aiqbench analysis
This research presents a hybrid framework that couples pre-trained numerics-informed neural networks (NINNs) with classical full-order models using the overlapping Schwarz alternating method to solve high-Peclet-number advection-diffusion problems. By holding NINN weights fixed during iteration, the approach effectively bridges data-driven approximations with traditional numerical solvers without requiring full-domain retraining. This methodology offers a promising pathway for scaling AI-based solvers in regimes where standard physics-informed networks struggle with accuracy or convergence.
Hybrid coupling with numerics-informed neural networks and the overlapping Schwarz alternating method
arXiv:2609.17841v1 Announce Type: new Abstract: We develop a hybrid modeling framework for coupling pre-trained numerics-informed neural networks (NINNs) with classical full order models (FOMs) using the overlapping Schwarz alternating method. We consider the two-dimensional advection-diffusion equation in the advection-dominated, Peclet-number 10^6 regime. We first demonstrate that, unlike the corresponding physics-informed neural network (PINN), a monolithic NINN can be accurately trained on our model problem without domain decomposition. We then employ overlapping multiplicative Schwarz as a deployment mechanism for coupling a pre-trained, subdomain-local NINN with a neighboring FOM, with the NINN weights held fixed throughout the Schwarz iteration. We consider two training approaches for the subdomain-local NINNs: a top-down approach, in which boundary data are obtained from a coupled Schwarz solve on the full domain with a FOM on each subdomain (FOM-FOM Schwarz), and a bottom-up approach, in which boundary traces are generated synthetically on the NINN subdomain without requiring any full-domain solves. The resulting hybrid NINN-FOM solutions agree closely with the corresponding FOM-FOM Schwarz solutions, with the top-down and bottom-up training approaches yielding comparable accuracy.
Abstract:We develop a hybrid modeling framework for coupling pre-trained numerics-informed neural networks (NINNs) with classical full order models (FOMs) using the overlapping Schwarz alternating method. We consider the two-dimensional advection-diffusion equation in the advection-dominated, Peclet-number 10^6 regime. We first demonstrate that, unlike the corresponding physics-informed neural network (PINN), a monolithic NINN can be accurately trained on our model problem without domain decomposition. We then employ overlapping multiplicative Schwarz as a deployment mechanism for coupling a pre-trained, subdomain-local NINN with a neighboring FOM, with the NINN weights held fixed throughout the Schwarz iteration. We consider two training approaches for the subdomain-local NINNs: a top-down approach, in which boundary data are obtained from a coupled Schwarz solve on the full domain with a FOM on each subdomain (FOM-FOM Schwarz), and a bottom-up approach, in which boundary traces are generated synthetically on the NINN subdomain without requiring any full-domain solves. The resulting hybrid NINN-FOM solutions agree closely with the corresponding FOM-FOM Schwarz solutions, with the top-down and bottom-up training approaches yielding comparable accuracy.
Machine Learning (cs.LG); Mathematical Physics (math-ph)
arXiv:2609.17841 [cs.LG]
(or arXiv:2609.17841v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.17841 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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