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Multipole Graph Neural Operator for Parametric Partial Differential Equations

Li, Zongyi and Kovachki, Nikola and Azizzadenesheli, Kamyar and Liu, Burigede and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima (2020) Multipole Graph Neural Operator for Parametric Partial Differential Equations. In: Advances in Neural Information Processing Systems 33 pre-proceedings (NeurIPS 2020). Advances in Neural Information Processing Systems .

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One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we purpose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.

Item Type:Book Section
Related URLs:
URLURL TypeDescription Paper
Li, Zongyi0000-0003-2081-9665
Kovachki, Nikola0000-0002-3650-2972
Azizzadenesheli, Kamyar0000-0001-8507-1868
Liu, Burigede0000-0002-6518-3368
Bhattacharya, Kaushik0000-0003-2908-5469
Stuart, Andrew0000-0001-9091-7266
Anandkumar, Anima0000-0002-6974-6797
Additional Information:Z. Li gratefully acknowledges the financial support from the Kortschak Scholars Program. A. Anandkumar is supported in part by Bren endowed chair, LwLL grants, Beyond Limits, Raytheon, Microsoft, Google, Adobe faculty fellowships, and DE Logi grant. K. Bhattacharya, N. B. Kovachki, B. Liu and A. M. Stuart gratefully acknowledge the financial support of the Army Research Laboratory through the Cooperative Agreement Number W911NF-12-0022. Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-12-2-0022. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
Funding AgencyGrant Number
Kortschak Scholars ProgramUNSPECIFIED
Bren Professor of Computing and Mathematical SciencesUNSPECIFIED
Learning with Less Labels (LwLL)UNSPECIFIED
Raytheon CompanyUNSPECIFIED
Microsoft Faculty FellowshipUNSPECIFIED
Google Faculty Research AwardUNSPECIFIED
Caltech De Logi FundUNSPECIFIED
Army Research LaboratoryW911NF-12-0022
Record Number:CaltechAUTHORS:20201106-120222366
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106492
Deposited By: George Porter
Deposited On:06 Nov 2020 23:08
Last Modified:02 Jun 2023 01:08

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