Up-Net: A Generic Deep Learning-Based Time Stepper for Parameterized Spatio-Temporal Dynamics

Stender, Merten; Ohlsen, Jakob; Geisler, Hendrik; Chabchoub, Amin; Hoffmann, Norbert; Schlaefer, Alexander

doi:10.2139/ssrn.4053304

Cited by 1 publication

(1 citation statement)

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“…Using machine learning (ML) to solve differential equations has become an active field of research. Some papers aim to entirely replace the numerical solver by neural networks [21,25]. Physics-informed neural networks (PINNs) [20], which use the residual of a partial differential equation (PDE) as well as boundaryand initial conditions in the loss function, are used in many applications.…”

Section: Related Workmentioning

confidence: 99%

Parareal with a Physics-Informed Neural Network as Coarse Propagator

Ibrahim,

Götschel,

Ruprecht

2023

Lecture Notes in Computer Science

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Parallel-in-time algorithms provide an additional layer of concurrency for the numerical integration of models based on time-dependent differential equations. Methods like Parareal, which parallelize across multiple time steps, rely on a computationally cheap and coarse integrator to propagate information forward in time, while a parallelizable expensive fine propagator provides accuracy. Typically, the coarse method is a numerical integrator using lower resolution, reduced order or a simplified model. Our paper proposes to use a physics-informed neural network (PINN) instead. We demonstrate for the Black-Scholes equation, a partial differential equation from computational finance, that Parareal with a PINN coarse propagator provides better speedup than a numerical coarse propagator. Training and evaluating a neural network are both tasks whose computing patterns are well suited for GPUs. By contrast, mesh-based algorithms with their low computational intensity struggle to perform well. We show that moving the coarse propagator PINN to a GPU while running the numerical fine propagator on the CPU further improves Parareal’s single-node performance. This suggests that integrating machine learning techniques into parallel-in-time integration methods and exploiting their differences in computing patterns might offer a way to better utilize heterogeneous architectures.

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Section: Related Workmentioning

confidence: 99%