Parallel integration speeds up neural solutions for time related problems
PaPi: Parallel Picard-Duhamel Integration for Fast Neuro-Spectral Architectures
Machine Learning
Summary
Solving complicated time-based equations with neural networks usually takes a long time because calculations must happen step by step. The authors introduce a new way called PaPi that allows many time steps to be done at once in parallel, making the process much faster. This method still keeps accurate results while reducing training time drastically, sometimes to under a second. PaPi uses some math tricks to combine ideas from different areas to achieve this speedup.
What this means in practice
- •For computational engineers: Speed up training and inference of neural models solving time-dependent PDEs in engineering simulations using parallel integration.
- •For software developers: Implement faster neural-network-based PDE solvers that reduce runtime in applications involving time-dependent physical phenomena.
Authors
Arthur Bizzi, Márcio Marques, Kleyton Da Costa, Vitor Balestro, Tiago Novello, João M. Pereira, Daniel Yukimura
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.