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Life sciences · Preprint
arXiv · August 19, 2026
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This is a preprint introducing two neural operator learning frameworks (DCNO and DGNO) for approximating convolution integrals via multi-stage training. The work is grounded in mathematical theory and numerical validation on synthetic problems, with no clinical, biological, or real-world application data presented in the abstract.
Preprint.
Both DCNO and DGNO achieve accuracy approaching machine precision under single float for convolution problems Multi-stage learning offers substantial efficiency gains for numerous queries or parametric variations compared to traditional solvers Frameworks extend to multi-input operator learning with variations in both density and kernel of convolution
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This is a methods paper presenting novel computational frameworks with numerical validation but no clinical, biological, or translational endpoint; it is fundamentally a machine-learning contribution without human or animal studies.
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Convolution integrals widely exist in applications, and to enable fast and accurate computations, this paper introduces two general multi-stage neural operator learning frameworks. The first, Deep Collocation Neural Operator (DCNO), is a supervised approach that iteratively refines the operator approximation by learning residuals from input-output data pairs. The second, Deep Galerkin Neural Operator (DGNO), is an unsupervised framework applicable when the target operator can be represented by a PDE, leveraging the weak form of the PDE residual for training. Both methods progressively construct basis operators through multiple training stages to enrich the approximation space, leading to significantly improved accuracy over standard one-shot operator learning. We provide theoretical analysis for their approximation capabilities and implement them for learning convolutions. Extensive numerical experiments demonstrate that both DCNO and DGNO achieve high accuracy, approaching machine precision under single float for convolution problems, and offer substantial efficiency gains for numerous queries or parametric variations compared to traditional solvers. We also extend these frameworks to handle multi-input operator learning scenarios involving variations in both the density and kernel of a convolution.
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