Title: Assessing the Reliability of Deep Learning-Driven Hybrid PDE Solvers: The Critical Role of Training and Update Mechanisms

Abstract:

Deep learning-based hybrid iterative methods (DL-HIMs) seek to accelerate convergence by merging the complementary spectral biases of neural operators with classical numerical solvers. However, their practical reliability in scientific computing is frequently questioned due to instances where these methods stagnate at false fixed points—scenarios characterized by vanishing neural updates despite large remaining physical residuals. This study demonstrates that the performance of DL-HIMs is critically dependent on both the training paradigms and update strategies employed, even when the underlying neural architecture remains constant.

By conducting a comprehensive analysis of two specific frameworks—a DeepONet-based Hybrid Iterative Numerical Transferable Solver (HINTS) and an FFT-based Fourier Neural Solver (FNS)—we reveal that substantial physical residuals often persist when training objectives fail to align with the dynamics of the solver and the underlying physics of the problem. Furthermore, we investigate the application of Anderson acceleration (AA), finding that its standard formulation is inadequate for handling nonlinear neural operators. To address this limitation, we propose Physics-Aware Anderson Acceleration (PA-AA), a novel approach that minimizes the physical residual directly rather than focusing on the fixed-point update. Our numerical experiments validate that PA-AA significantly reduces the number of iterations required to achieve reliable convergence. These results offer a definitive resolution to ongoing debates regarding AI-driven PDE solvers, establishing that reliability is determined not merely by architectural choices, but by the integration of physically informed training protocols and iterative design.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC