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Home / Proceedings / Proceedings of the AAAI Conference on Artificial Intelligence, 36 / No. 8: AAAI-22 Technical Tracks 8

Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs

February 1, 2023

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Authors

Nils Wandel

University of Bonn


Michael Weinmann

TU Delft


Michael Neidlin

Department of Cardiovascular Engineering, Institute of Applied Medical Engineering, RWTH Aachen University


Reinhard Klein

University of Bonn


DOI:

10.1609/aaai.v36i8.20830


Abstract:

Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the solution of PDEs based on a novel technique that combines the advantages of two recently emerging machine learning based approaches. First, physics-informed neural networks (PINNs) learn continuous solutions of PDEs and can be trained with little to no ground truth data. However, PINNs do not generalize well to unseen domains. Second, convolutional neural networks provide fast inference and generalize but either require large amounts of training data or a physics-constrained loss based on finite differences that can lead to inaccuracies and discretization artifacts. We leverage the advantages of both of these approaches by using Hermite spline kernels in order to continuously interpolate a grid-based state representation that can be handled by a CNN. This allows for training without any precomputed training data using a physics-informed loss function only and provides fast, continuous solutions that generalize to unseen domains. We demonstrate the potential of our method at the examples of the incompressible Navier-Stokes equation and the damped wave equation. Our models are able to learn several intriguing phenomena such as Karman vortex streets, the Magnus effect, Doppler effect, interference patterns and wave reflections. Our quantitative assessment and an interactive real-time demo show that we are narrowing the gap in accuracy of unsupervised ML based methods to industrial solvers for computational fluid dynamics (CFD) while being orders of magnitude faster.

Topics: AAAI

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HOW TO CITE:

Nils Wandel||Michael Weinmann||Michael Neidlin||Reinhard Klein Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs Proceedings of the AAAI Conference on Artificial Intelligence (2022) 8529-8538.

Nils Wandel||Michael Weinmann||Michael Neidlin||Reinhard Klein Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs AAAI 2022, 8529-8538.

Nils Wandel||Michael Weinmann||Michael Neidlin||Reinhard Klein (2022). Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs. Proceedings of the AAAI Conference on Artificial Intelligence, 8529-8538.

Nils Wandel||Michael Weinmann||Michael Neidlin||Reinhard Klein. Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs. Proceedings of the AAAI Conference on Artificial Intelligence 2022 p.8529-8538.

Nils Wandel||Michael Weinmann||Michael Neidlin||Reinhard Klein. 2022. Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs. "Proceedings of the AAAI Conference on Artificial Intelligence". 8529-8538.

Nils Wandel||Michael Weinmann||Michael Neidlin||Reinhard Klein. (2022) "Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs", Proceedings of the AAAI Conference on Artificial Intelligence, p.8529-8538

Nils Wandel||Michael Weinmann||Michael Neidlin||Reinhard Klein, "Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs", AAAI, p.8529-8538, 2022.

Nils Wandel||Michael Weinmann||Michael Neidlin||Reinhard Klein. "Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs". Proceedings of the AAAI Conference on Artificial Intelligence, 2022, p.8529-8538.

Nils Wandel||Michael Weinmann||Michael Neidlin||Reinhard Klein. "Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs". Proceedings of the AAAI Conference on Artificial Intelligence, (2022): 8529-8538.

Nils Wandel||Michael Weinmann||Michael Neidlin||Reinhard Klein. Spline-PINN: Approaching PDEs without Data Using Fast, Physics-Informed Hermite-Spline CNNs. AAAI[Internet]. 2022[cited 2023]; 8529-8538.


ISSN: 2374-3468


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