A FAST DIRECT SOLVER BASED NEURAL NETWORK FOR SOLVING PDES

Numerical solvers for large-scale linear systems and PDEs often face high computational costs, particularly when dealing with dense matrices or repeated solves for varying parameters. This paper investigates whether a neural network can learn the inverse of Hierarchically Off-Diagonal Low-Rank (HODLR) matrices to act as a fast, direct solver for linear and nonlinear PDEs, offering a more efficient alternative to traditional methods in parametric settings.

The authors design a neural network architecture inspired by the fast direct solver for HODLR matrices developed by Ambikasaran and Darve. HODLR matrices exploit the low-rank structure of off-diagonal blocks in a recursive binary partition to reduce computational complexity. The proposed network learns the inverse operation of these matrices by modeling the recursive factorization process. To handle nonlinear PDEs, the architecture is extended by replacing specific linear layers with deeper sub-networks, allowing the model to learn complex nonlinear solution operators.

The proposed neural network demonstrates strong performance across several benchmarks, including the Fredholm integral equation, the nonlinear Schrödinger equation, Burgers' equation, and steady-state Darcy flow. Key results include:

• Competitive Accuracy: The network achieves relative errors in the range of 10^-4 to 10^-2.
• Efficient Inference: Once trained, the network provides significantly faster inference times compared to classical numerical solvers, especially when amortizing the training cost over many parameter sweeps.
• The model generalizes well to unseen parameter values when trained on a diverse dataset.