PDEs by replacing linear layers with deep sub-networks.

Principles

• HODLR matrices efficiently represent large N-body problem matrices.
• Off-diagonal sub-matrices are low-rank across hierarchy.
• Deep sub-networks can extend linear solvers to nonlinear PDEs.

Method

The network learns HODLR matrix inverse using a fast direct solver, then replaces linear layers with deep sub-networks to learn nonlinear PDE solution operators.

In practice

• Solve Fredholm integral equations.
• Address nonlinear Schrödinger equation.
• Tackle Burgers' and Darcy's flow equations.

Topics

• Partial Differential Equations
• Neural Networks
• HODLR Matrices
• Fast Direct Solvers
• Numerical Analysis
• Neural Operators

Best for: AI Scientist, Research Scientist

Related on AIssential

• Reformulating Neural Operators in $d+1$ Dimensions for Embedding Evolution — Neural operators redefined in d+1 dimensions using Schrödingerisation better capture system evolution and achieve superior performance.
• Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System — PIBLS offers a backpropagation-free, least-squares optimization approach for PDEs, achieving faster, more accurate solutions than PINNs.
• A Machine Learning Approach to the Nirenberg Problem — A physics-informed neural network can effectively solve the Nirenberg problem and distinguish between realisable and non-realisable curvatures.
• Structure-Oriented Randomized Neural Networks for Poisson-Nernst-Planck and Poisson-Nernst-Planck-Navier-Stokes Systems — SO-RaNN uses randomized neural networks with structure-preserving corrections to solve complex PNP and PNP-NS systems.
• FLAME 2023: Diving into the Future of Fluid Dynamics and Machine Learning — AI techniques are transforming simulation by integrating physics-based modeling with data-driven machine learning.
• Learning Physical Operators using Neural Operators — A physics-informed framework improves neural operator generalization and temporal continuity for PDE solving via operator splitting.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.