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PaPi: Parallel Picard-Duhamel Integration for Fast Neuro-Spectral Architectures

Arthur Augusto Coutinho Bizzi, Márcio Marques, Kleyton da Costa, Vitor Balestro, Tiago Novello, João M. Pereira, Daniel Yukimura · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.28591 · cited by 0 works (OpenAlex) · open-access PDF

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Las arquitecturas neuroespectrales (NeuSA) son una clase de redes neuronales informadas por la física (PINNs) que han demostrado ser eficaces para resolver PDEs dependientes del tiempo en el dominio de frecuencia. Sin embargo, la formación e inferencia a menudo se basan en la integración numérica costosa del campo vector aprendido, típicamente utilizando el paso del tiempo secuencial Runge-Kutta, que limita el paralelismo temporal. Introducimos PaPi, un esquema de integración inspirado en flujos débiles no lineales que permite la inferencia paralelo-tiempo a través de la integración exponencial y un esquema Duhamel-Pardic. Mostramos que la Duhamel-Picarderation converge exponencialmente para todos los flujos, y proporciona límites de error explícitos que

Machine translation (Meta M2M100); the English original follows.

Abstract

Neuro-spectral architectures (NeuSA) are a class of physics-informed neural networks (PINNs) that have proven effective for solving time-dependent PDEs in the frequency domain. However, training and inference often rely on costly numerical integration of the learned vector field, typically using sequential Runge-Kutta time stepping, which limits temporal parallelism. We introduce PaPi, an integration scheme inspired by weakly nonlinear flows that enables time-Parallel inference through exponential integration and a Duhamel-Picard scheme. We show that the Duhamel-Picard iteration converges exponentially for all flows, and provide explicit error bounds that depend on how close the flow is to linear. Experiments show that PaPi accelerates training and inference by up to two orders of magnitude, achieving training times of under one second for low-dimensional PDE learning problems, while maintaining competitive predictive accuracy.

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Topics: Inference, Computer science, Numerical integration, Nonlinear system, Artificial neural network, Algorithm

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