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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

Japanese

Neuro-spectral architectures (NeuSA) は、周波数領域における時間依存性PDEsを解決するのに効果的であることが証明された物理学に知られているニューラルネットワーク(PINNs)のクラスであるが、トレーニングと推測はしばしば学習されたベクトルフィールドの費用がかかる数値統合に依存するが、典型的には、時間の並行性を制限する連続的なランゲー・カッタタイムステップを使用する。 我々はPaPiを導入し、流れが線形に近いかどうかに依存する明確なエラー境界を提供する。実験は、PaPiがトレーニングと推測を2つの順序まで加速させ、第2次低次元学習のためのトレーニング

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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