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Modeling Accurate Long Rollouts with Temporal Neural PDE Solvers
Phillip Lippe · Bastiaan Veeling · Paris Perdikaris · Richard E Turner · Johannes Brandstetter

Fri Jul 28 06:45 PM -- 07:00 PM (PDT) @
in New Frontiers in Learning, Control, and Dynamical Systems »
Event URL: https://openreview.net/forum?id=EGTY6V76b3 »

Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Recently, mostly due to the high computational cost of traditional solution techniques, deep neural network based surrogates have gained increased interest. The practical utility of such neural PDE solvers relies on their ability to provide accurate, stable predictions over long time horizons, which is a notoriously hard problem. In this work, we present a large-scale analysis of common temporal rollout strategies, identifying the neglect of non-dominant spatial frequency information, often associated with high frequencies in PDE solutions, as the primary pitfall limiting stable, accurate rollout performance. Based on these insights, we draw inspiration from recent advances in diffusion models to introduce PDE-Refiner; a novel model class that enables more accurate modeling of all frequency components via a multi-step refinement process. We validate PDE-Refiner on challenging benchmarks of complex fluid dynamics, demonstrating stable and accurate rollouts that consistently outperform state-of-the-art models, including neural, numerical, and hybrid neural-numerical architectures. Finally, PDE-Refiner's connection to diffusion models enables an accurate and efficient assessment of the model's predictive uncertainty, allowing us to estimate when the surrogate becomes inaccurate.

Author Information

Phillip Lippe (University of Amsterdam)
Bastiaan Veeling (University of Amsterdam)
Paris Perdikaris (University of Pennsylvania)
Richard E Turner (University of Cambridge)

Richard Turner holds a Lectureship (equivalent to US Assistant Professor) in Computer Vision and Machine Learning in the Computational and Biological Learning Lab, Department of Engineering, University of Cambridge, UK. He is a Fellow of Christ's College Cambridge. Previously, he held an EPSRC Postdoctoral research fellowship which he spent at both the University of Cambridge and the Laboratory for Computational Vision, NYU, USA. He has a PhD degree in Computational Neuroscience and Machine Learning from the Gatsby Computational Neuroscience Unit, UCL, UK and a M.Sci. degree in Natural Sciences (specialism Physics) from the University of Cambridge, UK. His research interests include machine learning, signal processing and developing probabilistic models of perception.

Johannes Brandstetter (Microsoft)

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